Electromagnetic Relations between Materials and Fields for Microwave Chemistry

Abstract

We consider the application of microwave energy to a material. The effects of the electromagnetic field on the material and of the material on the electromagnetic field will be described, focusing on the dielectric relaxation phenomenon of the liquid. The dielectric permittivity of mixtures is discussed by extending Debye relaxation to explain how the material behaves with respect to an electric field. We will also consider the energy that the electric field imparts to the material, both thermally and nonthermally. We will develop this relation and describe what form it should take if there is a nonthermal effect in the chemical reaction field under microwave irradiation.

1. Introduction

There has been much debate about whether microwave irradiation acts as heat in chemical synthesis or whether it has a nonthermal effect [1, 2, 3, 4, 5, 6]. This problem has been discussed in many cases based on changes in the reaction rate and in the selectivity of the difference between the results with and without the application of microwaves. What is particularly important here is whether the temperatures of the two conditions to be compared are exactly the same. In microwave irradiation, it is difficult to use a general thermometer such as a metal thermocouple or an alcohol thermometer. This is because the distribution of the electromagnetic field changes significantly due to the insertion of a metal material (thermocouple or mercury thermometer), or because the indicator material (alcohol) itself is heated. As an alternative, an indirect method such as measuring the temperature from radiation on the surface of the vessel is used. In the comparison between microwave irradiation and non-irradiation, if the measurement does not correctly indicate the internal temperature, the difference may be due to the microwave irradiation condition being higher than the non-irradiation condition.

The heat source in microwave irradiation is the loss of electromagnetic wave energy, i.e., loss of the electric or the magnetic fields due to undulation of the molecule itself. Therefore, the movement behavior varies depending on the molecular species of the irradiated material. A different momentum obtained for each molecule means that it is not in a thermal equilibrium state, meaning that it does not match the definition of temperature, which requires an isotropic equilibrium motion.

The aforementioned is an inductive argument that discusses differences due to microwave irradiation, such as changes in reaction rates and in selectivity. A lot of data are reported every year, but the interpretations are diverse, and there are cases where it is “both hard to explain and hard to ignore” [1].

On the other hand, in this review, the original physical meaning is examined based on the dielectric relaxation phenomenon and how the material behaves under microwave irradiation. Based on this, we will discuss deductions about what action should be generated if there is an effect other than heat based on principles, rather than data.

2. Material properties (relaxation properties)

2.1 Permittivity and refractive index

When an object is heated by irradiation with microwaves, the microwave energy is attenuated inside the object [7]. The Beer–Lambert law in optics can also be applied in the microwave region. The refractive index, n, and the attenuation factor, k, can be combined as a complex refractive index n*, as shown by Eq. (1):

$$n^{\ast }=n-\mathit{jk},$$ E1

where j is the square root of −1. When the wave has a cosine signal s(t, x) as a function of time t and position x, the n and k correspond to the propagation and attenuation velocities in the phasor formula (Eq. (2)) as shown in Figure 1 :

Figure 1.

Undulation and complex refractive index at t = 0.

$$s\left(x,t\right)=Acos\left(n^{\ast }x+\mathit{\omega t}\right)=Re\left({\mathit{Ae}}^{-j\left(n-\mathit{jk}\right)x}{e}^{-\mathit{j\omega t}}\right).$$ E2

Physical properties in the microwave region are indicated by the complex permittivity ε* [F/m] and complex permeability μ* [H/m]. The meanings of these terms can be explained by definition of the values of basic physical constants. The speed of light propagation through the vacuum (c = 2.99792458 × 10⁸ m/s) is defined value. After May 20, 2019, the magnetic constant μ ₀ changed from the defined value (4π × 10⁻⁷ H/m) to the experimentally determined value (1.25663706212 (19) × 10⁻⁶ H/m) [8]. The electric constant ε ₀ is derived from these constants (Eq. (3)):

$${\epsilon }_0=\frac{1}{{\mu }_0{c}^2}.$$ E3

When Eq. (3) is transformed to Eq. (4), c represents the reciprocal of the square root of ε ₀ and μ ₀:

$$c=\frac{1}{\sqrt{{\epsilon }_0{\mu }_0}}.$$ E4

To be precise, vacuum is not a material, but electromagnetic waves propagate through it. Since the same relationship applies to materials, they can be treated equally well in terms of mathematical expressions. The velocity v [m/s] of the electromagnetic wave propagating through a material is the reciprocal of square root of the product of the material’s permittivity ε [F/m] and permeability μ [H/m] (Eq. (5)):

$$v=\frac{1}{\sqrt{\mathit{\epsilon \mu }}}.$$ E5

Therefore, the refractive index n is obtained by Eq. (6):

$$n=\frac{c}{v}=\frac{\sqrt{\mathit{\epsilon \mu }}}{\sqrt{{\epsilon }_0{\mu }_0}}=\sqrt{{\epsilon }_r{\mu }_r}\text{where}\epsilon ={\epsilon }_0{\epsilon }_r,\mu ={\mu }_0{\mu }_r.$$ E6

Here, the relative permittivity ε _r [nd] and the relative permeability μ _r [nd] are coefficients based on ε ₀ and μ ₀, and n is dimensionless. In the following, the dimensionless value is expressed as [nd].

When attenuation of electromagnetic waves occurs in a material, the complex relative permittivity ε _r* [nd] and the complex relative permeability μ _r* [nd] are used. Therefore, the relationship with n* is as follows (Eq. (7)):

$$n^{\ast }=\sqrt{{\epsilon }_r^{\ast }{\mu }_r^{\ast }}\text{where}{\epsilon }_r^{\ast }={\epsilon }_r^{\prime }-j{\epsilon }_r^{″},{\mu }_r^{\ast }={\mu }_r^{\prime }-j{\mu }_r^{″}.$$ E7

The superscripts ‘and “indicate a real part and an imaginary part, respectively.

