Particle Diffusion Process with Artificial Control: Diffusion Metamaterials

Abstract

Diffusion acts as a fundamental process for both energy and mass movement, showcasing dynamics that differs from wave propagation. The emergence of metamaterials provides a robust platform for custom manipulations of mass diffusion, drawing on principles like transformation theory and scattering-cancellation theory. Such manipulations have set the stage for novel findings and pioneering applications. This chapter delves into a thorough analysis of the recent progress in diffusion metamaterials. Earlier studies are methodically categorized based on diffusion models, delving deeply into the related theories, applications, finite-element simulations, and experimental approaches of different mechanisms. The chapter wraps up with a scrutiny of the inherent challenges tied to the theoretical underpinnings and architectural design of diffusion metamaterials. Some of these challenges are seen as potentially mitigated by upcoming approaches, such as pseudo-conformal map** and machine-learning methodologies.

1 Opening Remarks

Metamaterials [1, 2] are meticulously designed structures, built from functional building blocks densely packed into an effective medium [3, 4]. Although the concept of methodically engineering artificial materials has historical precedence, their exceptional design and fabrication flexibility are remarkable. This adaptability is primarily due to the principles of effective materials based on sub-characteristic length [5] structures, differentiating them from other synthetic materials like photonic and phononic crystals [6]. For example, wave metamaterials [7] consist of structures built from units smaller than their respective characteristic wavelengths. Similarly, diffusion metamaterials [8,9,10,11], another branch of the metamaterial family, are designed with characteristic lengths in mind, distinguishing them from other thermal synthetic materials.

In 2006, transformation optics, drawing inspiration from Einstein’s general theory of relativity, linked virtual spatial distortions with the actual spatial inhomogeneities and general anisotropies of magneto-dielectric metamaterials [12] in Cartesian physical space. This approach offered a direct correlation between desired electromagnetic phenomena [13, 14] and the required material responses, providing a powerful tool for managing electromagnetic fields across various scales. Though metamaterials are often associated with negative refractive indices [1, 15, 16] and invisibility cloaking [17,18,19] in electromagnetism or optics [20, 21], the underlying concept of metamaterials reaches broader domains, including nonlinear optics [22, 23], acoustics [24], direct/alternating current [25, 26], and fluid dynamics [27, 28].

Heat transfer, a cornerstone of diffusion phenomena, governs energy transport and is generally propelled by spatial temperature disparities. The focal points of heat transfer research encompass temperature, heat flux management-ranging from heating to cooling, and energy harvesting [29, 30], which pertains to the conversion of thermal energy from sources like the Sun into other forms of energy. Recent advancements have seen the crafting of various thermal metamaterials to achieve superior functionalities [31, 32] using passive frameworks, including transformation theory [8, 33], scattering cancellation [34, 35], and topology optimization [36, 37]. In addition, there are active methods [38, 39] utilizing external triggers to ensure exceptional thermal manipulation. Notably, the inherent dissipative nature of diffusion aligns closely with non-Hermitian physics [40].

Managing mass flow is crucial for a multitude of processes in chemical and biomolecular sciences [41]. Microscopically, particle diffusion usually appears as Brownian motion [42], a fundamental concept in statistical physics [43, 44]. Decoding the intricate mechanisms underlying particle diffusion is essential for creating efficient transport materials and innovative therapeutic techniques. On a macroscopic scale, diffusion adheres to Fick’s law [45]. Given the similarities between particle diffusion and heat transport, the frameworks developed for thermal metamaterials are anticipated to be applicable to particle diffusion systems.

This chapter pivots on macroscopic particle diffusion, delving into theoretical designs, simulations, and experimental findings. In Sect. 17.2, traditional manipulation techniques within the convection-diffusion process are described. Subsequently, the spotlight shifts to chemical waves in Sect. 17.3, which are characterized by spatiotemporal [46] concentration variations. The chapter concludes with a discussion of the prevailing challenges and potential future directions in this field. It is hoped that this chapter elucidates the complexities of diffusion dynamics across diverse models and sheds light on outstanding challenges that merit future investigation.

2 Quasi-equilibrium Diffusion Model

2.1 General Transformation Theory

Metamaterials tailored for mass diffusion have garnered considerable interest since the debut of the particle cloak rooted in transformation theory. Their potential in fields like drug delivery underscores this interest. Numerous fundamental devices influencing mass diffusion have been documented, spanning from particle movement to light transmission. These devices encompass particle concentrators, electron cloaks, light-diffusive cloaks, and plasma rotators.

While the transformation theory doesn’t precisely correlate with Fick’s second equation, mass diffusion cloaking through the low-diffusivity approximation remains achievable. In 2013, Guenneau and colleagues [47] introduced a granular cloak grounded in transformation theory. Such a cloak holds potential, especially in safeguarding drugs during internal delivery. Their focus centered on the concentration of chemical species, potentially benefiting fields such as biophysics and bioengineering. They began by representing the convection-diffusion equation in a domain $\Omega$ as:

$$\begin{aligned} \frac{\partial c}{\partial t}=\sum _{i, j} \frac{\partial }{\partial x_i}\left( \kappa _{i j}(x) \frac{\partial c}{\partial x_j}\right) -\sum _i \frac{\partial }{\partial x_i} v_i c, \end{aligned}$$

(17.1)

where c denotes the mass concentration in biochemistry, as a function of time $t>0$, $\kappa$ signifies the chemical diffusion (in units of m$^3$ s$^{-1}$), and $v$ is the velocity field. After a variable change $x \rightarrow y$ governed by a Jacobian matrix $J$ where $J_{ij}=\partial y_i/\partial x_j$, the equation becomes:

$$\begin{aligned} \frac{1}{{\text {det}}_{i j}} \frac{\partial c}{\partial t}=\sum _{i, j, k, l} \frac{\partial }{\partial y_i}\left( \frac{1}{{\text {det}} J_{i j}} \mathrm {~J}_{i k} \kappa _{k l} J_{j l}^{T} \frac{\partial c}{\partial y_j}\right) -\sum _{i, j} \frac{1}{{\text {det}} J_{i j}} J_{i j}^{T} \frac{\partial }{\partial y_i} v_i c, \end{aligned}$$

(17.2)

The transformed diffusivity and velocity are defined as:

$$\begin{aligned} \left\{ \begin{aligned} \kappa ^{\prime }&=J \kappa J^{T}/\text {det}J\\ v^{\prime }&=J^{T}v/\text {det}J \end{aligned}\right. \end{aligned}$$

(17.3)

With Eq. (17.3), a diffusion cloak was constructed that rendered enclosed objects undetectable against a background. Spheres coated with concentric layers of homogeneous isotropic diffusivity were employed to simulate the desired anisotropic heterogeneous diffusivity. The study introduced a novel perspective on cloaking in the context of mass diffusion, with potential implications for fields such as bioengineering and chemical engineering.

