Abstract
Diffusionics, distinct from traditional physical laws, focuses on designing material parameters to actively control diffusion fields. The introduction of transformation theory provides a novel method to achieve active control of diffusion transport, leading to the design of devices with unique functions such as cloaks, concentrators, and rotators. However, materials corresponding to the parameters designed by transformation theory are challenging to find in nature. Therefore, the spatial arrangement of one or multiple materials to effectively achieve the desired parameters has become an alternative approach, indirectly spurring the development of metamaterials. This article reviews the fundamental theories and theoretical framework in diffusion science. We first introduce the basic concept of transformation theory, followed by a review of alternative theories such as effective medium theory and scattering cancellation theory. To study topological phenomena in diffusion systems and space-time modulated systems, the foundations of quantum mechanics, namely matrix mechanics and wave mechanics, are employed. Lastly, the article summarizes some challenges in diffusion science theory, which may be addressed by other methods in the future, such as transformation field methods and machine learning approaches.
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1.1 Opening Remarks
The development of physics has robustly characterized the microscopic and macroscopic mechanisms of diffusion systems. The first law of thermodynamics asserts the conservation of energy, while Fourier’s law articulates the connection between temperature gradients and heat flux. These laws are traditionally oriented towards describing thermal phenomena in natural contexts and do not inherently provide for the deliberate manipulation of artificial systems. By synthesizing the first law of thermodynamics with Fourier’s law, one can derive the governing equation for the temperature field, which is determined by its inherent structure and boundary conditions. On the one hand, boundary conditions typically stem from the ambient environment and are challenging to alter. On the other hand, the form of the heat conduction equation is intimately linked with variables such as density, specific heat capacity, and thermal conductivity. The task of actively modulating the temperature field, therefore, hinges on the strategic design of these parameters. Similar principles apply to other diffusion systems like the electric potential field in charge diffusion systems [1,2,3] and the concentration field in particle diffusion systems [4, 5], where field equations can be controlled exerted tailoring pertinent parameters. This encapsulates the core concept of diffusionics [6], which aims to steer diffusion fields through the deliberate adjustment of these parameters.
Altering the distributions of physical fields via modification of material properties is a familiar concept, yet deducing the spatial arrangement of such material parameters from a specified physical field configuration is a mathematically intricate inverse problem. Typically addressed through numerical methods, these problems demand considerable computational resources. A breakthrough came in 2006 when Pendry et al. [7] put forth a transformation theory that enables precise construction of material parameters to match desired field distributions. Drawing inspiration from Einstein’s general relativity, they formulated a mapping that transform a virtual space into real Cartesian space, thereby computing the necessary distribution of dielectric constants and magnetic permeability for the material. This transformation theory has been expanded to encompass various physical fields such as acoustics [8], thermotics [9,10,11,12,13,14], particle dynamics [15], and plasma physics [16]. The theory has provided a roadmap for material parameter manipulation to control diffusion fields at will. The resulting parameters, including diffusion coefficients, however, are often anisotropic, inhomogeneous, and sometimes singular, depending on the nature of the coordinate transformation. Recent advances have employed linear transformations [17] and pseudo-conformal transformations [18] to eliminate the inhomogeneity and anisotropy of the material parameters, significantly simplifying the challenge of experimental realization.
Despite these advancements, identifying natural materials that satisfy the stipulations of transformation theory remains a formidable challenge. The solution has been the development of metamaterials [19,20,21], engineered structures composed of multiple densely packed segments that collectively behave as an effective medium [22]. The design of metamaterials is anchored in the effective medium theory, with the Maxwell-Garnett formula, Bruggeman formula, and series-parallel formula serving as its foundational theoretical apparatus. Beyond the realm of metamaterial design, effective medium theory also facilitates the calculation of nonlinear effective thermal conductivities in random [23] and periodic particle systems [24], alongside the observation of enhanced effective nonlinear coefficients. The idea of scattering cancellation is also implicit in effective medium theory. For example, a metamaterial, through the spatial arrangement of several natural materials, placed in a background with the same effective thermal conductivity. This means that the scattering of these materials on the background field cancels each other out, yielding the distribution of the background field undisturbed. Recently, scattering cancellation theory [25,26,27,28] has emerged as an alternative to transformation theory, guiding the fabrication of thermal metamaterials with desired characteristics. Moreover, introducing effective convection through rotating disks [29,30,31] and direct heat source contact [32, 33] can also perturb the background temperature field, offering adjustable solutions adaptable to complex environments.
The vigorous expansion of topological physics [34, 35] in the last two decades has seen a multitude of topological phases of matter both theoretically conjectured and experimentally verified. Given that diffusion systems are inherently non-Hermitian, their governing equations can be likened to the Schrödinger equation, making the exploration of non-Hermitian physics and its topological characteristics [36] within macroscopic diffusion systems a tantalizing prospect. Analogies with the Schrödinger equation reveal that its potential energy term corresponds to the diffusion term of the diffusion equation, allowing the Bloch series expansion method to be aptly applied to spatiotemporally modulated diffusion systems [37].
This chapter explores diffusion systems, establishing the foundational theories and frameworks for manipulating diffusion fields. Section 1.2 sets out the theoretical underpinnings of transformation theory, exemplified through the heat conduction equation, proceeds with the implementation of various mappings, and then broadens the scope of transformation theory to encompass additional diffusion fields. Section 1.3 delves into the core equations of effective medium theory that are essential for actualizing the parameters dictated by transformation theory. Section 1.4 delineates the scattering cancellation theory, which bifurcates into passive and active strategies. The passive approach operates without the need for external energy inputs, distinguishing it from the active strategy which necessitates such inputs. Section 1.5 commences with an exposition on the fundamental approach to examining diffusion topological phenomena. This is achieved by discretizing the diffusion equation to draw parallels with the Schrödinger equation, subsequently facilitating the computation of energy bands and topological invariants. Following this, the section introduces the Bloch series expansion method, a valuable tool for addressing systems characterized by periodically modulated material parameters. The chapter concludes with Sect. 1.6, which synthesizes the content discussed throughout and offers insights into the prospective advancements in the field of diffusionics.
1.2 Transformation Theory
1.2.1 Foundation Framework
It is widely recognized that the unique determination of a physical field hinges on its governing equations coupled with boundary conditions. For illustrative purposes, let us consider the transient heat conduction process. The predominant equation governing this process is presented as:Â [2]
where \(\rho \) represents the mass density, c the heat capacity, \(\kappa \) the thermal conductivity, and Q the heat power density. This equation has different forms in different coordinate systems, and we’re going to expand it in any coordinates \(\{x_{u}\}\) (u can take 1, 2, and 3), consisting of a set of contravariant basis \(\left\{ \textbf{g}^u \right\} \). In general, the thermal conductivity is a tensor \(\kappa ^{uv}\), rendering the conductive term of the Eq. (1.1) as follows:
Here, g represents the determinant of the metric \(g_{ij}\) and the Christoffel symbol is defined as \(\Gamma _{k u}^k=\textbf{g}_k \cdot \partial \textbf{g}^u/\partial x_{k}\). So far, we have derived the general form of heat conduction equation in any coordinates,
Consideration is then given to a rectangular region with isotropic thermal conductivity \(\kappa \) within the Cartesian coordinate framework \(\{x_{u^{\prime }}\}\). The left and right boundaries are maintained at constant high and low temperatures, respectively, while the remaining boundaries are insulated. Illustrated in Fig. 1.1a are the uniform temperature gradients and isotherms within the domain. To fulfill particular requirements, such as those of a thermal guide capable of conducting heat flow uniformly, a coordinate transformation is performed:
Figure 1.1b depicts the transformed domain, showcasing isotherms (red line) and streamlines (yellow line) which run parallel and perpendicular to the radial direction, respectively. The transformation from Fig. 1.1a, b involves only a coordinate change, allowing the constitutive parameters in curvilinear coordinates to be expressed as:
By substituting Eq. (1.5) into Eq. (1.3), the component form of the heat conduction equation in the curvilinear coordinate system \(\{x_{u^{*}}\}\) is obtained and expressed as:
where \(g_{i^{*}j^{*}}=\frac{\partial x_{i^{\prime }}}{\partial x_{i^{*}}} g_{i^{\prime }j^{\prime }} \frac{\partial x^{j^{\prime }}}{\partial x^{j^{*}}}=(JJ^{\tau })^{-1}\), thus resulting in \(\sqrt{g^{*}}=\text {det}^{-1}(J)\). Thus, Eq. (1.6) reduces to
It is pertinent to note that these two coordinates describe the same temperature distribution (\(T(x_{1^{\prime }},x_{2^{\prime }},x_{3^{\prime }})=T(x_{1^{*}},x_{{2}^{*}},x_{{3}^{*}})\)), despite the isotherm shape appearing distinct in the curvilinear coordinates compared to Cartesian coordinates. In other words, while the shape plotted in the curvilinear coordinates is virtual, an equivalent temperature distribution is attainable in the physical space. This equivalence is founded upon the domain equation and boundary conditions that govern the temperature field distribution. To materialize such a concept, one need to consider a curved tube replicating the shape illustrated in Fig. 1.1b within the Cartesian coordinate system \(\{x_{\tilde{u}}\}\). Setting constant temperatures at both ends of this tube, with the remaining two walls maintained as adiabatic, keeping the same boundary conditions. Moreover, to align with the form of Eq. (1.7), there is a necessity to choose appropriate parameters that yield an analogous heat conduction equation within the Cartesian coordinate system, expressed as:
Comparing Eqs. (1.7) and (1.8), one derives the transformation relations for the constitutive parameters as \(\tilde{\kappa }^{\tilde{u} \tilde{v}}=J^{u^{*}}{}_{u^{\prime }}\kappa ^{u^{\prime } v^{\prime }}J^{v^{*}}{}_{v^{\prime }}/\text {det}(J)\), \(\tilde{\rho }\tilde{c}=\rho c/\text {det}(J)\), and \(\tilde{Q}=Q/\text {det}(J)\). These can be succinctly represented in matrix form as prescribed in [2]:
This set of relationships ensures the preservation of the heat conduction equation’s form post-coordinate transformation and affirms that the temperature profile in the virtual coordinate system \( T(x_{1^*}, x_{2^*}, x_{3^*}) \) can be manifested in the newly established Cartesian system \( \tilde{T}(x_{\tilde{1}}, x_{\tilde{2}}, x_{\tilde{3}}) \).