Discussions dealing only with dielectrics generally introduce important assumptions here. Since dielectrics often do not exhibit magnetism, the permeability is considered to be the same as that of vacuum, and μ _r * is set to 1−j0. Devices that measure permittivity (actually complex relative permittivity) often base their calculations on this assumption, so one should be careful when measuring materials with magnetism or high conductivity. Under this assumption, Eq. (7) is approximated by Eqs. (8) and (9):

$$n^{\ast }=\sqrt{{\epsilon }_r^{\ast }},$$ E8

$${\left(n-\mathit{jk}\right)}^2={\epsilon }_r^{\prime }-j{\epsilon }_r^{″}.$$ E9

Here, Eq. (1) is reviewed again. Since n is the ratio of the propagation velocity in the material to that in the vacuum, it can be regarded as the ratio of the wavelength λ ₀ [m] in the vacuum to the wavelength λ [m] in the material (Eq. (10)):

$$n=\frac{c}{v}=\frac{{\lambda }_0}{\lambda }.$$ E10

On the other hand, since attenuation does not occur in a vacuum, it is difficult to understand k as a ratio. Therefore, a distance δ [m] at which an electric field intensity E [V/m] becomes 1/e = 36.8% is used. Here, e is the Napier number and ω [rad / s] is the angular frequency. δ is called the skin depth and has dimensions of length. As δ decreases, the amount of attenuation increases, indicating a large k (Eq. (11)):

$$k=\frac{c}{\mathit{\omega \delta }}=\frac{{\lambda }_0}{2\mathit{\pi \delta }}.$$ E11

Furthermore, when Eq. (9) is transformed, Eqs. (12) and (13) are obtained:

$$n={\left[\frac{1}{2}{\epsilon }_r^{\prime }\left(\sqrt{1+{\left({\epsilon }_r^{″}/{\epsilon }_r^{\prime }\right)}^2}+1\right)\right]}^{1/2},$$ E12

$$k={\left[\frac{1}{2}{\epsilon }_r^{\prime }\left(\sqrt{1+{\left({\epsilon }_r^{″}/{\epsilon }_r^{\prime }\right)}^2}-1\right)\right]}^{1/2}.$$ E13

From these equations, n and k are obtained from the ε _r* of the material. When n is large, the microwave that has progressed is greatly refracted. In particular, a cylindrical vessel collects power at the center as in the case of a lens, and heating may proceed locally. Figure 2 shows an example simulating the electromagnetic field distribution of water in a cylindrical container. When the water temperature is uniform, the power loss density (PLD) is concentrated in the center [9].

Figure 2.

Simulation of the electromagnetic field of water in a cylindrical vessel in an oven-type furnace. A: Oven shape. B: PLD of 25°C water. C: PLD of 100°C water.

2.2 Penetration depth and skin depth

When the same material is irradiated with electromagnetic waves having the same frequency, the applied power intensity P [W/m³] is proportional to the square of the electric field intensity E [V/m]. When the attenuation is large, the microwave may not reach the deep part of the vessel. Although the skin depth, δ, has been described earlier, there is a penetration depth L [m] as a similar index [7]. The distance at which the power intensity P becomes 1/e due to the loss during propagation is expressed as L _1/e [m]. In this case, penetration depth L _1/e is half of the skin depth, δ. Microwaves are not absent beyond this depth.

There are two methods for describing L _1/e , as shown in Eqs. (14) and (15):

$$L_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$e$}\right.}=\frac{{\lambda }_0}{4\pi }{\left[\frac{2}{{\epsilon }_r^{\prime }\left(\sqrt{1+{\left({\epsilon }_r^{″}/{\epsilon }_r^{\prime }\right)}^2}-1\right)}\right]}^{1/2},$$ E14

$$L_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$e$}\right.}=\frac{{\lambda }_0}{2\pi \sqrt{{\epsilon }_r^{\prime }}tan\delta }.$$ E15

The δ in Eq. (15) is not a skin depth, but a value indicating dielectric loss in a narrow definition as tan δ (Eq. (16)):

$$tan\delta =\frac{{\epsilon }_r^{″}}{{\epsilon }_r^{\prime }}.$$ E16

Eq. (15) can be obtained from a modification in which the second and subsequent terms are ignored in the Maclaurin expansion when tan² δ → 0 in Eq. (14). Therefore, when using Eq. (15), it is assumed that tan δ → 0. On the other hand, during microwave heating, the material of tan δ → 0 does not heat, as will be described later. Therefore, if the target is not tan δ → 0, it is necessary to pay attention to whether the value based on the latter formula deviates from the premise.

2.3 Plotting on bode and Nyquist diagrams

The correlation with the horizontal axis representing frequency and the vertical axis representing complex permittivity is called a Bode diagram. The Bode diagram of the water is shown in Figure 3 , indicating that the dielectric constants ε _r′ and ε _r″ are functions of the (angular) frequency [10]. This is represented by the Eq. (17):

Figure 3.

Nyquist diagram and bode diagram of water (200 MHz–14 GHz).

$$\epsilon ={\epsilon }_0{\epsilon }_r^{\ast }\left(\omega \right)={\epsilon }_0\left\{{\epsilon }_r^{\prime }\left(\omega \right)-j{\epsilon }_r^{″}\left(\omega \right)\right\}.$$ E17

When the complex permittivity at each frequency is plotted on a Nyquist diagram with a real part on the horizontal axis and an imaginary part on the vertical axis, they draw a semicircular locus as shown in Figure 3 . Such behavior is called Debye relaxation. Debye relaxation is a behavior commonly found in nonionic liquid materials. Examples that cannot be applied include cases where the relaxation frequency is not single, and those where a conductive material is included (described at 2.10 and 2.11).

2.4 Relationship between the Debye relaxation formula and the bode/Nyquist diagram

In the previous section, we described how many liquids show a characteristically semicircular geometric locus due to Debye relaxation. We will return to the basics to explain why and what information can be gleaned below. The characteristic behavior in the microwave band is called dielectric relaxation. The deformation of electron clouds and molecular structures is a response in the UV and IR bands and is faster than in the microwave band. These contributions are prompt responses to undulated fields.

On the other hand, molecular orientation is a phenomenon based on rotation of an electric dipole. A large moment like a molecule causes a time delay in orientation with respect to field changes. The time delay referred to here is a phase delay and does not vibrate at a different period from that of the applied external field. Since the molecule cannot rotate if the external field vibration is too fast, but it can follow a too-slow external field vibration without time delay, the behavior is distributed around a specific vibration frequency [11, 12, 13, 14]. The prompt response is only a propagation delay and does not contribute to the loss. This is expressed as ε _r (∞) only in the real part. At the current time t, the application of an external electric field E(t) generates an electric flux density D(t) in the material.

