Diffusion Metamaterials for Plasma Transport

Abstract

Plasma technology has found widespread applications in numerous domains, yet the techniques to manipulate plasma transport predominantly rely on magnetic control. In this chapter, we present a streamlined diffusion-migration method to characterize plasma transport. Based on this framework, the viability of the transformation theory for plasma transport is demonstrated. Highlighted within are three model devices designed to cloak, concentrate, and rotate plasmas without significantly altering the density profile of background plasmas. Additionally, insights regarding potential implications for novel physics are discussed. This chapter aims to contribute to advancements in plasma technology, especially in sectors like medicine and chemistry.

1 Opening Remarks

Since ancient times, energy has always been one of the core needs for human development. Over the past few centuries, people have developed and used various sources of energy, such as fossil fuels, hydropower, and nuclear energy. However, with the growth of the world’s population and economy, the demand for energy has also been continuously increasing, posing significant challenges to global energy resources. Now, we face a deepening energy crisis, including the depletion of fossil resources and the severe environmental impacts of carbon emissions. Against this backdrop, the sustainability of energy and how to efficiently use existing energy resources have become hot topics of global concern, urgently requiring solutions. Therefore, energy management and regulation have become crucial. We need to explore various measures to achieve active control of energy, thereby using it more efficiently. In this process, the development of metamaterials has become a prominent field and is one of the key means to achieve active energy control.

Metamaterials refer to novel materials that achieve extraordinary properties through artificial structures. Their origin can be traced back to the phenomenon of negative refraction proposed by Professor Veselago in 1968 [1]. However, due to the limited material and technological levels at that time, this phenomenon went unverified experimentally for a long time. It wasn’t until the end of the 20th century that Professor Pendry’s team demonstrated the feasibility of negative refraction experimentally through metal wire arrays and split-ring resonator structures [2, 3]. Since then, the research on electromagnetic metamaterials has attracted widespread attention. Particularly, the establishment of transformation optics theory in 2006 provided a powerful tool for designing electromagnetic metamaterial devices with novel functionalities, enabling the free manipulation of electromagnetic wave propagation as desired [4, 5]. This marks a new height in mankind’s active control over physical fields, greatly propelling the development of the electromagnetic metamaterials field [6,7,8,9].

The successful application of metamaterials in wave systems laid a solid foundation for their expansion into diffusion systems. In 2008, Professor Huang Jiping’s team from Fudan University established the transformation thermotics in the heat conduction system and predicted the functionality of a thermal cloak [10], inaugurating the field of diffusion metamaterials [11,12,13]. Over the next decade, transformation theory continually developed in diffusion systems [14,15,16,17], with an increasing number of extended theories proposed, including effective medium theory [18,19,20,21,22], scattering cancellation theory, and optimization algorithms, among others [23, 24]. Accordingly, various diffusion metamaterial devices were designed or fabricated. Diffusion metamaterials have provided a novel approach for controlling the diffusion of energy and matter and have become a current hotspot in research. The in-depth study of diffusion metamaterials may also bring inspiration and vitality to classical thermodynamic and statistical physics [25,26,27] or soft condensed matter physics [28, 29].

Fig. 18.1

Schematic diagrams of plasma components and distribution

Plasma diffusion, as a specific form of mass diffusion, has seen limited progress in the research of metamaterials designed for its regulation. Plasma, often referred to as the fourth phase of matter, consists of a gaseous blend of free ions, electrons, and various reactive species, granting it strong electrical conductivity. While it’s not typically found on Earth’s crust, plasma can be synthetically produced by electrifying gases using direct/alternating currents or sources of radio/microwave frequencies. See Fig. 18.1. Given its distinct nature, plasma-based technologies hold significant importance across sectors including micro/nanoelectronics, chemical research, bio-medicine, aerospace, and materials science [30,31,32,33]. While there has been a plethora of theoretical and empirical research, mastering the control over plasma transportation remains a formidable task. Traditional methods of directing charged particles primarily rely on external magnetic fields, a straightforward approach that may curtail precise control. Over recent years, the transformation theory, which substitutes spatial transformation with material alteration, has gained prominence in wave and diffusion systems as an effective strategy to steer matter or energy trajectories [34,35,36]. Yet, its application in the domain of plasma transportation, which can be viewed as a distinct diffusion mechanism, remains unexplored. The intricate motion dynamics within plasmas might explain this gap. Only recently, reports have suggested the potential application of transformation theory to plasma transport [37]. Although this work merely used a toy model to describe the plasma diffusion process, it has paved the way for new possible approaches to control plasma.