(from Ref. [2], licensed under CC-BY 4.0)
Schematic diagram of the coordinate transformation process in cylinder structure. a, d, c, f Cartesian coordinates. b, e Curvilinear coordinates. The red (yellow) lines denote isotherms (stream). From a, d to b, e: only coordinate transformation. From a, d to c, f: only constitutive parameters’ transformation. The isotherm shape in c, f is the same as that in b, e, indicating that the effect of parameters’ transformation is equivalent to coordinate transformation.
1.2.2 Mapping Application
1.2.2.1 General Mapping
The manipulation of heat flux typically revolves around two key aspects: density and direction. Thermal concentrators, for example, are designed to enhance the heat flux density within a specified region, while thermal rotators are engineered to alter the direction of heat flow. This study integrates both functions within a singular device. As depicted in Fig. 1.1d, the Cartesian coordinate space is partitioned into four regions, each with uniform thermal conductivity \(\kappa \): Region I\(_1^{\prime }\) for \(r^{\prime } < R_1\); Region I\(_2^{\prime }\) for \(R_1 < r^{\prime } < R_2\); Region II\(^{\prime }\) for \(R_2 < r^{\prime } < R_3\); and Region III\(^{\prime }\) for \(r^{\prime } > R_3\). Subsequently, specific coordinate transformations are applied to these regions as follows:
Here, \(R_2\) spans from 0 to \(R_3\), resulting in three distinct devices: a rotating imperfect (or perfect) cloak for \(R_2 < R_1\) (\(R_2 = 0\)); a rotator for \(R_2 = R_1\); and a rotating concentrator for \(R_1 < R_2 < R_3\). Such transformations enable the compression and rotation of the temperature distribution in Regions I\(_1^{\prime }\) and I\(_2^{\prime }\) without perturbing the physical fields within the background (Region III\(^{\prime }\)). As shown in Fig. 1.1e, the isotherm labeled \(a^{\prime }-b^{\prime }-c^{\prime }-d^{\prime }-e^{\prime }-f^{\prime }\) transitions into the curvilinear shape \(a-b-c-d-e-f\) following the coordinate transformation. The Jacobian transformation matrix corresponding to Eqs. (1.10)–(1.12) is computed by
By inserting Eq. (1.13) into Eq. (1.9), the spatial distribution of the material parameters can be ascertained (Fig. 1.1f),
The transformation matrix is presented as follow:
where
1.2.2.2 Linear Mapping
This robust theoretical framework offers a roadmap for the precise manipulation of thermal fields. The trade-off, however, lies in the need for materials with anisotropic, inhomogeneous, and potentially singular thermal conductivities, a requirement that stems from the non-conformal and heterogeneous nature of the coordinate transformation employed. Addressing this challenge, Xu et al. [38] introduced an innovative approach to construct polygonal thermal harvesting devices—concentrators with homogeneous and nonsingular parameters—using a linear mapping function. The method for fabricating these polygonal concentrators is illustrated in Fig. 1 of Ref. [38]. To avoid disturbances in the background temperature distribution, one edge of the outer polygon is transformed to be congruent with the background. The transformation for the inner polygon involves a two-step process: firstly, rotation, and secondly, compression. In the initial step, the inner polygon is rotated around the central point \( O(0,0) \) relative to the outer polygon by an angle \( \pi /N \), where \( N \) is the number of polygon sides (\( N \ge 3 \)). In the subsequent step, the rotated inner polygon is compressed into a smaller physical space, and the intermediate area is expanded to recapture the heat flux. The intermediate region is partitioned into two distinct types of triangular elements by connecting the vertices of both polygons in an alternating sequence. Triangular elements sharing two vertices with the outer polygon are classified as type I, whereas those with only one vertex on the outer polygon are classified as type II. To illustrate this transformation succinctly, consider the regions \( A_0B_0A_{in} \) to \( A_0B_0A_0^{\prime \prime } \) and \( A_0A_{in}E_{in} \) to \( A_0A_0^{\prime \prime }E_0^{\prime \prime } \) as exemplars. Through these two transformation steps, the coordinates of the vertices for both the inner and outer polygons can be expressed accordingly:
with \( n \) denoting the sequential number of each adjacent element of the same type, respecting their geometric attributes. Given the linear correlation between the original and transformed elements, the intended harvesting region can be achieved by mapping two corresponding contiguous curves. Therefore, the transformation process for Type I is described as:
Utilizing Eq. (1.18), the Jacobian matrix pertinent to the polygonal coordinate transformation is derived as:
Taking Eqs. (1.9) and (1.19) together, the conductivity components of Type I can be written as follows:
The thermal conductivity for Type II can be calculated in the same way. These constant conductivity tensors can be obtained by alternating the arrangement of two layered isotropic thermal materials. Utilizing effective medium theory, the thermal conductivity components aligned with the principal axes in both parallel and perpendicular orientations were determined and an arbitrarily shaped polygonal thermal concentrator were realized [38]. Notably, the thermal field external to the heat harvesting region remains unperturbed, while a substantial amplification of the temperature gradient is observed within the harvesting zone. Moreover, additional research efforts [17, 39, 40] have employed a similar methodology for the creation of polygonal thermal cloaks, optical illusions, and acoustic rotators utilizing exclusively homogeneous materials.
1.2.2.3 Pseudo-conformal Mapping
The anisotropy of constitute parameters derived from transformation theory poses a significant challenge for experimental fabrication. This challenge was addressed in 2006 that Leonhardt et al. [41] proposed conformal optical mapping and deduced the isotropic refractive index transformation relation. An optical cloak requiring only isotropic materials was realized by using Zhuokousky mapping. Despite its utility, this method has yet to find application in diffusion systems, where the apparent diffusive nature and the unavoidable interface mismatch hinder its development. In a groundbreaking approach, Dai et al. [18] introduced the concept of diffusion pseudo-conformal mapping, as seen in [Fig. 1.2]. This concept reconciles the disparities between diffusion and wave propagation, achieving seamless interface matching. Starting with classical Zhukovsky mapping \( z_1=z+1/z \) and its inverse, where \( z_1=x_1+iy_1 \) and \( z=x+iy \) represent the auxiliary and physical complex planes, they ensured that the Cauchy-Riemann conditions were satisfied:
Substituting Zhukowski mapping and Eq. (1.21) into Eq. (1.9), the inherent parameter of thermal cloak is derived,
This mapping compressed an half-ellipse \(\textbf{E}^+{=}\left\{ z_1: 0 \leqslant x_1^2 / a^2{+}y_1^2 / b^2 \leqslant 1, y_1 \geqslant 0\right\} \) (Fig. 1.2b) into a half ring \(\textbf{A}^{+}=\left\{ z: R_1 \leqslant |z| \leqslant R_2, y\geqslant 0\right. \) (Fig. 1.2c), where \(R_1=1\) m, \(a=(R_2^2+R_1^2)/R_2\), and \(b=(R_2^2-R_1^2)/R_2\). Obviously, the outer boundary is not matched before and after this mapping unless the outer boundary is large enough \(R_2\gg R_1\). To address this issue, a further transformation is introduced:
This mapping transform the ellipse into a circular \(\textbf{D}^+=\left\{ z_0: 0 \leqslant z_0 \leqslant R_2, y_1 \geqslant 0\right\} \), whose outer boundary is the same as \(\textbf{D}^+\). Taking Eqs. (1.9) and (1.23) together, another set of constitutive parameter transformations is derived,
When a temperature gradient is applied along the \( x_0 \)-axis, only the thermal conductivity in the \( x_1 \)-axis is relevant. Combing Eq. (1.22) with Eq. (1.24), the conductivity for a bilayer cloak in the \( x \)-direction is thus given by:
Conversely, when the gradient is along the \( y_0 \)-axis, the conductivity for a zero index cloak in the \( y \)-direction is:
The numerical simulations depicted in Fig. 1.2d and e validate the effectiveness of both cloaking strategies. Additionally, an orthogonal relationship between the isotherms and the streamline patterns is observed.