The part of the electric flux density in the material D _p(t) (p: prompt) pertaining to prompt response is expressed by Eq. (18):

$$D_p\left(t\right)={\epsilon }_0{\epsilon }_r\left(\infty \right)E\left(t\right).$$ E18

On the other hand, the contribution of the delayed response includes not only the electric field E (t) at the current time t but also the influence of the electric field E (u) at the previous time u. Therefore, the electric flux density D _d (t) (d: delayed) of the dielectric following the delay is expressed by Eq. (19), integrated from start time 0 to the current time t:

$$D_d\left(t\right)={\int }_0^{t}E\left(u\right)f\left(t-u\right)\mathit{du}.$$ E19

In Eq. (19), f (t-u) represents a response function in terms of the previous time u and the current time t. From Eqs. (18) and (19), the electric flux density D (t) of the dielectric at the current time t is expressed by Eq. (20):

$$D\left(t\right)={\epsilon }_0{\epsilon }_r\left(\infty \right)E\left(t\right)+{\int }_0^{t}E\left(u\right)f\left(t-u\right)\mathit{du}.$$ E20

D(t) in Eq. (20) can also be expressed as the product of the applied electric field E(t) and the dielectric constant ε*(ω) of the material. Therefore, Eq. (21) can be obtained:

$$D\left(t\right)={\epsilon }_0{\epsilon }_r^{\ast }\left(\omega \right)E\left(t\right).$$ E21

Substituting Eq. (21) into Eq. (20) and rewriting the response function to f (x) yields Eq. (22):

$${\epsilon }_0\left\{{\epsilon }_r^{\ast }\left(\omega \right)-{\epsilon }_r\left(\infty \right)\right\}={\int }_0^{\infty }{e}^{-\mathit{j\omega x}}f\left(x\right)\mathit{dx}.$$ E22

Next, an appropriate expression is set for a response function. In Debye-type relaxation, Eq. (23) is used as a response function to Eq. (7):

$$f\left(x\right)={\epsilon }_0\left\{{\epsilon }_r\left(0\right)-{\epsilon }_r\left(\infty \right)\right\}\frac{e^{-x/\tau }}{\tau }.$$ E23

Here, τ [s/rad] is the relaxation time, which is the reciprocal of the angular frequency ω ₀ [rad/s] at which the vibration phase is delayed by π/2 (90 degrees). A very slowly undulating field is effectively the same as a static field. Therefore, the complex relative dielectric constant is also only the real part, and this is represented by ε _r(0). As shown in the Nyquist diagram of Figure 3 , ε _r′(ω), which is the real part of ε _r*(ω), is between ε _r(0) and ε _r(∞). Therefore, f (x) can be interpreted as having a proportional coefficient of ε _r(0)-ε _r(∞), which is the difference between the real parts.

When Eq. (23) is substituted into Eq. (22) and transformed, ε _r*(ω) is expressed by Eqs. (24)–(26). These are Debye’s dispersion equations:

$${\epsilon }_r^{\ast }\left(\omega \right)={\epsilon }_r\left(\infty \right)+\frac{{\epsilon }_r\left(0\right)-{\epsilon }_r\left(\infty \right)}{1+\mathit{j\omega \tau }};$$ E24

$${\epsilon }_r^{\prime }\left(\omega \right)=Re{\epsilon }_r^{\ast }\left(\omega \right)={\epsilon }_r\left(\infty \right)+\frac{{\epsilon }_r\left(0\right)-{\epsilon }_r\left(\infty \right)}{1+{\omega }^2{\tau }^2};$$ E25

$${\epsilon }_r^{″}\left(\omega \right)=Im{\epsilon }_r^{\ast }\left(\omega \right)=\frac{\left\{{\epsilon }_r\left(0\right)-{\epsilon }_r\left(\infty \right)\right\}\mathit{\omega \tau }}{1+{\omega }^2{\tau }^2}.$$ E26

Eliminating ω_τ from Eqs. (25) and (26) leads to the relationship of Eq. (27). Therefore, when ε _r′(ω) is on the horizontal axis and ε _r″(ω) is on the vertical axis, a semicircle is drawn as shown in Figure 3 :

$${\left\{{\epsilon }_r^{\prime }\left(\omega \right)-\frac{{\epsilon }_r\left(0\right)+{\epsilon }_r\left(\infty \right)}{2}\right\}}^2+{\epsilon }_r^{″}{\left(\omega \right)}^2={\left\{\frac{{\epsilon }_r\left(0\right)-{\epsilon }_r\left(\infty \right)}{2}\right\}}^2.$$ E27

From the aforementioned, if the measured values are plotted on a semicircle, it indicates that the material has a response that can be explained by Debye relaxation theory and is not in a special state.

2.5 Three angles on the semicircle: φ, θ, and δ

The dielectric loss is also written as tan δ. This is caused by the phase delay angle δ of the voltage and current when an alternating electric field is applied to the dielectric; it is defined by Eq. (16). The complex dielectric constant is calculated by measuring the loss and the propagation delay. The physical meaning of this value is a tangent, where δ is the angle between the horizontal axis and the line segment from the origin to the circumference ( Figure 4 ).

Figure 4.

Three angles on a semicircle.

Substituting Eqs. (25) and (26) into Eq. (16) shows that tan δ is a function of ω:

$$tan\delta =\frac{\left\{{\epsilon }_r\left(0\right)-{\epsilon }_r\left(\infty \right)\right\}\mathit{\omega \tau }}{{\epsilon }_r\left(0\right)+{\epsilon }_r\left(\infty \right){\omega }^2{\tau }^2}.$$ E28

If the central angle φ is determined at an arbitrary point on the semicircle when ε _r(0) is φ = 0°, ε _r(∞) shows φ = 180° [15]. Therefore, φ indicates a phase delay of molecular motion in the delayed response. The tan φ at an arbitrary frequency is obtained geometrically by the following equation:

$$tan\phi =\frac{{\epsilon }_r^{″}\left(\omega \right)}{{\epsilon }_r^{\prime }\left(\omega \right)-\frac{{\epsilon }_r\left(0\right)+{\epsilon }_r\left(\infty \right)}{2}}.$$ E29

Substituting Eqs. (25) and (26) into Eq. (29) eliminates ε _r(0) and ε _r(∞), as shown in Eq. (30). Therefore, the influence of φ upon temperature change means that it is based only on the change of the dielectric relaxation time τ when the frequency is constant:

$$tan\phi =\frac{2\mathit{\omega \tau }}{1-{\omega }^2{\tau }^2}=-\frac{1}{sinh\left(ln\left(\mathit{\omega \tau }\right)\right)}=-csch\left(ln\left(\mathit{\omega \tau }\right)\right).$$ E30

As shown in Figure 4 , if θ is defined as an angle formed by a straight line extending from the left intersection of the semicircle and the horizontal axis toward the measurement point and the horizontal axis, then θ is a circumferential angle of φ. Here, tan θ is expressed by Eq. (31):

$$tan\theta =\frac{{\epsilon }_r^{″}\left(\omega \right)}{{\epsilon }_r^{\prime }\left(\omega \right)-{\epsilon }_r^{\prime }\left(\infty \right)}.$$ E31

Substituting Eqs. (25) and (26) into Eq. (31) yields Eq. (32):

$$tan\theta =\mathit{\omega \tau };$$ E32

transforming it yields Eq. (33) for relaxation time τ [s/rad]:

$$\tau =\frac{1}{\omega }tan\theta =\frac{1}{\omega }\frac{{\epsilon }_r^{″}\left(\omega \right)}{{\epsilon }_r^{\prime }\left(\omega \right)-{\epsilon }_r\left(\infty \right)}.$$ E33

When calculating τ, in a region far from θ = 45°(φ = 90°), even if ω changes logarithmically, φ does not change significantly, and the calculation of τ has a large error. On the other hand, in the vicinity of φ = 90°, ε _r*(ω) changes sharply with respect to ω. Therefore, when calculating τ using Eq. (31), it is preferable to perform evaluation with a measurement point in the vicinity of 2θ = φ = 90°.