Following this, Zeng et al. [48] introduced an innovative technique to reduce steel corrosion in concrete. They designed a multilayered concrete cloak aimed at shielding against chloride ions. Simulation results indicated that the six-layer cloak effectively prevented the infiltration of chloride ions into the concrete. Experimental data corroborated that concentration gradients outside the cloak remained consistent, while gradients within the cloak were reduced.

The previous researches focused on geometrically isotropic structures, like circles, which might have limited their functional adaptability. Diversifying from this approach, Guenneau et al. [49] expanded the scope, moving from basic circular cylindrical designs to more complex geometries. They introduced a geometric transformation catering to cloaks of various shapes, denoted as:

$$\begin{aligned} \rho ^{\prime }=\frac{R_3(\theta )-R_1(\theta )}{R_3(\theta )}\rho +R_1(\theta ),0<\rho <R_3(\theta ), \end{aligned}$$

(17.4)

where $R(\theta )$ denotes a continuous function with a periodicity of 2$\pi$ and defines the cloak’s cross-section. Equation (17.4) transitions the point $\rho =0$ to the region $\rho ^{\prime }<R_1(\theta )$ and reshapes the space bounded by $0<\rho <R_3(\theta )$ into the region $R_1(\theta )<\rho ^{\prime }<R_3(\theta )$. By integrating Eqs. (17.3) and (17.4), the parameters for the arbitrary cloak were determined. Using this methodology, concentrators and rotators of various forms were crafted. The potential to design invisibility carpets optimized for diffusion processes was also explored.

Fig. 17.1

(from Ref. [50], licensed under CC-BY 4.0)

Coordinate transformation applications in mass separation metamaterials: a Cloaking: Transformation of a point into a circle with radius $R_1$, accompanied by adjacent region compression. b Concentration: A circular region defined by $0 < r < R_2$ is compressed to $0 < r < R_1$, succeeded by expansion of the neighboring annulus from $R_2$ to $R_1$. c Bi-Functional Multilayer Design: The structure incorporates four distinct materials, represented in dark blue, grey, light blue, and light green.

In 2016, Restrepo-Fl$\acute{\text {o}}$rez et al. [50] delved into extensive investigations, as illustrated in Fig. 17.1. Utilizing linear transformations, they designed both cloaks and concentrators. The cloaking transformation morphed a circle of a predetermined radius into an annulus. In contrast, the design of concentrators entailed two operations: compressing one circular region into another and subsequently expanding an adjacent area. Through experimental implementations, they introduced multilayered designs, utilizing concentric rings for cloaking and lamellar structures for concentration. The research illuminated the simultaneous manipulation of different particles, presenting a viable approach for mixture separation. It showcased the cloaking of N$_2$ and the concentration of O$_2$ over designated time frames, culminating in a stable state. The team further explored molecule cloaking and concentrating, emphasizing the significance of mass concentration discontinuities due to varied solubilities. Understanding these discontinuities is essential for the meticulous design and assessment of mass diffusion metamaterial devices.

The transformation theory also presents opportunities in the domain of metamaterial membrane designs. In a separate 2016 study, Restrepo-Fl$\acute{\text {o}}$rez et al. [51] leveraged coordinate transformations in tandem with metamaterial theory to enhance compound separation. By strategically designing spatial diffusivity for mass diffusion, they achieved localization of distinct compounds in specified regions, allowing for the separation of mixtures via a metamaterial membrane. Such an approach could hold relevance in various sectors necessitating separation processes.

2.2 Scattering Cancellation Theory

While prior discussions have underscored the potential of controlling mass diffusion through transformation theory, the inherent challenges of complex transformations, extreme parameters, and inhomogeneous functional regions pose considerable practical challenges. These issues may be more effectively circumvented by adopting the scattering cancellation theory for the design of metamaterial-based mass manipulations.

Li et al. [52, 53] considered a stable state without particle sources where the concentration distribution is governed by the equation:

$$\begin{aligned} \nabla \cdot (D \nabla c)=0, \end{aligned}$$

(17.5)

Here, $\nabla$ is the operator in the Cartesian coordinate system and D denotes diffusivity. For a sphere of radius $R_1$ encapsulated by a bilayer spherical cloak having an inner radius $R_2$ and an outer radius $R_3$, the concentration distribution within this sphere is:

$$\begin{aligned} c_1=-\sum _{l=1}^{\infty }\left[ A_l r^l+B_l r^{-(l+1)}\right] P_l(\cos \theta ) \quad \left( r \le R_1\right) , \end{aligned}$$

(17.6)

where $P_l(\cos \theta )$ is a lth-order Legendre polynomial, and $A_l$ and $B_l$ are constants yet to be determined. They also presented the concentration distribution for the bilayer cloak as:

$$\begin{aligned} \begin{aligned} c_2 & =-\sum _{l=1}^{\infty }\left[ C_l r^l+E_l r^{-(l+1)}\right] P_l(\cos \theta ) & \left( R_1<r \le R_2\right) \\ c_3 & =-\sum _{l=1}^{\infty }\left[ F_l r^l+G_l r^{-(l+1)}\right] P_l(\cos \theta ) & \left( R_2<r \le R_3\right) \\ c_4 & =-\sum _{l=1}^{\infty }\left[ H_l r^l+I_l r^{-(l+1)}\right] P_l(\cos \theta ) & \left( r>R_3\right) \end{aligned} \end{aligned}$$