(from Ref. [18], licensed under CC-BY 4.0)
Carpet cloaks. a–c show the geometric transformation to construct such a cloak. a, b, and c are the virtual space, auxiliary space and the physical space, respectively. d is the computed temperature profile of a normal bilayer cloak with an insulating inner layer. e is the computed temperature profile of a zero-index cloak with an constant-temperature inner layer (realized by an external source). Parameters: \(R_2=2R_1=2\) m, \(a=2.5\) m, and \(b=1.5\) m.
1.2.3 Extension to Other Diffusion Fields
1.2.3.1 Concentration Field
The heat conduction equation serves as an exemplar for demonstrating the invariance of physical laws under coordinate transformations. This invariance is equally significant when considering mass transport in systems critical for biochemical reactions, drug delivery, and particle separation. From a physics standpoint, the essence of mass transport is encapsulated by the convection-diffusion equation:
In this equation, c, t, \(\overleftrightarrow {D}\), and \(\vec {v}\) are representative of concentration, time, tensorial diffusivity, and advection velocity, respectively. According to the traditional transformation theory, the component forms of the diffusion-convection equations before \(\{x_{i}\}\) and after \(\{x_{i^{\prime }}\}\) the transformation are derived,
with the terms \(\partial x_i^\prime /\partial x_i\) and \(\partial x_j^\prime /\partial x_j\) representing components of the Jacobian transformation matrix \(J\). Equation (1.28) shows that the time-dependent term is preceded by an extra metric caused by a spatial transformation, which in turn cannot be absorbed by the physical quantity of the medium material. As a consequence, the transient convective-diffusion equation does not satisfy the condition for an invariant transformation form, revealing a discrepancy with traditional transformation theory, particularly for transient processes. Nevertheless, if the determinant of \(J\) is 1, the theory retains its rigor. To address the limitations of transformation theory concerning transient convective-diffusion, an approximate method is suggested, as demonstrated by Zhang et al. [42]. By multiplying the determinant of \(J\) to both sides of the relevant equation and thus negating the additional metric, the following parameter transformation relationship is derived:
where \(\tilde{D}\) and \(\tilde{v}\) are parameters in new Cartesian coordinates \(\{x_{\tilde{u}}\}\). The accuracy of this approximation is contingent upon the Jacobian matrix, which in turn depends on the specific nature of the spatial transformation. The magnitude of error introduced by this method is thus related to the system’s physical parameters and the concrete form of spatial transformation. To minimize this error, the optimization theory stipulates that both the diffusion rate and flow rate should be relatively low.
1.2.3.2 Thermoelectric Coupling Field
A steady-state thermoelectric transport process adheres to the governing equations as cited in the literature [2, 43]:
Here, \(T\) and \(\mu \) denote the spatially dependent temperature and electrical potential, respectively. The electrical conductivity \(\boldsymbol{\sigma }\), thermal conductivity \(\boldsymbol{\kappa }\), and Seebeck coefficient \(\boldsymbol{S}\) (with \(\boldsymbol{S^{\tau }}\) as its transpose) are second-order tensors. These equations are subsequently expressed in component form within an arbitrary coordinate system \(\{ x_{1},x_{2},x_{3} \}\), denoted as:
where indices \(u\), \(l\), \(v\), \(k\), \(m\), \(n\) assume the values 1, 2, 3, and \(g\) represents the determinant of the metric tensor \(g_{ij}\). Assuming the transformation from the Cartesian coordinate system \(\{x_{1^{\prime }},x_{2^{\prime }},x_{3^{\prime }}\}\) to this arbitrary system, one can invoke the coordinate transformation relation,
to rewrite Eq. (1.31) as:
In the context, \(\partial x_i^\prime /\partial x_i\) and \(\partial x_j^\prime /\partial x_j\) are components of the Jacobian-transformation matrix \(A\), and \(\delta _{u^{\prime }v^{\prime }}\) and \(\delta ^{u^{\prime }v^{\prime }}\) is equal to 1 (0) only if \(u^{\prime }=v^{\prime }\) (otherwise). Furthermore, the component form of thermoelectric equations in Cartesian coordinates are expressed as:
By comparing Eq. (1.33) with Eq. (1.34), the transformation relations of constitutive parameters in a new Cartesian coordinate \(\{x_{\tilde{u}}\}\) can be ascertained, denoted as:
The discussions above establish the invariance of the diffusion-convection and thermoelectric coupling equations under coordinate transformation. Equations (1.29) and (1.35) provide the foundational guidance for designing an array of diffusion metamaterials.
1.3 Effective Medium Theory
Linear mapping and conformal mapping can mitigate the singularity of parameters, yet their direct application using natural materials is impractical; thus, the necessity for an effective medium theory arises. This theory enables the realization of complex parameters through the spatial arrangement of two isotropic materials, most commonly in a layered alternating structure.
1.3.1 Classical Effective Medium Approximation Theories
1.3.1.1 Maxwell–Garnett Theory
Consider a two-component composite (refer to Fig. 1.3a), wherein numerous particles characterized by thermal conductivity \(\kappa _p\) and area fraction \(f_p\) for the 2D scenario (or volume fraction for 3D) are randomly dispersed within a matrix with thermal conductivity \(\kappa _m\). An external temperature gradient \(\nabla T_0\) is applied along the \(x_i\)-axis. Here, the subscripts \(p\) and \(m\) signify the particle and matrix, respectively. By invoking Fourier’s law, the thermal conductivity of the composite is calculated as:
where \(\langle J\rangle \) and \(\langle \nabla T \rangle \) denote the average heat flux and temperature gradient across the resultant composite particle structure. The average heat flux within the particle and matrix phases is represented as:
Resolution of a fundamental equation in thermal dynamics yields [45]
where \(L_{pi}\) denotes the shape factor along \(x_i\)-axis, quantifying the flattening of an ellipse. In a two-dimensional setting, this is expressed as \(L_{p1}=r_{p2}/\left( r_{p1}+r_{p2}\right) \) and \(L_{p2}=r_{p1}/\left( r_{p1}+r_{p2}\right) \), with \(r_{pi}\) as the half-axis length of elliptical particles corresponding to \(x_i\)-direction. The deviation of the shape factor from 0.5 correlates with the degree of ellipse flattening. In three dimensions, the shape factors are defined through:
Adapted from Ref. [44]
a Schematic diagram illustrating a randomly distributed mixture of particles \(\kappa _p\) within a matrix \(\kappa _m\). b Schematic representation of particles A and B dispersed within the matrix material. c Schematic of the layered structure showing an alternating, interleaved configuration of layers A and B.
with definitions of \(g\left( \rho _p\right) =\prod \limits _ir_{pi}\). Substituting Eqs. (1.37) and (1.38) into Eq. (1.36), the expression for the Maxwell–Garnett theory is obtained
For circular and spherical particles, the shape factors are 1/2 and 1/3, respectively. Consequently, the effective thermal conductivities for mixtures containing these particles are:
Since the weight \(f_p \varepsilon _{pi}\) of the particles is different from that of the matrix \(1-f_p\), Eq. (1.40) is suitable for calculating the effective thermal conductivity of asymmetric structures.