2 Transformation Theory for Plasma Transport

2.1 For Steady-State Plasma Transport

Compared to traditional diffusion systems, the transport in actual plasmas is more complex. This is because collisions between charged particles, and between charged and neutral particles, often result in reactions. Furthermore, the intrinsic electric field within the plasma can affect the movement of charged particles. Here, we do not consider collisions that result in nuclear reactions. Hence, the governing equation for the transport of charged particles in plasma can be written as:

$$\begin{aligned} \partial _{t} n+& \mathbf {\nabla }\cdot \Gamma =0, \end{aligned}$$

(18.1a)

$$\begin{aligned} \mathbf {\nabla }\cdot \Gamma =-\mathbf {\nabla }\cdot \left( D\cdot \mathbf {\nabla } n\right) & \pm \mathbf {\nabla }\cdot \left( \mu \boldsymbol{E}n\right) +\mathbf {\nabla }\cdot \left( \boldsymbol{v}n\right) , \end{aligned}$$

(18.1b)

where n, D, $\mu$, $\boldsymbol{E}$, and $\boldsymbol{v}$ represent the plasma density, diffusion coefficient, mobility, electric field, and advective velocity, respectively. The first term on the right-hand side of Eq. (18.1b) represents the diffusion term. The second term represents the mobility term, with the positive sign indicating the mobility of cations and the negative sign indicating the mobility of anions or electrons. The third term denotes the effect of convection. The transport equation for plasma also needs to be coupled with the Poisson’s equation:

$$\begin{aligned} \boldsymbol{\nabla }\cdot \boldsymbol{E}=\frac{e}{\epsilon _0}\left( n_i-n_e \right) , \end{aligned}$$

(18.2)

where e, $\epsilon _0$, $n_i$, and $n_e$ represent the elementary charge, permittivity of free space, cation density, and electron density, respectively. Assuming the plasma is electrically neutral internally, the right-hand side of the above equation is zero. For simplicity, we will not consider the advective process for now [38]. Therefore, Eq. (18.1) can be written as:

$$\begin{aligned} \partial _{t} n-\mathbf {\nabla }\cdot \left( D\cdot \mathbf {\nabla } n\right) \pm \mathbf {\nabla }\cdot \left( \mu \boldsymbol{E}n\right) =0. \end{aligned}$$

(18.3)

Note that in plasmas, the relationship between the diffusion coefficient and mobility is given by the Einstein relation:

$$\begin{aligned} \mu =\frac{De}{Tk_B}=\frac{D}{T_i}, \end{aligned}$$

(18.4)

where T is the particle temperature, $k_B$ is the Boltzmann constant, and $T_i=Tk_B/e$ is the reduced temperature in volts (V). Assuming the temperature is constant, the reduced temperature is also constant. Using the Einstein relation, we can simplify parameters and further reduce the plasma transport equation to:

$$\begin{aligned} \partial _{t} n-\mathbf {\nabla }\cdot \left( D \cdot \mathbf {\nabla } n\right) \pm \mathbf {\nabla }\cdot \left[ \left( \frac{D\cdot \boldsymbol{E}}{T_i}\right) n\right] =0. \end{aligned}$$

(18.5)

Next, we will study the transformation theory in plasmas based on the above equation. First, we discuss the transport under steady-state conditions.

In the steady state, where the plasma density does not vary with time, Eq. (18.5) becomes:

$$\begin{aligned} - \mathbf {\nabla }\cdot \left( D\cdot \mathbf {\nabla } n\right) \pm \mathbf {\nabla }\cdot \left[ \left( \frac{D\cdot \boldsymbol{E}}{T_i}\right) n\right] =0. \end{aligned}$$

(18.6)

To visualize the transformation results more intuitively, the equation above is converted into its component form in curved space [39]:

$$\begin{aligned} - \partial _i \left( \sqrt{A}D^{ij} \partial _j n\right) \pm \partial _i \left[ \left( \frac{\sqrt{A}D^{ij} E_j}{T_i}\right) n\right] =0. \end{aligned}$$

(18.7)

The curved space corresponds to coordinates $x_i$, with its covariant base vectors denoted as $\boldsymbol{a}_i$ and $\boldsymbol{a}_j$. In Eq. (18.7), A is the determinant of $\boldsymbol{a}_i\cdot \boldsymbol{a}_j$. The transformed Eq. (18.7) in physical space is expressed as:

$$\begin{aligned} -\partial _{i'}\left[ \frac{\partial x'_i}{\partial x_i}\sqrt{A}D^{ij} \frac{\partial x'_j}{\partial x_j}\partial _{j'} n\mp \frac{\partial x'_i}{\partial x_i} \left( \frac{\sqrt{A}D^{ij} E_j}{T_i}\right) n\right] =0, \end{aligned}$$

(18.8)