(17.7)

Given that $r \rightarrow \infty$, $c_4$ then equates to $-c_0 r \cos \theta$. Hence, it’s only essential to consider $l = 1$. If $r \rightarrow 0$, then $c_1$ remains finite, making $B_1=0$. The continuity conditions at boundaries are expressed as:

$$\begin{aligned} \begin{aligned} \left. c_i\right| _{r=R_1, R_2, R_3} & =\left. c_{i+1}\right| _{r=R_1, R_2, R_3},\\ \left. D_i \frac{\partial c_i}{\partial r}\right| _{r=R_1, R_2, R_3} & =\left. D_{i+1} \frac{\partial c_{i+1}}{\partial r}\right| _{r=R_1, R_2, R_3}, \end{aligned} \end{aligned}$$

(17.8)

where $D_i$ denotes the diffusivity of the central region, inner layer, outer layer of the bilayer cloak, and the background medium, with $i$ incrementing from 1 to 4. To maintain a uniform concentration within the sphere, they apply a inner layer with a zero diffusivity ($D_2=0$). Therefore, their focus shifts to the concentration distribution outside the cloak. Incorporating Eqs. (17.7) and (17.8) into Eq. (17.5), they derived:

$$\begin{aligned} H_1=c_0 R_2^3 \frac{D_3\left( 2 R_2^3-2 R_1^3\right) -D_4\left( 2 R_2^3+R_1^3\right) }{D_3\left( 2 R_2^3-2 R_1^3\right) +2 D_4\left( 2 R_2^3+R_1^3\right) } \end{aligned}$$

(17.9)

Ensuring $H_1$ equals 0 yields another parameter for the bilayer cloak:

$$\begin{aligned} D=\frac{2 R_2^3+R_1^3}{2\left( R_2^3-R_1^3\right) } D_0. \end{aligned}$$

(17.10)

In the case of 2D particle diffusion, this relationship can be rewritten as:

$$\begin{aligned} D=\frac{R_2^2+R_1^2}{R_2^2-R_1^2} D_0, \end{aligned}$$

(17.11)

where $D$ represents the diffusion coefficient of the bilayer cloak’s outer layer, and $D_0$ is the diffusion coefficient of the surrounding medium.

The discussion reveals that a bilayer cloak requires an inner layer with a zero index and an external layer whose index corresponds to the background medium. A fan-shaped concentrator is also introduced, grounded in scattering cancellation theory. This concentrator can centralize particle concentrations. A notable “plug and switch” metamaterial concept was introduced, which allows modular functional units to be integrated into a primary board, offering switchable functionalities. This design offers adaptability and reconfigurability, fitting a wide range of applications without the need for a complete system redesign. Potential uses include sustained drug delivery, catalytic amplification, and the creation of bio-inspired cytomembranes.

However, using monolayer anisotropic materials presents practical fabrication difficulties. In 2021, Zhou et al. [54] introduced a bilayer meta-device designed to manipulate binary masses (O$_2$ and N$_2$). This model was developed from a direct interpretation of the static Fick’s law, solely using homogeneous media. Numerical studies demonstrated the device’s ability to mask O$_2$ concentrations and concentrate N$_2$ concentrations in both constant and transient states. The minimal interference in external concentration fields highlighted the robustness of the device’s manipulation capabilities.

In environments such as vacuum and air, where light travels ballistically according to Maxwell’s equations, designing a universally effective invisibility cloak is challenging. Schittny et al. [55] explored environments that cause multiple light scatterings, like clouds or frosted glass, where light conforms to Fick’s diffusion equation. They constructed cylindrical and spherical invisibility cloaks, made from polydimethylsiloxane combined with melamine-resin microparticles. These cloaks, having hollow interiors, efficiently hid objects in a water-based diffusive setting across the entire visible spectrum. The discussions that followed emphasized the transient scenario, presenting experimental evidence of the effectiveness of diffusive light cloaking in both constant and quasi-static states. This research on light diffusion cloaks indicates that, in a diffusive light scattering environment, passive, broadband invisibility cloaking for all light directions and polarizations might be feasible.

Traditional mass separation methods are based on the assumption of isotropic membrane material, which restricts directional mass-flow guidance. In 2018, Restrepo-Fl$\acute{\text {o}}$rez et al. [56] presented a distinct separation device that uses an anisotropic polymeric membrane. These anisotropic membranes can be designed as multilayer structures, combining two isotropic materials with different diffusivities. The anisotropic technique achieved selectivities for O$_2$/N$_2$ separations, significantly outperforming the capabilities of leading isotropic polymers.

2.3 Transformation-Invariant Scheme

Traditional mass diffusion metamaterials have made remarkable strides, yet challenges persist when it comes to creating intelligent devices. Here, “intelligent" denotes a material’s ability to adaptively respond to environmental changes. A key limitation of conventional mass diffusion metamaterials is their rigidity; once their parameters undergo a transformation, they become static. This means that a device can function optimally in only one specific setting, restricting the broader applications of mass diffusion metamaterials. This underscores the critical need to explore mass diffusion metamaterials with adaptive capabilities. Transformation-invariant metamaterials present an intriguing avenue in this context.

Drawing inspiration from near-zero permittivity concepts, transformation-invariant metamaterials [57] first emerged within the realm of electromagnetics. These materials are predominantly recognized for their pronounced anisotropic properties. When applied to thermal diffusion, they manifest a pronounced anisotropic thermal conductivity. In the case of mass diffusion, the diffusion rate showcases this notable anisotropy: the rate is virtually zero in one direction while nearing infinity in another. Such distinctive characteristics ensure that transformation-invariant metamaterials maintain their robustness regardless of coordinate transformations. Consequently, they’ve attracted interest across electromagnetics, acoustics, and thermodynamics domains. Utilizing these materials to craft intelligent particle diffusion metamaterial shells holds promising potential.