From Eq. (32), ε _r″ (ω) is maximum at ω = 1/τ. In actual measurements, the function to be swept is often denoted by f[Hz] instead of ω[rad/s]. It should be noted that in many references, τ is sometimes written in terms of the reciprocal of the relaxation frequency f _c which is defined in Eq. (58). In this case, the unit of the derived τ is [s], which is 2π times τ[s/rad]. In Table 1 , the unit of τ is written in [s] instead of [s/rad].

Entry		f c [GHz]	τ [ps]	ε r(0)	ε r(∞)	ε r*_(2.45 GHz)		ε r*_(5.8 GHz)
						Calcd.	Found	Calcd.	Found
1	H2O	16.8	59.4	79.8	4.8	78.2−j10.7	78.9−j10.8	71.8−j23.1	72.0−j23.3
2	ProC	3.5	287	66.2	7.4	46.7−j27.6	46.5−j27.8	23.0−j26.0	22.9−j26.2
3	DMSO	7.7	130	47.6	7.9	44.0−j11.5	44.2−j11.1	33.2−j19.1	33.6−j19.0
4	DMAc	8.6	116	40.5	6.2	37.9−j9.0	37.8−j9.2	29.8−j15.9	29.7−j16.1
5	MeNO2	32.3	31.0	37.1	11.0	37.0−j2.0	37.4−j2.3	36.3−j4.5	36.4−j5.1
6	DMF	13.9	72.0	37.6	8.2	36.7−j5.0	36.9−j5.2	33.2−j10.5	33.0−j10.8
7	MeCN	39.2	25.5	35.6	12.2	35.5−j1.5	35.4−j1.8	35.1−j3.4	35.0−j4.4
8	Me-Im	5.0	198	38.0	5.5	31.8−j12.8	31.8−j12.6	19.5−j16.1	19.7−j16.2
9	NMP	6.8	148	33.4	6.0	30.2−j8.8	30.2−j8.7	21.8−j13.5	22.0−j13.7
10	PhNO2	3.5	285	35.0	4.5	25.0−j14.3	24.6−j14.3	12.7−j13.5	12.2−j13.4
11	MeOH	3.0	338	33.7	6.1	22.5−j13.6	22.2−j13.2	11.8−j11.2	11.6−j11.1
12	PhCN	4.5	223	26.0	4.4	21.1−j9.1	21.4−j8.7	12.5−j10.5	12.7−j10.6
13	Me2CO	44.4	22.5	21.2	5.8	21.2−j0.8	21.0−j1.0	21.0−j2.0	21.1−j2.4
14	MEK	24.2	41.3	18.7	9.0	18.6−j1.0	18.6−j1.1	18.1−j2.2	18.0−j2.4
15	PhCHO	5.2	191	18.9	4.0	16.2−j5.7	16.2−j5.7	10.7−j7.4	10.8−j7.5
16	DCE	15.4	65.1	10.4	5.2	10.2−j0.8	10.2−j0.9	9.7−j1.7	9.7−j1.9
17	CH2Cl2	—	—	9.3	8.5	—	9.3−j0.1	—	9.1−j0.5
18	EtOH	0.9	1100	25.8	4.6	7.2−j7.0	7.5−j7.0	5.1−j3.3	5.3−j3.4
19	MeI	17.9	55.9	7.3	5.7	7.3−j0.2	7.2−j0.2	7.2−j0.5	7.1−j0.6
20	AcOEt	13.0	76.6	6.3	4.6	6.2−j0.3	6.2−j0.3	6.0−j0.6	6.0−j0.7
21	PhCl	10.4	96.3	5.8	3.2	5.7−j0.6	5.7−j0.6	5.2−j1.1	5.2−j1.1
22	PhF	13.8	72.6	5.8	4.0	5.7−j0.3	5.7−j0.3	5.5−j0.6	5.5−j0.6
23	PhBr	7.5	134	5.8	3.1	5.6−j0.8	5.5−j0.7	4.8−j1.3	4.9−j1.3
24	CHCl3	11.1	89.9	5.1	3.9	5.0−j0.2	5.1−j0.0	4.8−j0.5	4.9−j0.4
25	PhI	5.2	192	5.0	3.0	4.7−j0.8	4.7−j0.7	3.9−j1.0	3.9−j1.0
26	Et2O	—	—	4.5	4.1	—	4.5−j0.1	—	4.4−j0.2
27	2-PrOH	0.4	2730	20.1	3.4	3.8−j2.4	4.1−j2.7	3.5−j1.0	3.6−j1.3
28	CPME	27.8	36.0	4.0	2.3	4.0−j0.1	3.9−j0.3	3.9−j0.3	3.9−j0.4
29	1-BuOH	0.3	3170	16.3	3.4	3.6−j1.6	3.9−j2.0	3.4−j0.7	3.5−j1.1
30	2-BuOH	0.3	3540	15.4	3.2	3.3−j1.4	3.6−j1.6	3.2−j0.6	3.3−j0.9
31	i-BuOH	0.2	4070	16.1	3.1	3.3−j1.3	3.5−j1.6	3.2−j0.5	3.2−j0.9
32	t-BuOH	0.3	2910	11.6	2.9	3.1−j1.2	3.3−j1.5	2.9−j0.5	3.0−j0.8
33	PhMe	—	—	2.5	2.4	—	2.5−j0.1	—	2.4−j0.1
34	PhH	—	—	2.4	2.3	—	2.4−j0.1	—	2.4−j0.0
35	c-Hex	—	—	2.2	2.1	—	2.1−j0.1	—	2.1−j0.0
36	n-Hex	—	—	2.0	1.4	—	2.0−j0.0	—	2.0−j0.0

Table 1.

Physical property values of several liquids at room temperature [16].

Entry 1: distilled water; 2: propylene carbonate; 3: dimethyl sulfoxide; 4: N,N-dimethylacetamide; 5: nitromethane; 6: N,N-dimethylformamide; 7: acetonitrile; 8: N-methylimidazole; 9: N- methyl-2-pyrrolidone; 10: nitrobenzene; 11: methanol; 12: benzonitrile; 13: acetone; 14: methyl ethyl ketone; 15: benzaldehyde; 16: 1,2-dichloroethane; 17: dichloromethane; 18: ethanol; 19: methyl iodide; 20: ethyl acetate; 21: chlorobenzene; 22: fluorobenzene; 23: bromobenzene; 24: chloroform; 25: iodobenzene; 26: diethyl ether; 27: 2-propanol; 28: cyclopentyl methyl ether; 29: 1-butanol; 30: 2-butanol; 31: isobutyl alcohol; 32: tert-butyl alcohol; 33: toluene; 34: benzene; 35: cyclohexane; 36: n-hexane. f _c: relaxation frequency obtained by fitting, τ: relaxation time, calcd. ε _r (f): calculated ε _r *, found: measured ε _r *.