Here, the physical space coordinates are denoted as $x'_i$. Thus, $\partial x'_i/\partial x_i$ and $\partial x'_j/\partial x_j$ are components of the Jacobian matrix $\textbf{J}$, and the determinant of this Jacobian matrix satisfies $\det \textbf{J}=1/\sqrt{A}$. This equation can then be rewritten as:

$$\begin{aligned} - \boldsymbol{\nabla }'\cdot \left( \frac{\textbf{J}D\textbf{J}^\tau }{\det \textbf{J}}\cdot \boldsymbol{\nabla }' n\right) \pm \boldsymbol{\nabla }'\cdot \left( \frac{\textbf{J}D\cdot \boldsymbol{E}}{\det \textbf{J}T_i}n\right) =0, \end{aligned}$$

(18.9)

We can simplify Eq. (18.9) by incorporating the extra metric through physical parameters, resulting in:

$$\begin{aligned} - \boldsymbol{\nabla }'\cdot \left( D'\cdot \boldsymbol{\nabla }' n\right) \pm &\boldsymbol{\nabla }'\cdot \left( \frac{D'\cdot \boldsymbol{E}'}{T_i}n\right) =0, \end{aligned}$$

(18.10a)

$$\begin{aligned} D'&=\frac{\textbf{J}D\textbf{J}^\tau }{\det \textbf{J}}, \end{aligned}$$

(18.10b)

$$\begin{aligned} \boldsymbol{E}'&=\textbf{J}^{-\tau }\boldsymbol{E}. \end{aligned}$$

(18.10c)

Here, $\boldsymbol{\nabla }'$ indicates the differential in the new coordinate system; $\textbf{J}^{\tau }$ is the transpose of $\textbf{J}$; $\textbf{J}^{-\tau }$ represents the inverse of $\textbf{J}^{\tau }$. Comparing Eqs. (18.6) and (18.10a) shows they have the same form, implying that the migration diffusion equation in the steady state strictly maintains its transformation form. Hence, we can design parameter control for steady-state plasma transport using Eqs. (18.10b) and (18.10c).

2.2 For Transient-State Plasma Transport

Although the transformation theory fits well with steady-state plasma transport, the situation under transient conditions is quite different. Re-examining Eq. (18.5) and processing it in a manner similar to the steady-state, we obtain:

$$\begin{aligned} \frac{1}{\det \textbf{J}}\partial _t n-\boldsymbol{\nabla }'\cdot \left( \frac{\textbf{J}D\textbf{J}^\tau }{\det \textbf{J}}\cdot \boldsymbol{\nabla }' n\right) \pm \boldsymbol{\nabla }'\cdot \left( \frac{\textbf{J}D\cdot \boldsymbol{E}}{\det \textbf{J}T_i}n\right) =0, \end{aligned}$$

(18.11)

Substituting Eqs. (18.10b) and (18.10c) into Eq. (18.11), we obtain:

$$\begin{aligned} \frac{1}{\det \textbf{J}}\partial _t n-\boldsymbol{\nabla }'\cdot \left( D'\cdot \boldsymbol{\nabla }' n\right) \pm \boldsymbol{\nabla }'\cdot \left( \frac{D'\cdot \boldsymbol{E}'}{T_i}n\right) =0, \end{aligned}$$

(18.12)

Comparing Eqs. (18.5) and (18.12), we notice an irreducible parameter in front of the time-dependent term. The transformed equation has changed its form, indicating that the transformation theory fails for transient plasma transport. Approximating Eq. (18.12), we get:

$$\begin{aligned} \partial _t n-\boldsymbol{\nabla }'\cdot \left( D''\cdot \boldsymbol{\nabla }' n\right) &\pm \boldsymbol{\nabla }'\cdot \left( \frac{D''\cdot \boldsymbol{E}''}{T_i}n\right) =0, \end{aligned}$$

(18.13a)

$$\begin{aligned} D''&=\textbf{J}D\textbf{J}^{\tau }, \end{aligned}$$

(18.13b)

$$\begin{aligned} \boldsymbol{E}''&=\textbf{J}^{-\tau }\boldsymbol{E}. \end{aligned}$$

(18.13c)

Where Eqs. (18.13b) and (18.13c) are parameter transformation rules. In this manner, the equation retains its transformation-invariant characteristics. Similar to the convection-diffusion equation, Eq. (18.13a) is an approximate form of Eq. (18.5). Only when $\det \textbf{J}=1$, does Eq. (18.13a) strictly transform to Eq. (18.5).