Fig. 17.2

(from Ref. [57])

Schematic diagram of chameleon-like a irregular concentrator and b circular rotator. The solid blue lines denote the mass flow.

2.3.1 Transformation-Invariant Approach in Mass Diffusion

In 2023, Zhang et al. [57] introduced designs for an irregular-shaped concentrator (Fig. 17.2a) and a circular rotator (Fig. 17.2b), specifically optimized to respond to environmental shifts. For the sake of clarity, the team assumed consistent solubilities for the chemical species and overlooked the effects of chemical potentials [58]. As a result, mass diffusion was predominantly directed by Eq. (17.5). Through the lens of transformation theory, the transparency characteristics of transformation-invariant metamaterials were detailed. Furthermore, it was demonstrated by them that the transparency maintained by these transformation-invariant metamaterials persists regardless of the specific finite coordinate transformation imposed. Given an arbitrary 2D coordinate transformation in cylindrical coordinates:

$$\begin{aligned} \left\{ \begin{aligned} r' &=f(r,\theta ),\\ \theta ' &=g(r,\theta ), \end{aligned} \right. \end{aligned}$$

(17.12)

where $f(r,\theta )$ and $g(r,\theta )$ are distinct functions of radius $r$ and angle $\theta$ and the Jacobian matrix ${\textbf {J}}$ can be expressed as:

$$\begin{aligned} {\textbf {J}}=\left[ \begin{matrix} \partial _rf &{} \partial _{\theta }f/r\\ r'\partial _rg &{} r'\partial _{\theta }g/r \end{matrix} \right] . \end{aligned}$$

(17.13)

The transformed diffusivity, $\boldsymbol{D}'$, is derived from the transformation theory:

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

(17.14)

where $\det {\textbf {J}}$ denotes the determinant of ${\textbf {J}}$ (${\textbf {J}}^{\tau }$ is its transpose). The components $D_{rr}$ and $D_{\theta \theta }$ represent the radial and tangential diffusivities. The eigenvalues of $\boldsymbol{D}'$ are then:

$$\begin{aligned} \lambda _1 &=\frac{D_{rr}}{\det {\textbf {J}}}\left[ (\partial _rf)^2+(r'\partial _rg)^2\right] , \end{aligned}$$

(17.15a)

$$\begin{aligned} \lambda _2 &\approx \frac{D_{\theta \theta }}{\det {\textbf {J}}}. \end{aligned}$$

(17.15b)

Given that for transformation-invariant metamaterials $D_{rr}\approx \infty$ and $D_{\theta \theta }\approx 0$, Eq. (17.15) can be simplified to:

$$\begin{aligned} \lambda _1 &\approx \infty , \end{aligned}$$

(17.16a)

$$\begin{aligned} \lambda _2 &\approx 0. \end{aligned}$$

(17.16b)

Equation (17.16) indicates that the eigenvalues are consistent across various coordinate transformations, emphasizing the robustness of transformation-invariant metamaterials against coordinate changes.

The coordinate transformation corresponding to the standard concentrator of uniform shape is provided by

$$\begin{aligned} \left\{ \begin{aligned} r' &=\frac{r_1}{r_m}r,~~~r'<r_1F(\theta ),\\ r' &=Ar+BF(\theta ),~~~r_1F(\theta )<r'<r_2F(\theta ),\\ \theta ' &=\theta , \end{aligned} \right. \end{aligned}$$

(17.17)

where $r_m$ is a constant. The function $F(\theta )$ is defined as $F(\theta )=1.2+0.5\sin (\theta )+0.5\cos (2\theta )$, with constants $A$ and $B$ given by $A=(r_2-r_1)/(r_2-r_m)$ and $B=(r_1-r_m)r_2/(r_2-r_m)$, respectively. Regarding the chameleonlike rotator, its coordinate transformation is given by

$$\begin{aligned} \left\{ \begin{aligned} r' &=r,\\ \theta ' &=\theta +\theta _0(r-r_2)/(r_1-r_2),~~~r_1<r<r_2, \end{aligned} \right. \end{aligned}$$

(17.18)

with $\theta _0$ representing the rotation angle. This transformation can be interpreted as a rotation of a series of circles with varying angles determined by their corresponding radii. By integrating Eqs. (17.14) and (17.18), the transformed diffusivity was derived.

Fig. 17.3

(from Ref. [57])

Simulations of the transformation-invariant (a1) concentrator and (b1) rotator with background diffusivity $10D_0$. Simulations of the normal-transformation (a2) concentrator and (b2) rotator with background diffusivity $10D_0$. (a3) Comparison between reference, transformation-invariant concentrator, and normal-transforming concentrator. (b3) Comparison between reference, transformation-invariant rotator, and normal-transforming rotator. TI and NT denote devices based on transformation-invariant denotes normal-transforming schemes, respectively. The green dashed squares in (a3)(b3) are enlarged images.

2.3.2 Finite-Element Simulation

To simulate the system, specific boundary conditions and parameters are established. The left boundary receives a high-concentration source, whereas the right boundary is exposed to a low-concentration source. Both the top and bottom boundaries are defined with no-flux conditions. The background diffusion rate is designated as $D_B = 10D_0$. The subsequent transformed diffusivity can be ascertained using Eqs. (17.14), (17.17), and (17.18).