2.6 Loss calculation formula

The propagation of the electromagnetic wave energy of the microwave is propagation of electromagnetic field vibration. Specifically, this vibration is caused by continuously and repeatedly converting the electric field energy into magnetic field energy and vice versa. The propagation equations are shown as follows (Eqs. (34)–(38)):

$$\mathrm{rot}\mathbf{H}=\mathbf{i}+\frac{\partial \mathbf{D}}{\partial t};$$ E34

$$\mathbf{B}=\mu \mathbf{H};$$ E35

$$\mathrm{rot}\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t};$$ E36

$$\mathbf{D}=\epsilon \mathbf{E};$$ E37

$$\mathbf{i}=\sigma \mathbf{E}.$$ E38

Here, E, H, D, B, and i are the electric field intensity [V/m], magnetic field intensity [A/m], electric flux density [C/m²], magnetic flux density [Wb/m²], and current density [A/m²], all of which are vector quantities. In addition, the physical property coefficient of the material is transferred as a permittivity ε [F/m], a conductivity σ [S/m], and a permeability μ [H/m]. The loss of propagation energy means that the vector quantity has been lost when converted to the other field. This loss is mainly regarded as a conversion to heat. The loss equation calculated based on the physical property values is derived below [11, 12, 17].

The applied E is expressed by using a phasor as follows:

$$E=E_0{e}^{\mathit{j\omega t}}.$$ E39

D in the dielectric produced by aligning the directions of the dielectric molecules undulates with a phase shift (delay) of δ. δ is a value expressed by Eq. (16) and is not φ, which is an undulation phase delay of the molecule:

$$D=D_0{e}^{j\left(\mathit{\omega t}-\delta \right)}.$$ E40

From Eq. (37), the product of E, ε ₀, and ε _r* gives D. Therefore, the phase delay of D can also be expressed by the complex permittivity (Eq. (41)):

$$D={\epsilon }_0{\epsilon }_r^{\ast }E.$$ E41

Substituting Eqs. (39) and (40) into Eq. (41) and converting the exponential function to a trigonometric function using Euler’s formula yields the Eqs. (42) and (43). Eq. (42) divided by Eq. (43) matches Eq. (16):

$$D_{0}cos\delta ={\epsilon }_0{\epsilon }_r^{\prime }{E}_0;$$ E42

$$D_{0}sin\delta ={\epsilon }_0{\epsilon }_r^{″}{E}_0.$$ E43

Next, the energy consumption per unit volume when the electric flux density of the dielectric changes by dD is determined as dU. This dU is obtained by the product of E and dD. When dD is integrated over one period of vibration (1 / f = 2π / ω), the energy consumed by the dielectric during one period is obtained as w,

$$w={\int }_0^{2\pi /\omega }\mathit{dU}={\int }_0^{2\pi /\omega }\mathit{EdD}={\int }_0^{2\pi /\omega }E\frac{\mathit{dD}}{\mathit{dt}}\mathit{dt}.$$ E44

In Eq. (44), E and D are the real parts:

$$E=E_{0}cos\mathit{\omega t};$$ E45

$$D=D_{0}cos\left(\mathit{\omega t}-\delta \right).$$ E46

After transforming Eq. (46) with the cosine difference formula, differentiating with t gives Eq. (47):

$$\frac{\mathit{dD}}{\mathit{dt}}=-{\mathit{\omega \epsilon }}_0{\epsilon }_r^{\prime }{E}_{0}sin\mathit{\omega t}+{\mathit{\omega \epsilon }}_0{\epsilon }_r^{″}{E}_{0}cos\mathit{\omega t}.$$ E47

By substituting Eq. (47) into Eq. (44), we obtain Eq. (48):

$$w={\mathit{\pi \epsilon }}_0{\epsilon }_r^{″}{E}_0^2.$$ E48

Since w is repeated f times per second (= ω/2π times), the energy W received by the dielectric from the electric field per unit volume / unit time is expressed by Eq. (49):

$$W=\frac{\omega }{2\pi }w=\frac{1}{2}{\mathit{\omega \epsilon }}_0{\epsilon }_r^{″}{E}_0^2=\mathit{\pi f}{\epsilon }_0{\epsilon }_r^{″}{E}_0^2.$$ E49

Rewriting Eq. (49) yields Eq. (50). Here, when E ₀ is rewritten to |E|, the electric field loss equation is obtained. In this paper, the dielectric loss based on ε obtained by Eq. (50) is defined as P _{ε_loss} [W/m³]:

$$P_{\epsilon \_\mathit{\text{loss}}}=\frac{1}{2}{\mathit{\omega \epsilon }}_0{\epsilon }_r^{″}{\left|E\right|}^2.$$ E50

When the frequency is very low and the change in the electric field is very slow, the molecules align their dipole moments in a direction that cancels the electric field without delay in proportion to the electric field strength, and this flux density can follow without delay. Since φ → 0 and δ → 0, ε _r″ → 0 from Eq. (43), P _{ε_loss} → 0 from Eq. (50), and the electromagnetic wave energy loss is small. On the other hand, if the frequency is very high and the change in the electric field is very fast, the movement of the molecules cannot respond to the alternating electric field, and the application direction reverses before aligning the dipole moments. As a result, since φ → π and δ → 0, ε _r″ → 0, so P _{ε_loss}→ 0 and the electromagnetic wave energy loss is small.

2.7 Difference between tan δ and ε _r″ in loss

If the target frequencies are the same, ω may be regarded as a constant. According to the aforementioned equation, the loss appears to be proportional to ε _r″, but the actual loss is not determined solely by the difference in ε _r″. The meaning of Eq. 50 indicates that if E is constant, the amount of power loss per unit volume is proportional to ε _r″. However, the propagation speed of electromagnetic waves decreases with the refractive index n, which varies with ε (and μ).