1.3.1.2 Bruggeman Theory
The Bruggeman theory provides an approach for evaluating the effective thermal conductivity of symmetric composite structures. Consider a medium with particles A and B, possessing thermal conductivities \(\kappa _A\) and \(\kappa _B\), and occupying area fractions \(f_A\) and \(f_B\), respectively. These particles are randomly distributed within a parent medium with thermal conductivity \(\kappa _m\), subject to an external temperature gradient \(\nabla T_0\) aligned along the x-axis as depicted in (Fig. 1.3b). The average heat flux and temperature gradient in this system adhere to the following relations:
where \(L_A\) and \(L_B\) denote the shape factors for particles A and B, respectively. Based on Eqs. (1.36) and (1.42), the thermal conductivity for the composite can be expressed as:
When considering particles A and B as a single entity, Eq. (1.43) simplifies to:
where \(f_{AB} = f_A + f_B\) represents the combined area fraction, and \(\kappa _{AB}\) is the effective conductivity of the two particle types. Combining Eqs. (1.43) and (1.44), the following relation is obtained:
To characterize the interaction between particles, it is posited that \(\kappa _e = \kappa _m\), which means that the effects of particles A and B on the matrix cancel each other out. This leads to the derivation of Bruggeman’s equation:
As Eq. (1.46) suggests, the contributions of both particle types to the overall thermal properties are balanced. Furthermore, extending Bruggeman’s theory to accommodate multi-component composites is straightforward, just utilizing the same methodology delineated herein.
1.3.1.3 Series and Parallel Formulas of Thermal Conductivity
The analysis of anisotropic effects in composite materials often employs a layered structure model. As illustrated in Fig. 1.3c, materials A and B are adjoined to form a stratified configuration, with thermal conductivities \(\kappa _A\) and \(\kappa _B\), respectively. Within this model, heat transfer along the x-direction is conceptualized as a serial coupling of A and B. In contrast, the y-direction heat transfer is analogous to a parallel coupling of the two materials. Initially, constant temperature boundary conditions \(T_L\) and \(T_R\) are imposed on the system’s left and right interfaces, respectively, while the remaining boundaries are insulated. Fourier’s law is then employed to define the x-direction thermal conductivity as follows:
where \(f_A\) and \(f_B\) are the area fraction of materials A and B. According to the normal direction boundary condition, \(\kappa _A \langle \nabla T_A\rangle =\kappa _B \langle \nabla T_B\rangle \), and Eq. (1.47) we obtain the series thermal conductivity,
Subsequently, constant temperature conditions \(T_t\) and \(T_b\) are applied to the top and bottom surfaces of the structure, with adiabatic conditions on the lateral edges. Fourier’s law facilitates the parallel formulas of thermal conductivity:
1.3.2 Model Application
1.3.2.1 Nonlinear Enhancement in Random Particle Composites
In the context of two-dimensional composites with circular inclusions, the effective thermal conductivity can be characterized using the Maxwell-Garnett (M &G) formula and Bruggeman’s formula as delineated by the following relations:
where \(f_i\) and \(f_h\) are area fractions of embedded particles or matrix material, and \(\kappa _i\) and \(\kappa _h\) denote corresponding thermal conductivity. For composite materials comprising nonlinear or temperature-dependent heat conduction constituents, Dai et al. [44] propose the following temperature dependence for their thermal conductivities:
where \(\kappa _{j0}\) is the temperature-independent linear part of the \(\kappa _j\) of the total thermal conductivity, \(\chi _j\) is the nonlinear coefficient, \(T_\text {rt}\) is some reference temperature, and \(\alpha \) is the power of the temperature-dependent dependence of the nonlinear thermal conductivity, which can be formally taken by any real number. In mathematical form, we can substitute Eq. (1.51) into Eq. (1.50) to obtain the temperature-dependent equivalent thermal conductivity \(\kappa _e(\kappa _i(T),\kappa _h(T),f_i)\). In scenarios where material nonlinearity is weak, the study focuses on two distinct cases: (I) The embedded particles are weakly nonlinear materials, whose thermal conductivity is denoted by \(\kappa _i=\kappa _{i0}+\chi _i(T+T_{\text {rt}})^\alpha \), and the host is a linear material with a thermal conductivity denoted by \(\kappa _h=\kappa _{h0}\); (II) The embedded particles are linear materials and their thermal conductivity is expressed as \(\kappa _i=\kappa _{i0}\); the host material is a nonlinear material and its thermal conductivity is expressed as \(\kappa _h=\kappa _{h0}+\chi _h(T+T_{\text {rt}})^\alpha \). The effective thermal conductivity \( \kappa _e \) can hence be approximated by:
Herein, the coefficient ratio \( c_j=\chi _e/\chi _j \) signifies the relative increase (or decrease) in nonlinearity of the system’s heat conduction. For convenience, subsequent formulations assume \( \alpha =1 \) and disregard the reference temperature \( T_{\textrm{rt}} \) to demonstrate that these parameters do not influence the theoretical calculation of the effective nonlinear coefficient or the proportionality factor for nonlinear enhancement via the effective medium theory. Under this framework, the nonlinear modulation coefficient \( c \) for both asymmetric and symmetric cases of I and II can be computed as:
The interest lies in instances where \( c > 1 \), indicating an amplified equivalent nonlinear coefficient for the composite. It is easy to see that Eq. (1.53) is only related to \(f_i\) and \(\kappa _{i.}/\kappa _{h0}\). In Fig. 1.4, the proportional coefficient c changes with \(f_i\) for the ratios of \(\kappa _{i0}/\kappa _{h0}\) of the above four expressions.
Adapted from Ref. [44]
The enhancement of the equivalent nonlinear coefficient is calculated according to the theory of effective medium. a M &G-Case I. b M &G-Case II. c Bruggeman-Case I. d Bruggeman-Case II.
According to Fig. 1.4, for the M &G model, a prerequisite for nonlinear enhancement is the embedding of linear particles within a nonlinear matrix, provided the ratio \( \kappa _{i0}/\kappa _{h0} \) exceeds unity, and the particle area fraction remains relatively low. It is critical to note that this constitutes a necessary condition. Analysis of additional data curves for various \( \kappa _{i0}/\kappa _{h0} \) ratios suggests a threshold value for nonlinear enhancement in the vicinity of 2.5. Conversely, Bruggeman’s theory also forecasts the possibility of nonlinear enhancement, contingent upon the linear component of the nonlinear material’s thermal conductivity being less than that of the linear material and a sufficiently large area fraction of the nonlinear constituent.
1.3.2.2 Nonlinear Enhancement in Periodic Particle Composites
In addition to their disordered counterparts, periodic composite structures play a pivotal role in the architecture of artificial crystals and have various applications in linear heat conduction, such as in thermal transparency devices [46], Janus thermal illusions [47], and phantom thermal diodes. Previous studies have indicated that effective medium theory correlates well with the linear thermal conductivity in composites of a periodic structure when the area fraction of the embedded particles is small. However, notable discrepancies arise as this fraction increases. Dai et al. [24] have employed the Rayleigh method, a first-principles approach, to calculate the equivalent thermal conductivity of periodic nonlinear thermal conductivity composites.
Adapted from Ref. [24]
Schematic diagram illustrating two classes of periodic composites: a nonlinear inclusions are periodically embedded in a linear (temperature-independent) host; b linear inclusions are periodically embedded in a nonlinear host. The nonlinear modulation coefficients of Cases c I and d II predicted by Rayleigh method.
They extend the analysis to two-dimensional composites with circular particles embedded in a matrix in a quadrangular lattice pattern. Corresponding to the examination of disordered structures, the two configurations depicted in Fig. 1.5 are considered: configuration I involves nonlinear particles embedded in a linear host material as shown in Fig. 1.5a, whereas configuration II involves linear particles in a nonlinear host material as illustrated in Fig. 1.5b. The derived expression for the linear effective thermal conductivity,
where \(\beta _1=3.31248\). This result is consistent with the calculation of the linear equivalent conductivity [48], confirming that a constant temperature differential does not influence the outcome of the equivalent thermal conductivity in the linear case. By integrating Eq. (1.51) into Eq. (1.54) and utilizing Taylor series expansion, the nonlinear modulation coefficient c is determined as:
To elucidate the conditions under which nonlinear enhancement occurs, Fig. 1.5c and d plot the variation of \(c=\chi _e/\chi _j\) (\(j=i\) or h) against \(f_i\) for values of \(\kappa _{i0}/\kappa _{h0}=0.02, 0.1, 1, 10,\) and 50. In both configurations corresponding to Fig. 1.5c and d, it is observed that \(c>1\). However, it is imperative to recognize that these conditions are merely sufficient. In the context of nonlinear particles embedded within a linear host material, a critical threshold for the enhanced ratio \(\kappa _{i0}/\kappa _{h0}\) is approximately 1/3.5, constrained by a maximum \(f_i\) value of \(\pi /4\). Conversely, when linear particles are embedded within a nonlinear host, the critical ratio \(\kappa _{i0}/\kappa _{h0}\) is about 2.5.