Figure 17.3a1 displays the concentration distribution along the x-direction for a translation-invariant concentrator. It becomes clear that the chameleon-like thermal concentrator functions efficiently in environments with high diffusivity. Conversely, as depicted in Fig. 17.3a2, the conventional concentrator is ineffective under such conditions. For a more granular assessment, data were extracted to facilitate a detailed quantitative analysis. The third row of Fig. 17.3a3 compares concentration data between the uniform background (serving as a reference line), the conventional concentrator, and the chameleon-like concentrator. Within this figure, the shell’s location is demarcated by a gray area. The concentration curve for the conventional concentrator deviates noticeably from the reference curve in the device’s surrounding background area. Importantly, within the device’s core area, when $D_B$=$10D_0$, the concentration gradient of the conventional concentrator is even inferior to the gradient of the reference curve. This observation underscores that the conventional concentrator not only disrupts the background but also falls short in its aggregation function. In stark contrast, the translation-invariant concentrator closely aligns its concentration curve with the reference line in the surrounding background area. Additionally, its internal concentration gradient consistently surpasses the background gradient, reinforcing theoretical predictions. In Fig. 17.3a3, the purple curve’s gradient surpasses that of the red curve, suggesting that even when the conventional concentrator operates as intended, its aggregation effect is still inferior to the chameleon-like concentrator. This phenomenon can be traced back to the inherent differences in aggregation degrees: while a conventional concentrator’s degree is influenced by $r_m/r_1$, the chameleon-like concentrator’s degree closely aligns with $r_2/r_1$. As a general observation, the aggregation degree of the conventional concentrator tends to be less pronounced compared to its chameleon-like counterpart.

Figure 17.3b1 showcases the chameleon-like rotator’s performance across varied backgrounds. The chameleon-like rotator adeptly adjusts to environmental shifts, altering the transport direction in its core without perturbing the surrounding environment. In contrast, the conventional rotator, demonstrated in Fig. 17.3b2, operates optimally only within specific background parameters. Any deviation from these conditions incapacitates the conventional concentrator. Furthermore, Fig. 17.3b3 highlights a superior concentration gradient within the core of the chameleon-like rotator compared to the background gradient, implying the dual functionality of rotation and aggregation. However, the conventional rotator is devoid of this aggregation capability. This distinction arises from the intrinsic aggregation capabilities of the transformation-invariant metamaterial shell. The rotational transformation remains non-intrusive to the shell’s native properties, echoing previous theoretical insights. Such findings accentuate the distinct attributes of the irregular chameleon-like concentrator and the annular chameleon-like rotator, both premised on the transformation-invariant metamaterial shell.

Fig. 17.4

(from Ref. [57])

a Diagrammatic figures and b simulation results of the experimental setup with chessboard structure.

2.3.3 Experimental Suggestion

Although nature seldom offers extremely anisotropic materials, mimicking such effects remains achievable using two materials with pronouncedly different diffusion rates. An implementable experimental methodology is also delineated. Taking the chameleon-like concentrator depicted in Fig. 17.4a as a representative example, testing under varied background conditions necessitates media with differing diffusion rates. Conventionally, this calls for procuring a multitude of materials, adding potential complexities to the experimental framework. As an alternative, leveraging a checkerboard design for the background medium is proposed, complemented by a layered approach for the concentrator’s construction. This design strategy requires only two materials with distinctly varied diffusion rates, with water and resin emerging as potential candidates. Figure 17.4b illustrates the concentration distribution of the translation-invariant concentrator. Despite fluctuations in the background diffusion rate, the layered concentrator maintains its functionality without noticeably impacting the background, elevating the internal concentration gradient. Such results suggest that even with a two-material-based simulation, the experimental apparatus exhibits optimistic outcomes. Nonetheless, a point of contention remains: the simulation results reflect a subdued concentration effect within the concentrator’s core, accompanied by slight background disturbances. These anomalies stem from inadequate background discretization and a rather dispersed distribution of inclusions, inducing pronounced localized discrepancies. A denser inclusion distribution is posited to refine the simulation results.

3 Non-equilibrium Diffusion Model

The precise regulation of mass transport plays a pivotal role in areas such as biochemical reactions, drug delivery, and particle separation. Certain drug components, for example, are designed to target specific cells exclusively. Yet, during the delivery phase, these components might be absorbed by non-target cells or even interact with chemical entities in the plasma, leading to diminished therapeutic efficacy. In industrial production and biochemistry, the tasks of particle separation, concentration, and detection are routine. This includes processes like the purification of hydrocarbon fuels from crude oil, the extraction of uranium from seawater, and monitoring harmful solutes in atmospheric or aquatic environments. From the standpoint of physics, the convection-diffusion equation typically underpins mass transport. Numerous methods, informed by this equation, have been suggested for its control, such as the scattering cancelation method and the transformation theory.

However, handling transient (or time-varying) mass transport presents challenges distinct from those of steady-state control. A paradigmatic example of this transient transport is the chemical wave [59], which are far from equilibrium [60], exhibiting concentration profiles characterized by spatiotemporal fluctuations. At its core, a chemical wave describes the dynamic variation in particle concentration across time and space. While the transformation theory fits well with steady-state mass transport, it falls short in transient situations. The primary reason is that the governing equation for transient mass transport doesn’t retain its structure during coordinate transformations. While research on transient heat diffusion provides some direction, the methodologies are often bound to specific geometric configurations. As a result, devising a theoretical framework to control chemical waves remains a substantial challenge.

3.1 Theoretical Foundation

To address this challenge, researchers [61] developed an optimized transformation-mass-transfer theory. This methodology enables the meticulous control over chemical waves and sheds light on their propagation properties. In the investigation of chemical processes, specific chemical components were set aside, focusing instead on concentration gradients as the predominant factors. This led to the utilization of the advection-diffusion equation to represent chemical waves:

$$\begin{aligned} \partial c/\partial t=\mathrm {\nabla }\cdot (\overleftrightarrow {D} \cdot \nabla c-vc) \end{aligned}$$

(17.19)

In this equation, c, t, $\overleftrightarrow {D}$, and v are representative of concentration, time, tensorial diffusivity, and advection velocity, respectively. Without compromising generality, a one-dimensional scenario was considered, characterized by scalar diffusivity $D$ and constant velocity $v$ along the $x$-axis. This simplifies Eq. (17.19) to:

$$\begin{aligned} \frac{\partial c}{\partial t}=D\frac{\partial ^2c}{\partial x^2}-v\frac{\partial c}{\partial x} \end{aligned}$$

(17.20)

For a chemical wave, considering the spatiotemporal variability in concentration, a plane-wave solution is appropriate:

$$\begin{aligned} c=c_ae^{i(\beta x-\omega t)}+c_r, \end{aligned}$$

(17.21)