Revisiting Eq. (18.11) and multiplying both sides by $\det \textbf{J}$, we get:

$$\begin{aligned} \partial _t n -\det \textbf{J}\left[ \boldsymbol{\nabla }'\cdot \left( \frac{\textbf{J}D\textbf{J}^\tau }{\det \textbf{J}}\cdot \boldsymbol{\nabla }' n\mp \frac{\textbf{J}D \cdot \boldsymbol{E}}{\det \textbf{J}T_i}n\right) \right] =0, \end{aligned}$$

(18.14)

Then, decomposing the differential terms and isolating $1/\det \textbf{J}$, we get:

$$\begin{aligned} \begin{aligned} &\det \textbf{J}\left[ \mathbf {\nabla }'\left( \frac{1}{\det \textbf{J}}\right) \cdot \left( \textbf{J}D\textbf{J}^\tau \cdot \mathbf {\nabla }'n\right) + \frac{1}{\det \textbf{J}}\mathbf {\nabla }'\cdot \left( \textbf{J}D\textbf{J}^\tau \cdot \mathbf {\nabla }'n\right) \right] \\ &\mp \det \textbf{J}\left[ \mathbf {\nabla }'\left( \frac{1}{\det \textbf{J}}\right) \cdot \left( \frac{\textbf{J}D\cdot \boldsymbol{E}}{T_i}n\right) + \frac{1}{\det \textbf{J}}\mathbf {\nabla }'\cdot \left( \frac{\textbf{J}D\cdot \boldsymbol{E}}{T_i}n\right) \right] =\partial _t n \end{aligned} \end{aligned}$$

(18.15)

Combining the terms that include $\mathbf {\nabla }'\left( 1/\det \textbf{J} \right)$, we get:

$$\begin{aligned} \partial _t n &=\mathbf {\nabla }'\cdot \left( \textbf{J}D\textbf{J}^\tau \cdot \mathbf {\nabla }'n\mp \frac{\textbf{J}D\cdot \boldsymbol{E}}{T_i}n\right) +\Delta , \end{aligned}$$

(18.16a)

$$\begin{aligned} \Delta &=\det \textbf{J}\mathbf {\nabla }'\left( \frac{1}{\det \textbf{J}} \right) \cdot \left( \textbf{J}D\textbf{J}^{\tau }\cdot \mathbf {\nabla }'n\mp \frac{\textbf{J}D\cdot \boldsymbol{E}}{T_i} n \right) . \end{aligned}$$

(18.16b)

We observe that if $\Delta$ is sufficiently small to be ignored, then Eq. (18.16a) reverts to Eq. (18.13a). Now, analyzing the specific expression of the error in Eq. (18.16b), which involves the Jacobian determinant, plasma diffusion coefficient, plasma density, temperature, and electric field. The spatial transformation determines the form of the Jacobian. Hence, $\Delta$ is related to the physical parameters of the system and the specific form of spatial transformation. To minimize the error, both diffusion coefficients and electric field need to be sufficiently small. Moreover, there exist special spatial transformations for which $\det \textbf{J}=1$, making the error $\Delta =0$. In this case, Eq. (18.13a) is strictly the same as Eq. (18.5).

3 Potential Applications for Transformation-Based Plasma Metamaterials

To validate our theory, we introduce three conceptual plasma devices: the plasma cloak, concentrator, and rotator. See Fig. 18.2. A unique feature they share is that while achieving their respective functionalities, they do not affect the plasma distribution in the background medium, which is quite distinctive in the plasma field. Specifically, the cloak can protect the core region, meaning that the background does not affect the core, and vice versa; the concentrator can concentrate plasma flow, enhancing the density gradient in the core region; the rotator can alter the propagation direction of the plasma in the core region. Next, we will detail the coordinate transformation relations for implementing the functionalities of these three devices, starting with the cloak.

Fig. 18.2

(from Ref. [37])

Schematic diagrams of the three plasma devices. From left to right: cloak, concentrator, and rotator. The blue solid lines represent the direction of the plasma flow. The side length of the square background in the model is set to $l=0.12$ m. The diffusivity of the background is set as $D_0=9.2\times 10^{-7}$ m $s^{-1}$, and the field strength is $E_{x0}=1.04\times 10^4$ V m$^{-1}$. Other parameter values: $r_1=0.020$ m, $r_2=0.030$ m, $r_m=0.025$ m, $\theta _0=\pi /3$, and the reduced temperature is set to $T_0=2.0$ V.

3.1 Cloak

To realize the plasma cloak, we can express the coordinate transformation relationship from the virtual space $r_i$ to the physical space $r'_i$ as

$$\begin{aligned} \begin{aligned} r' &=\frac{r_2-r_1}{r_2}r+r_1,\\ \theta ' &=\theta , \end{aligned} \end{aligned}$$

(18.17)

where $r_1$ and $r_2$ represent the inner and outer diameters of the cloak, respectively. Identical to the transformation relationship of the chemical wave cloak in the previous work, the spatial transformation here also expands the point at the center in the virtual space into an inner circle, thereby compressing the outer circle into a ring with an inner diameter equivalent to the radius of the small circle. We can then compute the Jacobian matrix based on Eq. (18.17), expressed as

$$\begin{aligned} {\textbf {J}}= \begin{bmatrix} \frac{r_2-r_1}{r_2} &{} 0\\ 0 &{} \frac{\left( r_2-r_1\right) r'}{r_2\left( r'-r_1 \right) } \end{bmatrix} . \end{aligned}$$

(18.18)