1.4 Scattering Cancellation Theory
The strictness of the transformation thermotics ensures its powerful ability to manipulate the thermal field, but the parameters predicted by the theory are often non-uniform and anisotropic, which brings great challenges to the practical preparation. The linear/conformal mapping and effective medium theory previously discussed provide theoretical frameworks for designing the parameters in the production of transformation optics devices. In pursuit of alternative methodologies, some scholars have applied scattering cancellation theory to directly resolve the governing equations and determine the distribution of the physical field. This approach utilizes isotropic homogeneous media to replace the complex components of the original structure, thereby replicating the external field distribution. It is acknowledged that obstacles or external heat sources (besides the uniform gradient field) disrupt the background temperature field. Depending on the need for external stimuli, thermal metamaterials are typically categorized into passive or active designs.
1.4.1 Passive Scheme: No External Energy Input
1.4.1.1 Circular Shape
Xu et al. [49, 50] examined a core-shell configuration with inner and outer radii denoted by \(r_c\) and \(r_s\) in (Fig. 1.6a), respectively. The thermal conductivities of the core and shell are \(\boldsymbol{\kappa }_c\) and \(\boldsymbol{\kappa }_s\), respectively, and both are anisotropic,
The metric of polar coordinates is
By incorporating Eqs. (1.56) and (1.57) into Eq. (1.3) and neglecting both the time-dependent term and the internal heat source term, the heat conduction equation in polar coordinates is given by:
The general solution for Eq. (1.58) is expressed as:
where \(u_i^\pm =\pm i\sqrt{\kappa _{\theta \theta }/\kappa _{\textrm{rr}}}\). Due to the symmetry inherent in the core-shell structure, Eq. (1.59) can be simplified as follows:
where \(A_0\) is a constant set to zero for convenience, \(A_1\) and \(B_1\) are two constants to be determined, and \(u^\pm \) are respectively \(\pm \sqrt{\kappa _{\theta \theta }/\kappa _{rr}}\) in 2D and \(-1/2\pm \sqrt{1/4+2\kappa _{\theta \theta }/\kappa _{rr}}\) in 3D. Then, the temperature distributions in the core (\(T_c\)), shell (\(T_s\)), and matrix (\(T_m\)) are represented as follows:
where \(A_c\), \(A_s\), \(B_s\), and \(B_m\) are four constants determined by boundary conditions, and \(A_m\) is the applied temperature gradient. In 2D, \(u_{c}^+=\sqrt{\kappa _{c\theta \theta }/\kappa _{crr}}\), \(u_{s}^\pm =\pm \sqrt{\kappa _{s\theta \theta }/\kappa _{srr}}\), and \(u_{m}^-=-1\); in 3D, \(u_{c}^+=-1/2+\sqrt{1/4+2\kappa _{c\theta \theta }/\kappa _{crr}}\), \(u_{s}^\pm =-1/2\pm \sqrt{1/4+2\kappa _{s\theta \theta }/\kappa _{srr}}\), and \(u_{m}^-=-2\).
The boundary conditions necessitate the continuity of temperature and normal heat flux, expressed as follows:
where \(\kappa _m\) is the thermal conductivity of the matrix. Substituting Eq. (1.61) into Eq. (1.62) yields
Solving these equations enables the analytical determination of coefficients \(A_c\), \(A_s\), \(B_s\), and \(B_m\). The condition of scattering cancellation is met when the influence of the core-shell structure is nullified, denoted by \(B_m=0\). Applying this condition, one can deduce the necessary thermal conductivity:
where \(\kappa _e\) signifies the effective thermal conductivity of the core-shell configuration. This computational approach extends beyond a single-shell model. For configurations with n shells, the effective thermal conductivity can be derived using \(2n+2\) boundary conditions analogous to Eq. (1.62), where \(n+1\) equations describe the temperature continuity, and the other \(n+1\) equations indicate the normal heat flux continuity. An alternative methodology involves a recursive, shell-by-shell calculation of the effective thermal conductivity. For example, Eq. (1.64) can be employed to compute the effective thermal conductivity of the core combined with the first shell, \(\kappa _{e1}\). Thereafter, the core and the first shell are treated as a new core with thermal conductivity \(\kappa _{e1}\). Subsequently, Eq. (1.64) can be applied again to calculate the effective thermal conductivity of this new core in conjunction with the second shell, \(\kappa _{e2}\), and the process is repeated for all n shells. Notably, Li et al. [51] have crafted components characterized by diverse diffusion rates employing scattering cancellation theory. The insertion of these varied components enables switching between cloaking and concentrating functions in the concentration field. This innovative design boasts adaptability and reconfigurability, rendering it applicable across a broad spectrum of uses without necessitating a comprehensive redesign of the existing system.
(from Ref. [50])
Schematic diagram of two representative schemes. a Isotropic geometry with anisotropic thermal conductivity; b Anisotropic geometry with isotropic thermal conductivity.
1.4.1.2 Elliptical Shape
In the study of geometrically anisotropic confocal core-shell structures [52, 53], we assume isotropic thermal conductivities for the core, \(\kappa _c\), and the shell, \(\kappa _s\), as depicted in Fig. 1.6b. Anisotropy in the shell results in the loss of a degree of design freedom compared to isotropic materials, often necessitating an additional shell layer for compensation. Without loss of generality, let us first discuss the calculation method of the single shell, which can be naturally generalized to the double shell or multiple shell. Within a Cartesian coordinate system denoted by \(x_i\), a two-dimensional scenario permits i to assume the values of 1 and 2, while a three-dimensional scenario allows i to be 1, 2, and 3. The semi-axes of the core and shell in the \(x_i\) direction are represented by \(r_{ci}\) and \(r_{si}\), respectively. The transformation from Cartesian coordinates \(x_i\) to elliptic or ellipsoidal coordinates \(\rho _j\) is governed by the relation:
where j takes on values of 1, 2, and 3 for three-dimensional spaces and 1 and 2 for two-dimensional spaces. The coordinate \(\rho _1\), which is greater than \(-r_{ci}^2\), functions similarly to the radial coordinate in spherical systems. For instance, the shell’s inner and outer boundaries correspond to \(\rho _1=\rho _c\) (which equals zero) and \(\rho _1=\rho _s\), respectively. In the presence of a uniform thermal field along the \(x_i\) axis, thermal conduction equation can be expressed in the elliptical (or ellipsoidal) coordinate system as [45]
with a definition of \(g\left( \rho _1\right) =\prod \limits _i\left( \rho _1+r_{ci}^2\right) ^{1/2}\). For three dimensions, \(4 \pi g\left( \rho _{1}=0\right) {/} 3=4 \pi r_{c 1} r_{c 2} r_{c 3} / 3\) (or \( 4 \pi g\left( \rho _{1}=\rho _{s}\right) / 3\) \(=4 \pi r_{s 1} r_{s 2} r_{s 3} / 3\)) represents the volume of the core (or the core plus the shell). For two dimensions, \(\pi g\left( \rho _{1}=0\right) =\pi r_{c 1} r_{c 2}\left( \right. \)or \(\left. \pi g\left( \rho _{1}=\rho _{s}\right) =\pi r_{s 1} r_{s 2}\right) \) denotes the area of the core (or the core plus the shell). The temperature distributions within the core (\(T_{ci}\)), the shell (\(T_{si}\)), and the matrix (\(T_{mi}\)) along the \(x_i\) axis are formulated as:
with a definition of \(\phi _i\left( \rho _1\right) =\int _{\rho _c}^{\rho _1}\left[ \left( \rho _1+r_{ci}^2\right) g\left( \rho _1\right) \right] ^{-1}d\rho _1\). The constants \(A_{ci}\), \(A_{si}\), \(B_{si}\), and \(B_{mi}\) are ascertainable through the continuity conditions of temperature and normal heat flux. Since the temperature distribution in the background should be undistorted, we take \(B_{mi}=0\) and leads to the following expression for the effective thermal conductivity \(\kappa _e\):
Here, \(f=g\left( \rho _c\right) /g\left( \rho _s\right) =\prod \limits _ir_{ci}/r_{si}\) and \(L_{wi}\) is the shape factor of the ellipse along \(x_i\)-direction, which can refer to the definition of Eq. (1.39). This method can be easily extended to calculate the effective thermal conductivity of a core-shell structure with n shells, as we did for the geometrically isotropic case.