Here, $c_a$, $\beta$, $\omega$, and $c_r$ symbolize the concentration amplitude, wave number, circular frequency, and reference concentration of the chemical wave, respectively. It is observed that with a predetermined frequency of the concentration source, the concentration amplitude of the chemical wave diminishes upon propagation. This observation leads to the interpretation of the wave number $\beta$ as a complex entity given by $\beta =k+i\alpha$, where both $k$ and $\alpha$ are real quantities. Therefore, Eq. (17.21) can be equivalently represented as $c=c_a{e^{-\alpha x}e}^{i(kx-\omega t)}+c_r$. Evidently, $\alpha$ designates the spatial dissipation of the chemical wave, while $k$ stands for the real wave number. Incorporating this into Eq. (17.20) results in:

$$\begin{aligned} \left\{ \begin{aligned} \alpha =&\frac{-2v+\xi }{4D}\\ k=&\frac{\sqrt{v^4+16D^2\omega ^2}-v^2}{16D^2\omega }\xi \end{aligned}\right. \end{aligned}$$

(17.22)

with $\xi =\sqrt{2v^2+2\sqrt{v^4+16D^2\omega ^2}}$. The consideration is limited exclusively to forward propagation, thus retaining only the positive solution. An evident increase in $\omega$ correlates with a rise in either $\alpha$ or $k$. Such a concentration profile might be conceptualized as a chemical wave. However, a comprehensive chemical wave model would be more sophisticated than the currently presented framework. Nevertheless, this basic model can shed light on the regulation of chemical waves, especially when utilizing the subsequent optimized-transformation theory crafted for transient mass transfer.

To elucidate the transformation rule, the component form of the advection-diffusion equation in curvilinear space was articulated using the corresponding coordinate $x_i$:

$$\begin{aligned} \sqrt{G}\partial _tc=\partial _i\left( \sqrt{G} D^{ij}\partial _jc-{\sqrt{G} v}_ic\right) , \end{aligned}$$

(17.23)

where $G$ symbolizes the determinant of $G_i \cdot G_j$, with $G_i$ and $G_j$ being the covariant bases of the curvilinear space. This equation was then manifested in the physical space $x_i^\prime$:

$$\begin{aligned} \sqrt{G}\partial _tc=\partial _{i^\prime }\frac{\partial x_i^\prime }{\partial x_i}\left( \sqrt{G} D^{ij}\frac{\partial x_j^\prime }{\partial x_j}\partial _{j^\prime }c-{\sqrt{G} v}_ic\right) , \end{aligned}$$

(17.24)

In this context, $\partial x_i^\prime /\partial x_i$ and $\partial x_j^\prime /\partial x_j$ are components of the Jacobian-transformation matrix $J$. The determinant of $J$ is represented as $1/\sqrt{G}$, which provides:

$$\begin{aligned} \frac{1}{\text {det}J}\partial _t c=\nabla ^{\prime } \cdot (\frac{J \overleftrightarrow {D} J^{\tau }}{\text {det}J} \nabla ^{\prime } c-\frac{Jv c}{\text {det} J}), \end{aligned}$$

(17.25)

where $J^\tau$ signifies the transpose of $J$. Intriguingly, when transformed parameters $J \overleftrightarrow {D} J^{\tau } / \text {det}J$ and $Jv / \text {det}J$ are denoted as $\overleftrightarrow {D}^{\prime \prime }$ and $v^{\prime \prime }$ respectively, a complete substitution from space transformation to material transformation remains elusive, attributed to the lack of a physical quantity preceding $\partial _t c$. Hence, the form-invariance for transient mass transfer under a coordinate transformation is confined to cases where $\text {det}\,J = 1$.

Fig. 17.5

(from Ref. [61])

a–d Schematic diagrams of cloaking, concentrating, rotating, and separating in two dimensions, respectively.

To circumvent this constraint, an approximation was adopted, which excludes the $1/\text {det} J$ term before $\partial _t c$, refining the equation to:

$$\begin{aligned} \partial _t c={\nabla }^{\prime } \cdot (J \overleftrightarrow {D}J^{\tau } \nabla ^{\prime }c-Jvc). \end{aligned}$$

(17.26)

It should be highlighted that Eq. (17.26) serves predominantly as an approximation to Eq. (17.25), given the position-dependent attribute of $\text {det} J$. Yet, subsequent simulations affirmed the feasibility of this approximation. As a result, the transformed parameters $\overleftrightarrow {D}^{\prime }$ and $v^{\prime }$ can be deduced as:

$$\begin{aligned} \left\{ \begin{aligned} \overleftrightarrow {D}^{\prime }&=J \overleftrightarrow {D} J^{\tau },\\ {v}^\prime &={Jv}. \end{aligned}\right. \end{aligned}$$

(17.27)

Therefore, it can be inferred that Eq. (17.26) maintains form invariance under coordinate transformation by approximation. The performance of Eq. (17.27) in simulations aligns with the optimized transformation-mass-transfer theory.

3.2 Model Application

Utilizing the optimized transformation mass transfer theory, four functional devices have been conceptualized as tangible illustrations for regulating chemical waves. These devices are tailored to achieve specific functions: cloaking, concentrating, rotating, and separating chemical waves. Figure 17.5 provides schematic representations of the four device models. Chemical waves are introduced from the left end of the rectangular medium, propagating along the x-direction, and flow through the functional devices without disturbing their waveforms. Maintaining the waveform of the chemical waves in the surrounding background medium is a critical feature of devices designed using transformation theory. Notably, the shielding device offers protection to entities within the core region from external material influences; the concentrator funnels chemical waves into the core, amplifying the concentration gradient; the rotator adjusts the chemical wave’s propagation direction within the core, facilitating a seamless rotation angle adjustment; the separator has different guiding effects on different particles, realizing the separation effect.