Next, based on Eqs. (18.13b) and (18.13c), the transformation parameters required for the cloak can be calculated as

$$\begin{aligned} D''&=\left[ \begin{array}{cc} D_0\left( \frac{r_2-r_1}{r_2}\right) ^2 &{} 0\\ 0 &{} D_0\left[ \frac{\left( r_2-r_1\right) r'}{r_2\left( r'-r_1 \right) }\right] ^2 \end{array} \right] , \end{aligned}$$

(18.19a)

$$\begin{aligned} \boldsymbol{E}''&=\left[ \begin{array}{cc} \frac{r_2}{r_2-r_1}E_r \\ \frac{r_2\left( r'-r_1 \right) }{\left( r_2-r_1\right) r'}E_t \end{array} \right] , \end{aligned}$$

(18.19b)

where $D_0$ is the diffusion rate of the background, and $E_r$ and $E_t$ are the radial and tangential components of the background electric field $\boldsymbol{E}_0$, respectively. In this model, the background electric field is set in the x direction, so $\boldsymbol{E}_0=\left[ E_{x0},\,0\right] ^\tau$ V m$^{-1}$. Thus, based on the conversion relationship between Cartesian coordinates and cylindrical coordinates, the specific forms of $E_r$ and $E_t$ are

$$\begin{aligned} \boldsymbol{E}_0=\left[ \begin{array}{cc} E_r \\ E_t \end{array} \right] =\left[ \begin{array}{cc} E_{x0}\cos \theta \\ -E_{x0}\sin \theta \end{array} \right] . \end{aligned}$$

(18.20)

Since the transient mass transfer theory is an approximation, the electric field is not continuous at the boundary between the cloak and the background medium. One potential solution is to artificially control the potential at this boundary. Then, with Eq. (18.19), we can achieve transient plasma cloaking.

3.2 Concentrator

For the concentrator, the spatial transformation corresponding to the coordinate transformation can be designed as,

$$\begin{aligned} \begin{aligned} r' &=\frac{r_1}{r_m}r,\qquad r<r_m\\ r' &=\frac{r_1-r_m}{r_2-r_m}r_2+\frac{r_2-r_1}{r_2-r_m}r,\quad r_m<r<r_2\\ \theta ' &=\theta . \end{aligned} \end{aligned}$$

(18.21)

$r_m$ is a constant between $r_1$ and $r_2$ (represented by a dashed line in Fig. 18.2). Similarly, this spatial transformation can be understood in virtual space, where a circle with radius $r_2$ is divided into two parts by a circle with radius $r_m$. One part is the inner circle with a radius of $r_m$ and the remaining part is an annulus with an inner diameter of $r_m$ and an outer diameter of $r_2$. The radius of the inner circle is then compressed to $r_1$ and the inner diameter of the annulus is stretched to $r_1$.

For convenience, let $p=\frac{r_2-r_1}{r_2-r_m}$, $q=\frac{r_1-r_m}{r_2-r_m}r_2$, and $f=\frac{r_1}{r_m}$. Based on Eq. (18.21), the Jacobian matrix of the concentrator is obtained as,

$$\begin{aligned} {\textbf {J}}_1 &=\left[ \begin{matrix} f &{} 0\\ 0 &{} f \end{matrix} \right] ,~~~~~~~r'<r_1 \end{aligned}$$

(18.22a)

$$\begin{aligned} {\textbf {J}}_2 &=\left[ \begin{matrix} p &{} 0\\ 0 &{} \frac{r'p}{r'-q} \end{matrix} \right] ,~~r_1<r'<r_2 \end{aligned}$$

(18.22b)

According to Eqs. (18.13b) and (18.13c), the transformation parameters of the concentrator at $r'<r_1$ ($D_1''$ and $\boldsymbol{E}_1''$) and $r_1<r'<r_2$ ($D_2''$ and $\boldsymbol{E}_2''$) are obtained as,

$$\begin{aligned} D_1''& =\left[ \begin{array}{cc} D_0f^2 &{} 0\\ 0 &{} D_0f^2 \end{array} \right] , \end{aligned}$$

(18.23a)

$$\begin{aligned} \boldsymbol{E}_1''&=\left[ \begin{array}{cc} E_r/f\\ E_t/f \end{array} \right] , \end{aligned}$$

(18.23b)

$$\begin{aligned} D_2''&=\left[ \begin{array}{cc} D_0p^2 &{} 0\\ 0 &{} D_0\left( r'p/\left( r'-q\right) \right) ^2 \end{array} \right] , \end{aligned}$$

(18.23c)

$$\begin{aligned} \boldsymbol{E}_2''&=\left[ \begin{array}{cc} E_r/p \\ E_t\left( r'-q\right) /r'p \end{array} \right] . \end{aligned}$$

(18.23d)

The efficiency of the concentrator is related to the value of $r_m/r_1$. The larger the $r_m$, the higher the concentration efficiency, corresponding to a larger density gradient. This can be understood from the physical image of the space transformation. When $r_m$ is larger, the compressed area of the inner circle is larger, which corresponds to a higher degree of concentration. With Eq. (18.23), we can achieve the transient plasma concentration.