1.4.1.3 Irregular Shape
The challenge of solving the steady-state heat conduction equation for boundaries with irregular geometries is significant, as it complicates the analytical approach. Traditional scattering cancellation theories often struggle to maintain an undisturbed background thermal field in such cases. Addressing this, Xu et al. [54] introduced a concept referred to as a thermal supercavity, as illustrated in Fig. 1.7. This design adheres to the thermal uniqueness theorem, which suggests that matching the boundary conditions of the shell with those of the surrounding environment will preserve the undisturbed nature of the background temperature field. The assumption is made that the thermal gradient in the background, \(\nabla T_0\), is oriented along the x-direction, and the thermal conductivity of the shell in the y-direction is taken to be infinitely large. Contrasting with a conventional circular cloak (see Fig. 1.7a), the shell in this scheme (see Fig. 1.7b) ensures that the temperature gradient component along the y-direction is negligible. Furthermore, the cavity’s interior is filled with air and objects, implying an equivalent thermal conductivity close to zero. By considering the conservation of heat flux across the shell and background interface, the following relationship is established,
where \(\text {Gradient }T_s\) denotes the horizontal temperature gradient within the shell, and \(\kappa _b\) represents the thermal conductivity of the background. The functions f(x) and g(x) describe the lengths of the cavity and shell at the position x, respectively, as depicted in Fig. 1.7c and d. Consequently, the thermal conductivity of the shell, comprising an anisotropic material, is derived as:
(from Ref. [54])
Schematic diagram of a the traditional circular cloak and b circular supercavity. c shows an example of square supercavity, whose details are indicated in d. Brown lines and red arrows represent isotherms and heat flux, respectively. Temperature distribution e without cavity and with f cavity.
Equation (1.70) facilitates the design of a shell with tailored properties, ensuring its invisibility within the thermal landscape, as presented in Fig. 1.7e and f. Subsequently, Wang et al. [55] expanded upon this approach within the conduction-radiation domain, developing thermal cloaks and sensors for irregular-shaped objects.
1.4.2 Active Scheme: External Energy Input
The above discussion has centered on passive heat transfer mechanisms. However, in many practical applications, the introduction of external energy inputs can unlock additional functionalities. This discussion will delve into two types of external energy inputs: one achieved through the rotation of a rotor to induce heat flow, and the other by means of direct contact with an external heat source.
1.4.2.1 Introduce Thermal Convection by Rotating Particle
Xu et al. [29] investigate a scenario involving a rotating particle characterized by an angular velocity \(\Omega \), radius R, and thermal conductivity \(\kappa _p\), situated within a finite matrix with thermal conductivity \(\kappa _m\). This setup is visualized in Fig. 1.8. For a steady-state and passive system, the governing equation is given by
where \(D=\kappa /\rho c\) is thermal diffusivity and \(v=\Omega r\) is linear velocity. Utilizing the metric of polar coordinate (Eq. (1.57)), Eq. (1.71) can be rewritten as:
The temperature profile \(T_p\) in the particle has the form of
where F(r) and \(G(\theta )\) correspond to the radial and angular distribution functions, respectively. On substituting Eq. (1.73), the resulting equation is:
Considering that G varies periodically with \(\theta \) and transitions smoothly in this system, the assumption that \(G = \exp (i\theta )\), with i being the imaginary unit, is made. This leads to a tractable solution of Eq. (1.74), wherein F satisfies:
By performing Variable substitution \(r=\sqrt{D/\Omega }x\), Eq. (1.75) reduces to
where the solution is a combination of the first-order Kelvin functions:
In scenarios where the velocity is significantly high, and consequently, x is large, the following expression is derived:
where M(x) signifies the amplitude of f(x), and \(\phi (x)\) encapsulates the influence of rotation.
Adapted from Ref. [29]
The schematic diagram of rotating particle embed in a finite matrix.
The replacement of a rotating particle with a stationary one with an effective thermal conductivity \(\kappa _p\) leads to the heat conduction equation without rotation given by
The general solution to this equation is expressed as:
where \( T_0 \) represents a reference temperature, here set to zero for simplicity, and \( A \) and \( B \) are constants. The temperature distributions in the matrix (\( T_m \)) and in the particle (\( T_p^* \)) are described by
where \(B_m\) and \(A_p^{*}\) are two constants determined by the boundary conditions, and \(A_m\) is the applied temperature gradient. The continuity of temperatures and heat fluxes at the boundary is characterized by
Substituting Eq. (1.81) into Eq. (1.82), the relationship is derived as follow:
It is worth mentioning that \( T_p^{*} \) generally differs from \( T_p \), yet their difference can be minimized by a specific condition. They define the difference as
According to the minimum entropy production principle, the condition \( \phi (x(R))=\pi /4 \) should be satisfied to attain the minimum value of this integral, resulting in
Reconsidering the rotating particle, and the boundary conditions are
Substituting Eqs. (1.78) and (1.81) into Eq. (1.86), one can derive
The substitution of Eqs. (1.83) and (1.85) into Eq. (1.87) yields
which can be simplified to:
Building on Eq. (1.89), Xu et al. [30] realized an analog thermal material whose effective conductivity can be in-situ tuned from near-zero to near-infinity. Moreover, Li et al. [31] introduced a rotating ring to realized effective infinite thermal conductivity, demonstrating a thermal zero-index cloak theoretically and experimentally. Compared with traditional thermal cloaks, this metadevice can operate in a background of high thermal conductivity, and the sheltered object maintains a high sensitivity to external temperature changes.
1.4.2.2 Introduce Energy by Contacting External Heat Sources
A thermal dipole can be analogized to an electric dipole, comprising both a heat source and a cold source. As a result of this thermal dipole, the generated thermal fields can assume a comparable shape. Xu et al. [32] introduced the concept of a thermal metamaterial that is assisted by thermal dipoles, as depicted in Fig. 1.9.
(from Ref. [32])
Finite-element simulations in the presence of a, d an external uniform thermal field, b, e a thermal dipole, and c, f an external uniform thermal field and a thermal dipole together.
Consider a circular particle with thermal conductivity \(\kappa _p\) and radius \(r_p\) embedded in a matrix with thermal conductivity \(\kappa _m\), as illustrated in Fig. 1.9. When a uniform thermal field \(G_0\) is applied, the temperature distribution within the matrix is disrupted due to the disparity in thermal conductivities between the particle and the matrix. Drawing on the two-dimensional temperature general solution provided by Eq. (1.60), the temperature distributions for the particle \(T_{pe}\) and the matrix \(T_{me}\) can be expressed as follows:
where \(G_0\) denotes the thermal gradient at infinity and \(T_r\) represents the reference temperature. Constants \(A_1\) and \(A_2\) are ascertainable via the boundary conditions:
From these conditions, one can solve for \(A_2\), hence depicting the temperature distribution of the matrix as presented in (Fig. 1.9a and b):
On the other hand, the temperature distribution induced by a thermal dipole can be characterized as:
with the corresponding boundary conditions:
Therefore, \(B_2\) is determined and the temperature distribution for the matrix is acquired as shown in (Fig. 1.9c and d),
where \(M=Ql\) is the dipole moment. According to the superposition principle, the total temperature distribution \(T_s\) (Fig. 1.9e and f) in the matrix is
To maintain a uniform thermal field in the matrix, the second term to the right of Eq. (1.96) must be nullified, which leads to determining the appropriate thermal dipole moment:
where \(A=\pi r_p^2\) denotes the area of the particle. After applying Eq. (1.97), the distortion could be reduced to ensure a uniform temperature gradient. These discoveries are poised to facilitate advancements in thermal management applications, including infrared signature minimization, thermal protection, and infrared camouflage. Moreover, various scholars have employed linear heat sources with either a constant temperature [33] or constant gradient [56] to neutralize external fields. The findings have potential applications in other Laplace fields, including direct current [1] and fluid dynamics [57].
1.5 Special Theories
1.5.1 Topology-Related Theory: Geometric Phases and Edge State
Topology, initially a geometrical concept, has extended its reach to elucidate electron transport phenomena in condensed matter physics due to its formal similarity with the Schrödinger equation [58]. Over the last two decades, topological physics has experienced a surge in growth, leading to the theoretical prediction and experimental discovery of numerous topological matter phases. The wave equation’s resemblance to the Schrödinger equation has also stimulated research interests in various wave systems, encompassing electromagnetic waves [59] and acoustic waves [60].