Figure 5 shows a model in which microwaves pass through media in the order of air, water, and air. Here, for the sake of simplicity, the electric flux is indicated by a straight arrow, and the reflected wave at the boundary is not considered. Wavelength reduction is considered first [18]. The electric flux density in air is D ₁ [C/m²], and that in water is D ₂ [C/m²]. Since wavelength shortening occurs in water with a large ε _r′, the interval (density) of arrows in D ₂ increases to (ε _r)^0.5 times in D ₁ according to Eq. (8). Next, the electric field strength is considered. The electric field strength in air is E ₁ [V/m], and that in water is E ₂ [V/m]. In water where ε _r′ is large, the dipole of water cancels the applied electric field so that the intensity decreases to 1/ε _r according to the constitutive Eq. (37). Combining these two effects, the electric field strength E ₂ in water attenuates to 1 / (ε _r)^0.5 times the electric field strength E ₁ in air as shown in Eq. (51):

Figure 5.

Model structure of heating considerations. Area 1:Air, area 2:Water, area 3:Air.

$$\left\{\begin{array}{l}{D}_2=\sqrt{{\epsilon }_r^{\prime }}{D}_1\\ D_1={\epsilon }_0{E}_1\\ D_2={\epsilon }_0{\epsilon }_r^{\prime }{E}_2\end{array}\right.\text{therefore}{E}_2=\frac{1}{\sqrt{{\epsilon }_r^{\prime }}}{E}_1.$$ E51

In actual examinations, it is difficult to measure the electric field strength inside the irradiation target. Therefore, irradiation with a predetermined irradiation power is performed. Thus, the following interpretation is derived:

1. Eq. (52) indicates that the loss is proportional to ε _r″ when the applied electric field is constant:

   $$P_{\epsilon \_\mathit{\text{loss}}}=\frac{1}{2}{\mathit{\omega \epsilon }}_0{\epsilon }_r^{″}{\left|E_2\right|}^2.$$ E52

2. Eq. (53) indicates that the loss is proportional to tan δ when the applied power is constant:

   $$P_{\epsilon \_\mathit{\text{loss}}}=\frac{1}{2}{\mathit{\omega \epsilon }}_0{\epsilon }_r^{″}{\left|\frac{E_1}{\sqrt{{\epsilon }_r^{\prime }}}\right|}^2=\frac{1}{2}{\mathit{\omega \epsilon }}_{0}tan\delta {\left|E_1\right|}^2.$$ E53

The applied electric field and applied power referred to here are the net electric field and power applied to the materials. As will be described later, not all irradiation power is always applied. The irradiated and reflected powers can be measured in area 1. The passing through power can be measured in area 3.

2.8 Maximum of tanδ

From the Nyquist diagram, the maximum value of tan δ is the tangent point through the origin. Therefore, it is obtained from Eq. (54):

$$tan{\delta }_{\max }=\frac{{\epsilon }_r\left(0\right)-{\epsilon }_r\left(\infty \right)}{2\sqrt{{\epsilon }_r\left(0\right){\epsilon }_r\left(\infty \right)}}.$$ E54

Furthermore, ε _r′(ω) and ε _r″(ω) when tan δ is maximal are represented by Eqs. (55) and (56):

$${{\epsilon }_r^{\prime }}_{tan\delta max}=\frac{2{\epsilon }_r\left(0\right){\epsilon }_r\left(\infty \right)}{{\epsilon }_r\left(0\right)+{\epsilon }_r\left(\infty \right)},$$ E55

$${{\epsilon }_r^{″}}_{tan\delta max}=\frac{{\epsilon }_r\left(0\right)-{\epsilon }_r\left(\infty \right)}{{\epsilon }_r\left(0\right)+{\epsilon }_r\left(\infty \right)}\sqrt{{\epsilon }_r\left(0\right){\epsilon }_r\left(\infty \right)}.$$ E56

The tan θ when tan δ becomes maximum is defined as tan θ _tanδmax. This is geometrically determined from Figure 4 . The angular frequency ω _tanδmax at this time is given by Eq. (32). When both are combined, Eq. (57) is derived:

$$tan{\theta }_{tan\delta max}={\omega }_{tan\delta max}\tau =\sqrt{\frac{{\epsilon }_r\left(0\right)}{{\epsilon }_r\left(\infty \right)}}.$$ E57

The frequency at which ε _r″ is maximized is defined as f _c. Since tanθ is 1 at f _c, 2πf _c τ is equal to 1 from Eq. (32) and Figure 4 . Therefore, Eq. (57) becomes Eq. (58) by f _tanδmax and f _c:

$$f_{tan\delta max}=\sqrt{\frac{{\epsilon }_r\left(0\right)}{{\epsilon }_r\left(\infty \right)}}{f}_c.$$ E58

The semicircle in the Nyquist diagram shows that the maximum values of ε _r″ and tan δ do not match. For example, water has a maximum value of ε _r″ at 18 to 22 GHz, but the maximum value of tan δ is on the higher frequency side. Since 2.45 GHz is much smaller than these, it appears to be less efficient at heating water. However, when the amount of absorption is large, attenuation occurs rapidly in the surface and does not penetrate into the inner side. Therefore, it cannot be said that a larger loss is always effective for heating the inside to a wide area.

2.9 Changes in ε _r* with temperature

The complex dielectric constant according to Debye relaxation can be generalized by obtaining ε _r(0), ε _r(∞), and τ by measuring in a wideband and fitting to a semicircle. From this relationship, it can be seen that the temperature, frequency, and complex permittivity have the following relationship:

1. When the temperature rises, the molecule becomes disturbed. As a result, the external response amount ε _r′(ω) decreases. This also applies to ε _r(0).

2. When the temperature rises, the intermolecular bond becomes weaker and the relaxation time τ becomes shorter. As a result, the peak value of ε _r″(ω) shifts to the high frequency side.

3. If the dielectric loss is based on the response-phase difference φ, ε _r′(ω) and ε _r″(ω) are linked by the Kramers-Kronig relations and are not independent.

As shown in Figure 6 , the Bode diagrams of ε _r′(ω) and ε _r″(ω) shift from 1 → 2 → 3 with temperature rising [19]. Therefore, in the Nyquist diagram, the central angle φ indicated by the irradiation frequency f _i can be classified into six types depending on the position [11, 12, 13].

Figure 6.

Area division of the bode diagram. Left: area division; Right: temperature rising.

Type I is the case where the relaxation frequency f _c of the irradiated material before heating is much larger than f _i, and the φ of the irradiated material before heating is in the vicinity of 0π/8 to 2π/8. As the temperature rises, φ approaches 0 and the amount of loss decreases.

Type II is the case where f _c is slightly larger than f _i, and φ indicates 2π/8 to 4π/8. Since ε _r″(ω) is large, the temperature rises rapidly. However, as the temperature becomes high, φ decreases and the temperature rise rate greatly decreases.

Type III is when f _i is close to or slightly smaller than f _c, and φ is in the vicinity of 4π/8 to 5π/8. Since ε _r″(ω) is large as a whole, the temperature rise rate is high and ε _r″(ω) rises until φ becomes π/2 and then falls.