3.3 Rotator

For the rotator, the spatial transformation corresponding to the coordinate transformation is given by,

$$\begin{aligned} \begin{aligned} r' &=r,\\ \theta ' &=\theta +\theta _0,~~~r<r_1\\ \theta ' &=\theta +\theta _0\frac{r-r_2}{r_1-r_2},~~~r_1<r<r_2 \end{aligned} \end{aligned}$$

(18.24)

where $\theta _0$ is a constant rotation angle. This spatial transformation can be understood as a series of circles with radius $r\in [r_1,\, r_2]$ rotating around their center, with their rotation angles linearly varying with their sizes.

For convenience, let $g=\theta _0/\left( r_1-r_2\right)$. The Jacobian matrix of the rotator can then be obtained as,

$$\begin{aligned} {\textbf {J}}_1 &=\left[ \begin{matrix} 1 &{} 0\\ 0 &{} 1 \end{matrix} \right] ,~~~~~~~r'<r_1 \end{aligned}$$

(18.25a)

$$\begin{aligned} {\textbf {J}}_2 &=\left[ \begin{matrix} 1 &{} 0\\ r'g &{} 1 \end{matrix} \right] .~~~r_1<r'<r_2 \end{aligned}$$

(18.25b)

Noting that $\det \textbf{J}_1=\det \textbf{J}_2=1$, the Eq. (18.13) becomes an exact solution. Similarly, using Eqs. (18.13b) and (18.13c), the transformation parameters for the rotator in the regions $r'<r_1$ ($D_1''$ and $\boldsymbol{E}_1''$) and $r_1<r'<r_2$ ($D_2''$ and $\boldsymbol{E}_2''$) can be computed as,

$$\begin{aligned} D_1''& =\left[ \begin{array}{cc} D_0 &{} 0\\ 0 &{} D_0 \end{array} \right] , \end{aligned}$$

(18.26a)

$$\begin{aligned} \boldsymbol{E}_1''&=\left[ \begin{array}{cc} E_r\\ E_t \end{array} \right] , \end{aligned}$$

(18.26b)

$$\begin{aligned} D_2''&=\left[ \begin{array}{cc} D_0 &{} D_0r'g\\ D_0r'g &{} D_0\left[ \left( r'g\right) ^2+1\right] \end{array} \right] , \end{aligned}$$

(18.26c)

$$\begin{aligned} \boldsymbol{E}_2''&=\left[ \begin{array}{cc} E_r-r'gE_t\\ E_t \end{array} \right] . \end{aligned}$$

(18.26d)

In practical simulations, the $\boldsymbol{E}''$ must be rotated. Since the parameters after transformation are taken from the new coordinate system, and the new coordinates are rotated by a certain angle relative to the old coordinates, the Eq. (18.26) cannot be used directly if the system’s coordinate system is not adjusted. The core parameters in the old cylindrical coordinate system should be,

$$\begin{aligned} D_1'' =\left[ \begin{array}{cc} D_0 &{} 0\\ 0 &{} D_0 \end{array} \right] ,\quad \boldsymbol{E}_1''=\left[ \begin{array}{cc} E_r\left( \theta -\theta _0\right) \\ E_t\left( \theta -\theta _0\right) \end{array} \right] , \end{aligned}$$

(18.27)

and the shell parameters are,

$$\begin{aligned} D_2''=\left[ \begin{array}{cc} D_0 &{} D_0rg\\ D_0rg &{} D_0\left[ \left( rg\right) ^2+1\right] \end{array} \right] , \quad \boldsymbol{E}_2''=\left[ \begin{array}{cc} E_r\left( \theta -\theta _r\right) -rgE_t\left( \theta -\theta _r\right) \\ E_t\left( \theta -\theta _r\right) \end{array} \right] , \end{aligned}$$

(18.28)

where $E_r\left( \theta -\theta _0\right)$ and $E_t\left( \theta -\theta _0\right)$ represent the values of $E_r$ and $E_t$ at $\theta =\theta -\theta _0$; $E_r\left( \theta -\theta _r\right)$ and $E_t\left( \theta -\theta _r\right)$ represent the values of $E_r$ and $E_t$ at $\theta =\theta -\theta _r$; and $\theta _r=g\left( r-r_2\right)$. $E_r$ and $E_\theta$ are determined by Eq. (18.20). The transformation form of the diffusion rate is not affected by the rotation of the coordinate system, because rotation of the scalar D has no meaning. Up to this point, we have derived the parameter transformation rules for regulating the plasma, namely the cloak, the concentrator, and the rotator, which are (18.19), (18.23), (18.27) and (18.28). Finite-element simulation is used to verify the model.