For cloaking applications, a coordinate transformation maps the virtual space, described by $(r, \theta )$, to the physical space defined as $\left( r^\prime ,\theta ^\prime \right)$. This transformation is governed by the relations $r^\prime =ar+b$ and $\theta ^\prime =\theta$. The constants are given as $a=\left( r_2-r_1\right) /r_2$ and $b=r_1$. In this context, $r_1$ and $r_2$ delineate the radii of the cloak’s inner and outer boundaries, respectively, as depicted in Fig. 17.5a. The Jacobian transformation matrix, J, is derived as:

$$\begin{aligned} J_2=\text {diag}\left[ {ar}^\prime ,\ a/\left( r^\prime -b\right) \right] \end{aligned}$$

(17.28)

For the concentrator, the coordinate transformation is characterized by $r^\prime =hr$ for $r<r_m$, $r^\prime =nr+m$ for $r_m<r<r_2$, and $\theta ^\prime =\theta$ for $0<r<r_2$. The parameters are defined as: $h=r_1/r_m$, $n=\left( r_2-r_1\right) /\left( r_2-r_m\right)$, $m=\left( r_1-r_m\right) r_2/\left( r_2-r_m\right)$, with $r_m$ being an intermediate radius between $r_1$ and $r_2$ as shown by the dashed line in Fig. 17.5b. Consequently, the Jacobian-transformation matrices for the concentrator in the domains $r^\prime <r_1$ and $r_1<r^\prime <r_2$ are given by:

$$\begin{aligned} \begin{aligned} {J}_1&=\text {diag}\left[ h,h\right] ,\\ {J}_2&=\text {diag}\left[ n,r^\prime n/\left( r^\prime -m\right) \right] . \end{aligned} \end{aligned}$$

(17.29)

It is essential to recognize that these coordinate transformations exclusively modify the radius, kee** the angle invariant. To induce a rotation in the chemical waves, a refined theory was applied. The corresponding coordinate transformations for this rotation are $r' = r$ for the domain $0 < r < r_2$, $\theta ' = \theta + \theta _0$ for $r < r_1$, and $\theta ' = \theta + g(r-r_2)$ for $r_1 < r < r_2$, as illustrated in Fig. 17.5c. Here, $g$ is defined as $\theta _0/(r-r_2)$, with $\theta _0$ denoting the rotation angle. The resultant Jacobian transformation matrices are:

$$\begin{aligned} \begin{aligned} {J}_1&=\text {diag}\left[ 1,1\right] , (r' < r_1) \\ J_2 &= \begin{bmatrix} 1&{} 0 \\ r' g&{} 1 \end{bmatrix} , (r_1 < r' < r_2). \end{aligned} \end{aligned}$$

(17.30)

Then, they introduce an innovative chemical wave separator that transforms both the radius and the angle simultaneously. As depicted in Fig. 17.5d, a combined chemical wave comprising particles A and B enters from the top, moves past the separator (represented by grids), and eventually splits into separate channels, exiting from the bottom. During this progression, particles A and B are distinctly channeled into separate regions. The underlying principles of this separation mechanism are discussed in subsequent sections.

For an in-depth analysis, the separator is divided into two distinct regions. The rectangular section on the right is referred to as area-1, while the remaining portion is designated as area-2. The dimensions of area-1 are characterized by a length of $x_0$ and a width of $y_0$. To ensure clarity and visual coherence, a transformation in Cartesian coordinates is utilized. As such, the coordinate transformation for point A in area-1 is defined by $x^\prime =\left( y/\left( 2y_0\right) +1\right) -x_0y/\left( 2y_0\right)$ and $y^\prime =y$. Similarly, for point B, the transformation is expressed as $x^\prime -x_0=\left( y/(2y_0)\ +1\right) +x_0y/\left( 2y_0\right)$ and $y^\prime =y$. The corresponding Jacobian matrices are detailed as:

$$\begin{aligned} \left\{ \begin{aligned} {J}_{1A}&=\left( \begin{matrix}\left( y^\prime +2y_0\right) /\left( 2y_0\right) &{}\left( x^\prime -x_0\right) /\left( y^\prime +2y_0\right) \\ 0&{}1\\ \end{matrix}\right) ,\\ {J}_{1B}&=\left[ \begin{matrix}\left( y^\prime +2y_0\right) /\left( 2y_0\right) &{}x^\prime /\left( y^\prime +2y_0\right) \\ 0&{}1\\ \end{matrix}\right] , \end{aligned}\right. \end{aligned}$$

(17.31)

In this context, ${J}_{1A}$ represents the Jacobian-transformation matrix for point A in area-1, while ${J}_{1B}$ denotes that for point B. With the Jacobian-transformation matrices for these four models established, the substitution of Eqs. (17.28), (17.29), (17.30), and (17.31) into Eq. (17.27) allows for the computation of the desired diffusivities and velocities.

Fig. 17.6

(from Ref. [61])

Simulation results of a, b cloaking, c, d concentrating, and e, f rotating. (a), (c), (e) Concentration profiles at 240 s. Comparison between reference and (b) cloaking, (d) concentrating, and (f) rotating at 240 s.

3.3 Finite-Element Simulation

To validate the proposed theory, simulations were conducted using COMSOL Multiphysics through finite element analysis. The boundary conditions for cloaking, concentration, and rotation are depicted in Fig. 17.6a–f. A periodic concentration source is described as $c_b = c_a \cos (\omega _0 t) + c_r$ where $c_a = 0.5$ mol m$^{-3}$, $\omega _0 = \pi /10$ s$^{-1}$, and $c_r = 1$ mol m$^{-3}$. This source was introduced on the left boundary (highlighted in red). The decay rate, as determined by Eq. (17.22), is influenced by $\omega$. Consequently, $\omega _0$ was selected as $\pi /10$ s$^{-1}$ to assure minimal dissipation. The right boundary was defined as an open boundary (marked in blue). Moreover, isolated and no-flow conditions were set for both the top and bottom boundaries. In Fig. 17.6a, the inner boundary received similar conditions. The diffusivity and advection velocity of the background were allocated values of $D_0$ and $v_0$ respectively. All parameters were deduced from Eqs. (17.27)–(17.30). The simulation results for cloaking, concentration, and rotation are featured in Fig. 17.6. Notably, in Fig. 17.6a, an obstacle is effectively cloaked, preserving concentration profiles in the background and upholding the chemical wave’s integrity. The tangential diffusivity component significantly exceeds its radial counterpart when $r' \approx r_1$, directing chemical waves around the core region. This outcome aligns with the simulation findings. In Fig. 17.6b, the dashed line signifies concentration distribution in a pure background, while the solid line represents the same in the presence of a cloak. Detailed concentration outcomes are depicted in Fig. 17.6c–d. There is a pronounced concentration gradient, as shown in Fig. 17.6d where the solid line appears denser than its dashed counterpart in the core. Figure 17.6e showcases the rotation’s impact on a chemical wave at 240 s, with the chemical wave in the core being deflected counter-clockwise by $\pi /6$, leaving the background concentration profile unaffected. For comparison, Fig. 17.6f illustrates the concentration distributions in both a pure background and one with a rotator.