Type IV is when f _c is smaller than f _i, and φ indicates 5π/8 to 7π/8. The tan δ is a large region, and as temperature rises, φ approaches 4π/8, so ε _r″(ω) further increases. Thermal runaway due to uneven heating may occur.

Type V is the case where f _c is much smaller than f _i, and φ indicates 7π/8 to 8π/8. Although it has the property that the temperature rises due to irradiation and the amount of absorption increases, there is a case where the irradiation frequency is too high, such that there is a lot of transmission and heating is not sufficient.

Type 0 is not shown in this figure. This corresponds to the case where contributions to the original dielectric loss, such as that from nonpolar molecules, are very small.

When the irradiated f _i is considerably smaller than f _c, the change in ε _r′ (ω) and ε _r″(ω) due to temperature rising is classified as type I, but when it irradiates a considerably larger f _c, it becomes a V type. Therefore, this classification also means that the classification changes depending on the irradiation frequency, even for the same material. As an example, the physical property values of several liquids and the value of τ calculated from a semicircle are shown in Table 1 .

2.10 Liquid mixture

For the relaxation time τ in the system to show only one value, it is necessary for the entire material to be in a uniform state. This occurs when there is only one kind of electric dipole that controls the dielectric constant, and the surrounding molecules that control the relaxation time are also uniform. This means that τ is distributed according to the δ function [20]. In the mixed liquid, there are various types of molecules exhibiting dielectric loss and various types of surrounding molecules. Therefore, the relaxation time is not always one. In the model, when the ε _r*(ω) of a system having the same ε _r(0) and ε _r(∞), different τ is calculated. Figure 7 shows that wideband complex permittivity plot is separated into two semicircles when the difference between the two τ is large. On the other hand, when the difference in τ is not large, the distortion of the semicircle is small. In the case of simple two-component mixing, the trajectory shown in the Nyquist diagram is considered to be similar to any in Figure 7 . However, there is little distortion in the measurement range as shown in Table 1 , and there is almost a single semicircle. This can be regarded as a response in which ε _r(0), ε _r(∞), and τ show one average value. Furthermore, when τ is evaluated with respect to the mixing ratio, it continuously changes according to the composition, but it is not always shown on the straight line in terms of arithmetic mean. However, there is also a behavior that protrudes upward and downward.

Figure 7.

Nyquist diagrams with two τ. Line 1: zero difference in τ, Line 2: small difference in τ ∼ Line 6: large difference in τ.

Figure 8 shows the Argan diagram as a complex relative permittivity at 2.45 GHz by the reflection probe method [21, 22, 23]. One line indicates the nine mixtures made with a volume ratio of 9:1 to 1:9 between two pure liquids. If the ε _r* obtained in the mixed sample obeys the additive property, the connection should be a straight line, but in many cases a curve is shown. This is because ε _r(0), ε _r(∞), and τ change due to the mixing of the two materials, such that ε _r′ and ε _r″ do not always attain a single arithmetic average value.

Figure 8.

Argan diagrams of liquid mixtures. 1: Water; 2: Propylene carbonate; 3: Dimethyl sulfoxide; 7: Acetonitrile; 11: Methanol; 13: Acetone; 18: Ethanol; 27: 2-propanol; 28: Cyclopentyl methyl ether.

A plot of the responses at other frequencies on the Argan diagram is also shown in Figure 8 . Apparently, these figures have changed greatly. However, as shown in point A, ethanol:water = 4:6 mixed solution, acetonitrile:propylene carbonate = 2:8 mixed solution, and acetone:propylene carbonate = 2:8 mixed solution had close ε _r* values, regardless of frequency. Similar intersection points were also observed in Group B (acetone:propylene carbonate = 9:1, cyclopentyl methyl ether:acetonitrile = 3:7) and Group C (ethanol:propylene carbonate = 7:3, methanol = 10). This means that if the average ε _r(0), ε _r(∞), and τ obtained by mixing are close, ε _r*(ω) matches, and therefore the same ε _r* is shown at different frequencies. Thus, when the liquid mixture characteristic is shown as a “train map,” the intersection corresponding to the “transfer station” does not change even if the frequency does.

2.11 Involvement of conductivity σ

The conductive material is heated by the conductive loss [10, 24, 25, 26, 27]. This property agrees with the dielectric loss in that it is proportional to the square of the electric field strength. These are losses to the electric field and not to the magnetic field. Since ω is not included in the conduction loss equation, it occurs even when the frequency is zero. Conduction loss is a phenomenon in which materials with a single charge, that is, positive and negative ion atoms (or molecules), are accelerated in opposite directions by application of an electric field. An increase in the distance between the counterions means that the electrostatic potential of the material is increased. Also, if ions that should linearly move with constant acceleration are decelerated to constant speed, this means that resistance has made ions motion isotropic, or the surrounding molecules have received the kinetic energy of ions. This means that the current has been converted to Joule heat. It is clear that such a conduction loss differs from dielectric loss in which the charge distance in the molecule is constant. Therefore, in the partial dielectric including conductivity, the conductive and dielectric losses appear separately. For example, the Nyquist diagrams of 0.1-mol/L = NaCl aqueous solution are shown in Figure 9 . Comparing to Figure 3 , it is shown that the locus is changed by adding NaCl due to conductive loss.

Figure 9.

Nyquist and bode diagrams of 0.1-Mol/L NaCl aqueous solution.

In many cases, only one parameter σ is used for the discussion of conductive loss. However, in the Nyquist diagram, σ must be a complex number with real and imaginary parts. If these parts have the same value, the low-frequency part shown in Figure 9 should be a straight line with a 45° slope, but the measured value is not. Therefore, from this figure, when discussing losses in the microwave band, conductivity must also be considered to have a complex value.

The power density w* [J/m³] of one cycle can be calculated from the current density i(t) = i ₀ sinωt with amplitude i ₀ [A/m] and the electric field intensity E(t) = E₀ sinωt with amplitude E₀ [V/m]. From the definition of the complex conductivity σ* [S/m] = σ′−jσ″, Eq. (38) is deformed as Eq. (59):

$$i\left(t\right)={\sigma }^{\ast }E\left(t\right).$$ E59

Here, w* is shown as Eq. (60):

$$w^{\ast }={\int }_0^{2\pi /\omega }i\left(t\right)E\left(t\right)\mathit{dt}=\frac{1}{2f}{\sigma }^{\ast }{E}_0^2.$$ E60

Since w* is repeated f times per second, the energy W received by the conductive material from the electric field per unit volume/unit time is expressed by Eq. (61):

$$W^{\ast }=\frac{1}{2}{\sigma }^{\ast }{E}_0^2.$$ E61

Here, σ* is a complex value and describes mobility and loss at the same time. When the real part is mobility and the imaginary part is loss, the conduction loss can be described in the same way as Eq. (50), and Eq. (62) is obtained:

$$P_{\sigma \_\mathit{\text{loss}}}=\frac{1}{2}{\sigma }^{″}{\left|E\right|}^2.$$ E62