3.4 Simulation Verification

The models for the three devices have been presented in Fig. 18.2. In order to visually depict the transient distribution of the plasma, a periodically fluctuating plasma source $n_b$ is chosen to impose on the left boundary of the square background. The specific expression for this is given by:

$$\begin{aligned} n_b = n_1 \cos (\omega _0 t) + n_0, \end{aligned}$$

(18.29)

where $n_1 = 5.0 \times 10^{15}~\text {m}^{-3}$, $\omega _0 = \frac{2\pi }{10}~\text {s}^{-1}$, and $n_0 = 1.0 \times 10^{17}~\text {m}^{-3}$. The right side of the background is set as the outflow boundary, while the top and bottom boundaries are set as no flux. For the shield, its internal boundary should also be specifically set as no flux. In this context, the diffusion coefficient in the background is set as a constant $D_0$, and the electric field is set as a uniform electric field along the x-direction, represented by $\boldsymbol{E}_0$. Then all the parameters can be designed according to the above transformation rules, and the simulation results of cloaking, concentrating, and rotating are shown in Figs. 18.3, 18.4 and 18.5, respectively.

Fig. 18.3

(from Ref. [37])

Simulation results of the cloak at transient states. (a1)–(a3) Density profiles for pure background at 10 s, 22 s, and 40 s, respectively. (b1)–(b3) Density profiles for background with an obstacle at 10 s, 22 s, and 40 s, respectively. (c1)–(c3) Density profiles for background with the cloak at 10 s, 22 s, and 40 s, respectively.

Fig. 18.4

(from Ref. [37])

Simulation results of concentrator and rotator at transient states. (a1)–(a3) Density profiles for the concentrator at 10 s, 22 s, and 40 s, respectively. (b1)–(b3) Density profiles for the rotator at 10 s, 22 s, and 40 s, respectively.

Figure 18.3 illustrates the transient simulation of plasma transport under three conditions, namely, transporting in a pure background medium (set as the reference), in a background medium with a bare obstacle, and in a background medium with an obstacle covered by the cloak. The columns from left to right are screenshots of distributions of the plasma density at 10 s, 22 s, and 40 s, respectively. Due to the boundary condition of harmonically oscillating density, the plasma streams forward in a wave-like form. Moreover, the amplitude attenuation of the plasma flow reflected from the figures is caused by the diffusion, whose decay rate is codetermined by the oscillation frequency, diffusivity, and electric field. As a result, suitable values are carefully chosen to make the results more intuitive. The cloak designed with the transformation theory helps to cancel the scattering induced by the obstacle. Therefore, the density profiles of the background plasma keep nearly undisturbed, which shows the validity of the theory.

Fig. 18.5

(from Ref. [37])

(a1)–(a3) Color mapping of density profiles at 40 s with a cloak, concentrator, and rotator, respectively. (b1)–(b3) Comparisons between density profiles in the pure background (reference) and those with a cloak, concentrator, and rotator, respectively. The grey dashed lines denote the position of the devices. The data are extracted along the yellow dashed line (y $=$ 0) in (a1)–(a3).

The transient simulation results for the concentrator and rotator are shown in Fig. 18.4. The first row of snapshots shows the converging effect of the gradient of plasma density. In addition, as a determinant of the converging effect, a bigger ratio ($r_m/r_1$) would bring a higher converging effect. And the maximum ratio is $r_2/r_1$. For the rotator, the rotation of plasma flow appears in Fig. 18.3b1–b3. Linearly deflecting concentric circles in the virtual space can account for the gradual deflection of the density profiles. The target rotation angle in the core region is determined by $\theta _0$ in Eqs. (18.24). Particularly, $\det \textbf{J}=1$ for rotators helps to completely eliminate the disturbance to background plasma density.

To further explore the performance of the devices, the density values along a horizontal line (denoted by the yellow dashed lines in Fig. 18.5) from the results at 40 s are extracted and the density distribution of functional devices is compared with that of reference. See Fig. 18.5b1–b3. Two regions should be remarked. One is the core region of the device, the other is the background. All the red dashed lines in Fig. 18.5b1–b3 denote the data of the reference, while the blue dotted lines represent the data of the cloak, concentrator, and rotator, respectively. In Fig. 18.5b1, it is clear that the data are well overlapped in the background, and the plasma is excluded well from the core region. Moreover, the relative difference in the plasma density in the background region was less than 0.15%. In Fig. 18.5b2, the dotted line is indeed denser than the dashed line in the core region without being seriously dislocated in the background. And the relative difference was less than 0.13%. In Fig. 18.5b3, the relative difference was less than 0.01% which is far smaller than the value of the cloak or concentrator. As mentioned above, the accurate transformation form of Eq. (18.13a) may account for this nearly zero difference. Overall, the simulation can confirm the feasibility and reliability of the theory.