1.5.1.1 Geometrical Phase
Phase, a pivotal concept in physics, has significantly influenced topology studies. While the geometric phase has been investigated within micro heat transfer domains, discerning the geometric phase at the macroscopic scale has been challenging due to the divergent mechanisms—phonon-based micro heat transfer versus the macroscopic Fourier law. Research efforts have thus been channeled towards exploring the wave-like nature of temperature fields, aiming to unveil the geometric phase of macroscopic heat transfer by encircling anomalies in parameter space.
Basic heat transfer properties of macroscopic heat conduction-convective systems. a Three-dimensional ring model. b Simplified two-dimensional models. c Decay rate -Im (w) and frequency Re (w) as a function of velocity u. d Illustration of five representative eigenstates of the system. The phase shift for \(\Psi _1\) (or \(\Psi _{1}^{\prime }\)) is depicted as \(\pi /4\) (or \(3\pi /4\)), while \(\Psi _2\), \(\Psi _3\), and \(\Psi _3^{\prime }\) exhibit a phase difference of \(\pi /2\). The temperature profiles on the left and right for each eigenstate correspond to the thermal distributions in the lower and upper sections of the rings, respectively. Adapted from Ref. [61]
In pursuit of understanding the thermal geometric phase, Xu et al. [61] scrutinized anomalies within heat transfer systems. They considered a configuration involving two rings moving at equal velocities but in opposite directions, \((+u,-u)\), flanking a stationary ring with inner and outer radii \(r_1\) and \(r_2\), respectively, as illustrated in Fig. 1.10b. This figure demonstrates the 3D model projection along the yz plane, where the internal surface length is \(l=2\pi r_1\) and the width of the moving ring (intermediate layer) is denoted by w (\(w_i\)). Temperatures for the upper ring, lower ring, and intermediate layer are represented by \(T_1\), \(T_2\), and \(T_i\) correspondingly. The macroscopic heat conduction-convection process adheres to an energy conservation equation detailed as:
where \(D_1\) (\(=D+d\)), \(D_2\) (\(=D-d\)), and \(D_i\) signify the thermal diffusivities of the upper ring, lower ring, and middle layer, respectively. The quasi-one-dimensional model presupposes \(l\gg w\), allowing for the assumption that temperature variations along the z axis are negligible. With heat exchange occurring between the two moving rings through the intermediate layer, the energy conservation equation can be reformulated as:
Incorporating the wave-like properties of temperature fields introduces phase characteristics, which can be mathematically represented as:
Where \(A_1\) and \(A_2\) denote the amplitudes of the temperature field in the upper and lower rings, respectively. \(\beta \) is the wave number, \(\omega \) is the frequency, and \(T_\text {r}\) is the reference temperature, which can be regarded as 0. By substituting the Eq. (1.100) into the Eq. (1.99) and discussing the special case of \(d = 0\), i.e. \(D_1 = D_2 = D\), the following relationship is derived:
In the context, \(|\boldsymbol{\psi }\rangle =\left[ A_1,A_2\right] ^{\dagger }\) represents the eigenstate with \(\dagger \) indicating transposed conjugation. The Hamiltonian \(\boldsymbol{H}\) and the corresponding eigenvalues are expressed as:
where \(D=\kappa _1/(\rho C)=\kappa _2/(\rho C)\). As the parameter u increases from zero to infinity, the system transitions between two distinct states. The exceptional point, denoted as \(u_{\text {EP}} = h/\beta \), marks the boundary between these states. For values of u less than \(u_{\text {EP}}\), the frequency \(\omega _{\pm }\) is purely imaginary, as illustrated in Fig. 1.10c. This corresponds to a scenario where the temperature fields of the two moving rings exhibit a static phase difference, merely decaying over time without any propagation, as depicted in \(\psi _1\) and \(\psi _1^{\prime }\) within Fig. 1.10d. Upon reaching the velocity \(u = u_{\text {EP}}\), the disparity between \(\omega _{+}\) and \(\omega _{-}\) vanishes, as shown in Fig. 1.10c. At this point, the eigenstates share a phase difference of \(\pi /2\) and continue to decay without propagation, as shown in \(\psi _2\) and \(\psi _2^{\prime }\) in Fig. 1.10d. As u surpasses \(u_{\text {EP}}\), the real parts of the frequencies \(\omega _{\pm }\) emerge and become significant, as indicated in Fig. 1.10c. Consequently, the eigenstates maintain a \(\pi /2\) phase difference, exhibiting both decay and propagation. The direction of motion is governed by the ring which manifests a greater amplitude in its temperature field, as demonstrated in \(\psi _3\) and \(\psi _3^{\prime }\) in Fig. 1.10d.
Subsequently, simulations are conducted to examine the thermal geometric phase,
which aligns with analogous findings in quantum systems. The simulations encompass a cyclic path of velocity depending on time, described by the Hamiltonian \(\boldsymbol{H} (u(t))\). Five distinct loop paths are investigated, as depicted in Fig. 1.11a. The initial condition sets the velocity at \(u = 0\) mm/s and the state at the eigenstate 0, corresponding to the eigenvalue \(\omega _-\). In this regime, since the velocity is below the exceptional point, the eigenvalue is purely imaginary, signifying the absence of an accumulated phase difference. The evolutionary trajectories for both routes ultimately revert to their initial states, as demonstrated in Fig. 1.11b through Fig. 1.11f. Additionally, Xu et al. [62] explored the geometric phases in particle diffusion systems.
Adapted from Ref. [61]
Thermal geometry phase simulation results. a Five different evolutionary routes. b, c, d, e and f respectively show the Max (\(T_1\)) and Max (\(T_2\)) trajectories corresponding to the five evolutionary routes.
1.5.1.2 Topological Edge State
The preceding analysis demonstrates that a macroscopic thermal system, undergoing convection and conduction, can exhibit geometric phase, characteristic of a non-Hermitian system. Recently, other pivotal concepts such as topological invariance and bulk-edge correspondence have been investigated through the discretization of diffusion processes. Yoshidae et al. [36] integrated Fick’s law with the continuity equation to derive the one-dimensional diffusion equation for a continuous scalar field, given by
where \(\partial _{t(x)}\) represents the derivative with respect to time t (spatial coordinate x).
(from Ref. [36], licensed under CC-BY 4.0)
a System composed of two sites coupled with the diffusion coefficient D. b One-dimensional chain under the periodic boundary condition for \(L_x=10\). c Sketch of the model under the fixed boundary condition for \(L_x=6\). d Spectrum of \(\tilde{H}_\text {SSH}\) for \(D=1\) and \(L_x=240\). e The time-evolution of in the case for \((D,D^{\prime })=(1, 0.2)\) [\((D,D^{\prime }) = (0.2,1)\)], where the system is topologically nontrivial (trivial). The function \(-(D + D^{\prime })t\) is plotted with a black line.
The diffusion Eq. (1.104) was then discretized, establishing an analogy between diffusion phenomena and a tightly bound model in quantum systems. As depicted in Fig. 1.12a, consider a system composed of two sites where the discrete field values \(\phi _0\) and \(\phi _1\) are defined. By applying Fick’s law, the flux from site 0 to 1 is expressed as \(J_{0 \rightarrow 1}= -D (\phi _0-\phi _1)\), assuming a unit-length distance between the sites. Consequently, the temporal evolution of the vector \(\vec {\phi }=(\phi _0,\phi _1)^T\) is governed by:
Extending this framework to a one-dimensional chain with \(L_x\) sites, as illustrated in Fig. 1.12b, the time evolution of the vector \(\vec {\phi }=(\phi _0,\phi _1,\dots ,\phi _{L_x-1})^T\) is similarly described as:
The discrete diffusion equation reveals that the dynamics of classical diffusion can be encapsulated within a tightly bound model, suggesting that diffusion systems could serve as a novel platform for topological investigations beyond the realm of quantum systems.