Fig. 17.7

(from Ref. [61])

Simulation Analysis of Separated Chemical Waves from Particles A and B: (a1)–(a3) Wave propagation in a pure medium at 6, 12, and 18 s, respectively. (b1)–(b3) Propagation of particle-A waves with a separator in the background medium at 6, 12, and 18 s, respectively. (c1)–(c3) Propagation of particle-B waves with a separator in the background medium at 6, 12, and 18 s, respectively. (d1) Concentration difference, $\bigtriangleup c$, of Particle A [or Particle B] between (b1) and (a1) [or between (c1) and (a1)] along the x-axis at 6 s, based on data from the x-directed green dashed lines in (b1) and (a1) [or in (c1) and (a1)]. Likewise, (d2) and (d3) display $\bigtriangleup c$ at 12 and 18 s, respectively.

The separator’s model setup differs slightly from the preceding model. In the background medium, the diffusivity and convection velocities are denoted as $D_{s0}$ and $v_{s0}$ respectively. The geometric parameters incorporated include $m_1$ and $m_2$. A concentration $c_{sb} = c_{sa} \cos (\omega _{s0} t) + c_{sr}$, with parameters $c_{sa} = 0.5$ mol m$^{-3}$, $\omega _{s0} = \pi$ s$^{-1}$, and $c_{sr} = 1$ mol m$^{-3}$, was introduced at the top boundary. The bottom boundary was marked as open, with isolated and no-flow conditions determined for the other sides. From Fig. 17.7a1–c3, it is evident that the same separator produces distinct guiding effects on different particles. Chemical waves of particle A converge on both sides, while those of particle B converge in the middle, achieving the separation of the two particles. Notably, the chemical waves of particles A and B retain their original phase information post-separation, marking a significant characteristic of this separator. Additionally, as time progresses, the separated chemical waves tend to diffuse towards each other, resulting in their recombination. For a more comprehensive understanding of the separation efficiency, quantitative data comparisons were made. As depicted in Fig. 17.7d1–d3, the concentration difference $\Delta c$ between post-separation and pre-separation chemical waves of particles at 6, 12, and 18 s is displayed. Blue and red dot arrays represent the differences for particles A and B, respectively, with $\Delta c= c_A - c_{reference}$ and $\Delta c=c_B - c_{reference}$. Figure 17.7d1 reveals that the concentration of both particle types drops sharply at the interface of their respective aggregation regions, and is nearly zero in the opponent’s aggregation region, attesting to the high purity of the freshly separated chemical waves and underscoring the superior performance of this separator. This validates the efficacy of the optimized transformation mass transfer theory when both radius and angle are varied simultaneously.

Regarding the experimental device, it was suggested to uniformly divide the separator into neatly arranged small segments. Each segment has a consistent diffusion rate parameter, which equals the value of the original separator’s diffusion rate at the center point of that segment. In this manner, the originally spatially continuous diffusion rate is replaced by a stepwise-changing diffusion rate. Clearly, this segmented approach serves as an approximation. Its accuracy depends on the granularity of the segmentation; the smaller the segments, the closer the outcome approaches the ideal scenario.

4 Conclusion and Outlook

This chapter provides a comprehensive examination of the evolution of metamaterials in particle dynamics, drawing inspiration from recent advancements in electromagnetism and thermotics. The foundational particle diffusion model, governed by the convection-diffusion equation, is elucidated. Owing to a restricted array of transformable parameters, the transformation theory does not fit perfectly with the transient governing equation. However, manifestations such as mass diffusion cloaking, concentrating, and rotating are feasible through the low-diffusivity approximation. Two significant challenges are evident. Firstly, the applicability of current mass-diffusion metamaterials is limited by their flexibility in dynamic environments. Addressing this, transformation invariant metamaterials have been utilized to craft adaptable chameleon-like metashells. These innovative metashells can autonomously adjust their effective parameters in response to environmental fluctuations without any energy consumption. The second challenge pertains to the typically anisotropic and singular parameters arising from the transformation, making experimental implementations intricate. In response, a myriad of research efforts have explored the scattering cancellation theory in particle diffusion systems, leading to the development of a range of mass diffusion meta-devices, encompassing cloaks, rotators, and separators. The article then delves into the chemical-wave model, characterized by its non-equilibrium state and concentration profiles that display spatiotemporal alterations. The propagation traits of this model can be manipulated via the transformation transient-mass transfer theory, leading to the proposition of four functional devices optimized for chemical wave management.

While there have been significant strides in the realm of particle diffusion metamaterials, several paramount challenges and queries remain unaddressed. Transformation thermotics and scattering cancellation are conventionally considered distinct methodologies for devising diffusion metamaterials. Yet, parameters derived from the former display anisotropy and inhomogeneity, whereas the latter grapples with transient particle diffusion and complex geometric designs. Of late, innovative solutions have surfaced to tackle these challenges. The introduction of the diffusive pseudo-conformal map** [62] stands out, bridging the gap between diffusion and waves while ensuring optimal interface coherence. Noteworthy advancements in numerical techniques, including topology optimization algorithms [36, 37] and machine-learning strategies [63], provide sophisticated methodologies for crafting metamaterials apt for a diverse array of applications.