The progression in plasma physics has paved the way for novel technologies and methods, finding cutting-edge applications in biomedicine, the crystal industry, and materials science [31]. See Fig. 18.6. We envisage several potential applications for devices crafted based on transformation theory. Consider the cloak, which has an isolated core region, as a prime candidate for safeguarding healthy tissue during plasma treatments of infected wounds. In catalyst development, a plasma flow convergence, characterized by a higher density of active particle clusters, augments the interaction between the plasma and the catalyst. This makes the concentrator an ideal tool to enhance catalytic performance. Furthermore, in aerospace, the concentrator might hold promise in elevating the efficiency of plasma-assisted engines. Beyond the uses already mentioned, the principles of coordinate transformation could be harnessed to achieve plasma separation or guidance, proving valuable for plasma etching or depositing. Additionally, transformation theory could contribute to the development of plasma metamaterials intended for electromagnetic wave manipulation [40, 41].

Fig. 18.6

Adapted from Ref. [12]

Applications of plasmas.

Indeed, the proposed approach rooted in transformation theory holds merit. Even with the inherent challenges in manifesting the transformed diffusivities and electric fields, alternative techniques can be employed to achieve similar outcomes. Plenty of research has delved into customizing particle diffusivities. For instance, the scattering cancellation method allows for the creation of a bilayer diffusive cloak using two homogeneous materials [42]. The complex diffusivity might be attainable through the effective medium theory [43] or even machine learning techniques [44]. When it comes to electric field manipulation, insights from studies on electrostatic and magnetic cloaks could be illuminating [45, 46].

While the future is ripe with possibilities, it also presents its fair share of challenges. Under broader circumstances, we must factor in the effects of magnetic fields and gas-phase reactions within the plasma. Truly exerting control over diffusivities and electric fields is a daunting task due to the intricate interplay between charged particles and electromagnetic fields. As such, it becomes paramount to incorporate alternative theoretical models or methodologies, such as the particle-in-cell/Monte Carlo collision model [47] or the nonequilibrium Green’s function approach [48]. Additionally, plasma temperatures tend to fluctuate over time or space, especially in transient states, leading to a shift in transformation principles. In some scenarios, advection may play a role in plasma transport. Accounting for this term could diversify plasma regulation techniques. Furthermore, the emerging focus on spatiotemporal modulation in heat diffusion could provide new insights into plasma physics [49]. In conclusion, refining the transformation theory for plasmas requires heightened focus, research, and dedication.

4 Potential Impacts for Novel Physics

The realm of plasma transport transformation, being in its nascent stage, harbors a vast expanse of uncharted physics awaiting exploration. Among the intriguing avenues is the challenge of nonreciprocal plasma transport. Given the consistent display of space-inversion symmetry in diffusion equations, pinpointing nonreciprocal mechanisms within plasma transport proves daunting. An effective approach might involve the application of spatiotemporal modulations or nonlinear parameters, as found in the dynamics of heat and particle diffusion [50,51,52,53,54]. Another captivating area is topological plasma transport [55]. With lessons drawn from well-established mechanisms in topological particle and heat diffusion [56,57,58,59,60,61,62,63,64], researchers could embark on a variety of experimental strategies. For instance, formulating a plasma analog of the Su-Schrieffer-Heeger model might shed light on the intricate interplay between bulk-boundary and the emergence of edge states [65]. Additionally, the exploration into the presence of exceptional points and geometric phases promises rich insights. In a linear context, the use of Chern numbers can act as descriptors for the unconventional topologies found in plasma. Furthermore, the introduction of elements such as flow shear or a density gradient could lead to non-Hermiticity in systems, thereby highlighting plasma as a key medium for delving into the complexities of non-Hermitian physics. Given these vast possibilities, we anticipate a slew of groundbreaking discoveries in this domain in the near future.

5 Conclusion

In this chapter, a toy model, specifically the diffusion-migration model, has been utilized to elucidate plasma transport. This investigation has highlighted the practicality of the transformation theory. It was revealed that the transformed diffusion-migration equation maintains form-invariance at steady states but deviates at transient states. However, it was also illustrated that by setting minimal diffusivities, the transformed transient equation can attain an approximate form-invariance. Based on this, three conceptual model devices were conceptualized, serving as plasma cloaks, concentrators, or rotators for transient plasma transport. Such findings can potentially expand methods for manipulating plasma flow and offer applications across multiple sectors, including medicine and aerospace. In conclusion, controlling plasma may serve as a nexus between diffusion and wave metamaterials. Although there are evident differences between diffusion and wave metamaterials, simultaneous control over diffusion and wave propagation signifies a notable progression. Plasma transport predominantly follows a diffusion process, but plasmas are frequently harnessed to regulate electromagnetic waves due to their distinctive permittivity. As such, the convergence of diffusion and wave propagation within a singular system can be a precursor to the emergence of new physics.