The study progresses by analyzing a one-dimensional system with dimeric compounds (Fig. 1.12c) that corresponds to the Su-Schrieffer-Heeger (SSH) model of quantum systems. Following the procedure used to derive Eq. (1.106), the equation satisfied by \(\vec {T}=(T_{0A},T_{0B},T_{1A},\cdots ,T_{L_{X}-1B})^{\textrm{T}}\) is obtained,
It can be found that \(\hat{H}_\text {SSH}\) is similar to the one-dimensional SSH model in the quantum system, only the onsite energy is shifted. Therefore, the topological nature of the one-dimensional SSH model is preserved. Focusing on the bulk properties under periodic boundary conditions applied to the structure in Fig. 1.12c, the Hamiltonian matrix \(\hat{H}_\text {SSH}\) in momentum space is represented by:
where k is the Bloch vector. \(\hat{h}_{\textrm{SSH}}(k)\) represents the gap and maintains chiral symmetry, potentially harboring topologically non-trivial features indicated by the winding number:
This winding number quantifies the encirclements made by a non-diagonal element of \(\hat{h}_{\textrm{SSH}}(k)\) around the origin in the complex plane and is hence an integer. Figure 1.12d illustrates the spectrum of \(\hat{H}_\text {SSH}\) under fixed boundary conditions, indicating the presence of localized edge states (or absence thereof) corresponding to the winding number \(W = 1\) (\(W = 0\)), with each state represented by a blue dot for different values of \(D^\prime \). In Fig. 1.12e, the temporal evolution of the temperature at the edge \((ix, \alpha ) = (0, A)\) is plotted. Here, the temperature \(T_{0A}\) exhibits exponential decay for \(t \le 2\tau \), with a half-life \(\tau = 1/(D + D^\prime ) = 0.83\) influenced by the edge state when \((D,D^\prime ) =(1,0.2)\). This deviates from the standard exponential decay near \(t = 0.5\) and results in a shorter half-life compared to when \((D,D^\prime ) =(0.2, 1)\).
1.5.2 The Bloch Series Expansion Method
The heat conduction equation has been previously compared to the Schrödinger equation, with the conduction term of the former analogous to the potential energy term of the latter. According to Bloch’s theorem, the periodic potential field and wave function can be expanded into a series of periodically modulated plane waves. Analogously, in the context of heat conduction, this corresponds to the periodic modulation of thermal conductivity, resulting in temperature fluctuations [64, 65]. The Bloch series expansion method is adept at addressing such problems.
Adapted from Ref. [37]
a Fizeau drag of light. b Failure of direct thermal simulation due to lack of macroscopic heat momentum. c Diffusion Fizeau drag in spatiotemporal thermal metamaterials, based on thermal Willis coupling. d Numerical results of the speed ratio of \(\eta =|v_f/v_b|\) as a function of \(2\pi \Gamma =\phi \epsilon u_y d/D_0\), with constant \(\Lambda =\Delta _{\rho }\cos \theta /\Delta _{\kappa }\).
A significant instance of this analogy is observed in the context of diffusion Fizeau drag within spatiotemporal thermal metamaterials. The Fizeau drag phenomenon, originally discovered by Fizeau in 1851, describes the differential speeds of light traveling with and against a current of water, as illustrated in Fig. 1.13a. This behavior has been comprehensively explained by relativistic kinematics and observed in various moving media or spatiotemporal media, attributable to the nonlinear motion of Dirac electrons. In simpler terms, while photons interact with momentum in a water flow and polarons in an electric current, heat does not carry momentum. Therefore, even when advection is introduced, the temperature field propagates at the same speed in both directions (Fig. 1.13b), although the amplitude [63] of the temperature field in the opposite direction may vary.
To achieve diffused Fizeau drag, Xu et al. [37] engineered a spatiotemporal thermal superlattice to study the diffusion of Fizeau drag in heat conduction, as depicted in (Fig. 1.13c). They introduced periodic inhomogeneity into a porous medium as described by:
where \(\Delta _\rho \) and \(\Delta _\kappa \) represent the modulation amplitudes of density and thermal conductivity, respectively; \(G=2\pi /d\) is the wave number for modulation; d is the modulation wavelength along the horizontal axis; \(\xi = x+\zeta y\) denotes the generalized coordinate; \(\zeta = d/h\) is the ratio with h as the vertical height; and \(\theta \) is the phase difference in modulation. To eliminate any asymmetry in the amplitude of the temperature field caused by horizontal advection, the consideration is directed towards normal advection \(u_y\), which typically does not lead to asymmetry along the horizontal axis. The heat transfer equation governing the inhomogeneous medium is then articulated as:
Consider the upward advection velocity denoted as \(u_y\), Eq. (1.112) is expanded to:
According to Bloch’s theorem, they introduce a wavelike temperature field with spatial modulation,
where \(s=0, \pm 1, \pm 2,\dots ,\pm \infty \) and \(F_0=1\). Subsequent expression of the partial derivatives of temperature in relation to time and space yields:
Using Fourier expansions, Eq. (1.110) can be written as:
where \(\rho _{\pm 1}=e^{\pm \text {i}\theta }\rho _0 \Delta _{\rho }/2\) and \(\kappa _{\pm 1}=\kappa _0 \Delta _{\kappa }/2\) are defined. By substituting Eqs. (1.114) and (1.115) into Eq. (1.112), the component form of the governing equation is represented as:
To derive the numerical results, the series is truncated for \(s = 0, \pm 1, \dots , \pm 10\), positing that \(F_{|s|>10}= 0\). This results in twenty-one equations that facilitate the resolution of the unknown quantities for \(\beta \) and \(F_{|s|\le 10}\). With \(\beta \) determined, the propagation speed of the wavelike temperature fields is computed as \(v=\omega /\text {Re}\left[ \beta \right] \). Furthermore, the speed ratio \(\eta =|v_f/v_b|=|\text {Re}\left[ \beta _b\right] /\text {Re}\left[ \beta _f\right] |\) is derived.
The terms \(v_f\) and \(v_b\) correspond to the forward and backward propagation speeds of the temperature field. The magnitude of the diffused Fizeau drag, denoted by \(\eta \), is predominantly dependent on three non-dimensional parameters: \(2\pi \Gamma = \phi \epsilon u_y d/D_0\), \(\Lambda = \Delta \rho \cos \theta / \delta \kappa \), and \(\zeta = d/h\). The influence of \(\Lambda \) on the diffused Fizeau drag, for a fixed \(\zeta \) of 0.2, is explored. It is found that both \(2\pi \Gamma = 0\) and \(2\pi \Gamma \rightarrow \infty \) yield \(\eta = 1\), signifying the necessity of vertical advection for modulation, but a larger \(\Lambda \) does not always correspond to enhanced performance. Figure 1.13a delineates two categories of curves: Type I consistently maintains an \(\eta \) above 1, as demonstrated by the top three curves; Type II displays an initial \(\eta \) above 1 that subsequently diminishes below this threshold, as depicted by the bottom three curves. A critical juncture at \(\Lambda = 1\) marks the transition between Type I and II, as illustrated by the third curve from the top. These two types of curves highlights the flexible control achievable over the diffused Fizeau drag. The Bloch series expansion method introduced earlier can be applied to other wave-like systems, such as chemical waves [66].
1.6 Conclusion and Outlook
Over the past decade, since the advent of transformation optics, the domain has evolved to encompass diffusion systems, extending from static passive configurations to dynamic active ones, from pure heat conduction to the intertwined dynamics of heat conduction-convection and thermoelectric phenomena, and from linear to nonlinear realms. Moreover, the ascent of transformation theory has catalyzed the refinement of other theoretical approaches relevant to diffusion processes, such as effective media theories, scattering cancellation techniques, non-Hermitian topologies, and series expansions. While these theoretical methods were exhaustively explored prior to the inception of transformation theory, the emerging trend of deploying composite methods to actively steer diffusion fields is unprecedented.
Within the foundational theoretical scaffolding of diffusion studies, a plethora of new functionalities have been actualized, and numerous novel phenomena have been unveiled. Nonetheless, several pivotal challenges and questions persist unaddressed. A primary concern is that certain some governing equations do not satisfy transformation invariance before and after coordinate transformations; that is, transformable parameters remain elusive, as deliberated in Sect. 1.2.3.1. Therefore, it is meaningful to Enhance the universality of transformation thermotics to accommodate more intricate and varied diffusion fields. The second challenge involves refining a more comprehensive effective medium theory within thermotics to simplify the complexity inherent in parameterization. Although theoretical advancements in effective thermal conductivity have been considerable, computational strategies, such as machine learning [67] and topological optimization, are increasingly recognized for their superiority in devising complex structures. The third challenge is that the scattering cancellation approach is constrained to diffusion fields with simplistic boundaries and falters with more complicated configurations. The recent introduction of the transformation field method [68] promises to circumvent the intricacies associated with solving complex boundary problems. The final challenge lies in elucidating the topological characteristics within the diffusion process. Despite the proliferation of scholarly articles, translating these insights into practical applications necessitates further in-depth investigation.
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Zhuang, P. (2024). Diffusionics: Basic Theory and Theoretical Framework. In: Diffusionics. Springer, Singapore. https://doi.org/10.1007/978-981-97-0487-3_1
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