Comparing Eqs. (50) and (64), it can be seen that σ″ and ωε ₀ ε _r″ have the same dimensions. Therefore, the loss equation of the combined electric field is Eq. (63):

$$P_{E\_\mathit{\text{loss}}}=\frac{1}{2}\left({\sigma }^{″}+{\mathit{\omega \epsilon }}^{″}\right){\left|E\right|}^2.$$ E63

The apparent relative permittivity measured in Figure 9 is the sum of the permittivity term and the conductivity term. This response is represented by ψ _r * (ω). Here, since σ _r* has the same dimension as ωε _r*, ψ _r*(ω) is expressed by the following Eq. (64):

$${\psi }_r^{\ast }\left(\omega \right)=\frac{{\sigma }^{\ast }\left(\omega \right)}{{\mathit{\omega \epsilon }}_0}+{\epsilon }_r^{\ast }\left(\omega \right)=\frac{{\sigma }_r^{\ast }\left(\omega \right)}{\omega }+{\epsilon }_r^{\ast }\left(\omega \right).$$ E64

When the complex conductivity σ* [S/m] = σ′−jσ″ is determined in the same manner as the complex conductivity ε* [F/m] = ε′−jε″, the complex relative permittivity ε _r* [nd] = ε*/ε ₀, the complex relative conductivity is derived as σ _r* = σ*/ε ₀ = σ _r’ − jσ _r″. Here, σ _r* has the same dimensions as ωε _r*, [rad/s]. Although the international annealed copper standard (IACS) is defined as 58 MS/m as a standard for conductivity, σ _r* indicates a ratio to ε ₀, rather than IACS.

In the previous diagram of the NaCl aqueous solution, the Nyquist diagram shows a semicircle in the high-frequency band, so this region has dielectric properties. On the other hand, in the low-frequency region, a locus different from a semicircle based on the movement of ions is shown. This means the loss due to the phase delay is small and mainly due to the motion of the ionic molecules. Whether the heat generation behavior at the irradiated frequency is mainly caused by dielectric or conductive loss cannot be distinguished by measurement at one frequency. Unless it is a Nyquist or Bode diagram, the contribution ratio of the dielectric and conductive losses cannot be separated from the locus.

3. Extraordinary microwave effect

In many discussions of microwave chemistry, temperature and heat are very important. It is shown that the loss of electromagnetic energy is based on the imaginary part, i.e. ε _r″ or σ _r″, as shown in Eq. (65). On the other hand, it is stated that energy is accumulated in a material with a large real part because propagation delay occurs, and substantial loss can be obtained by the tan δ term as shown in Eq. (53). Since microwave energy penetrates the material, the amount of energy on the spot can be increased without changing the amount of material. Therefore, it must be remembered that there is a high energy in the field, even if there is no conversion to heat. From this viewpoint, we considered what behavior should be taken if there is a nonthermal effect when microwave energy is applied. This section discusses deductions based on principles, not induction based on data [28, 29].

3.1 Classification of microwave effects

When the chemical reaction field under microwave irradiation is different from the non-irradiation condition, its form can be classified into four types.

1. Fast reaction rate (acceleration);

2. Slow reaction rate (deceleration);

3. Different products (selective production);

4. Different consumptions (selective consuming).

The first is an example in which the reaction speed increases when microwave energy is applied, and as a result, the reaction’s end time is shortened. Since the reaction rate can be significantly accelerated by increasing the temperature, one can always discuss whether the temperature was accurately measured or whether the actual reaction field temperature was high. The second is an example opposite to the first. This phenomenon appears when the actual reaction field temperature is low, but this case has little detailed discussion. If phase transition is included in the category, supercooling and overheating phenomena correspond to this. The third is an example in which products differ depending on the presence or absence of microwave irradiation when multiple products are considered. This happens when there are multiple reaction paths, one of which is particularly accelerated. This includes cases where only intermediates are obtained in a multistep reaction. The fourth is an example of a case with multiple substrates, and a reaction of the specific substrates is prioritized. In any of these cases, the reaction rate is considered to have changed due to the application of microwave energy. That is,

1. The target reaction speed increased;

2. The inhibition reaction speed increased;

3. The reaction speed for obtaining target products increased;

4. The reaction speed for consuming specific substrates increased.

All four of these interpretations mean that the specific reaction rate was increased by the application of microwave energy. From the aforementioned, when the chemical reaction under microwave irradiation is different from that in the non-irradiation case, if it is not a thermal effect, an equation for changing the reaction rate must be derived even if the temperature T is constant. The following describes the possibility of such an expression.

3.2 Real and imaginary parts

Eq. (63) describes the attenuation of the energy of the electric field. Considering the propagation here, it is an expression in which the imaginary part is changed to a real part. This means that the propagation equation is derived as Eq. (65):

$$P_{E\_\mathit{\text{prop}}}=\frac{1}{2}\left({\sigma }^{\prime }+{\mathit{\omega \epsilon }}^{\prime }\right){\left|E\right|}^2.$$ E65

Since the conductive material can be regarded as a special state of the dielectric, discussion of the material including σ will be omitted hereafter. Energy of electromagnetic waves vibrate between electric and magnetic fields as follows equations (Eqs. (66)–(69)). Herein, the magnetic field loss and the propagation energy P_H are also expressed based on the complex permeability μ* = μ′−jμ″ [H/m]:

$$P_{E\_\mathit{\text{prop}}}=\frac{1}{2}{\mathit{\omega \epsilon }}^{\prime }{\left|E\right|}^2;$$ E66

$$P_{E\_\mathit{\text{loss}}}=\frac{1}{2}{\mathit{\omega \epsilon }}^{″}{\left|E\right|}^2;$$ E67

$$P_{H\_\mathit{\text{prop}}}=\frac{1}{2}{\mathit{\omega \mu }}^{\prime }{\left|H\right|}^2;$$ E68

$$P_{H\_\mathit{\text{loss}}}=\frac{1}{2}{\mathit{\omega \mu }}^{″}{\left|H\right|}^2.$$ E69

Therefore, propagation of electromagnetic waves occurs according to the following stages:

• E-stage 1: Electric field energy given in the system is accumulated as propagation amount P _{E_prop}, and loss amount P _{E_loss} is converted into isotropic thermal motion of the molecules;

• E-stage 2: Propagation amount P _{E_prop} changes to magnetic field energy;

• H-stage 1: The magnetic field energy given in the system is accumulated as propagation amount P _{H_prop,} and the loss amount P _{H_loss} is converted into isotropic thermal motion of the molecules;

• H-stage 2: Propagation amount P _{H_prop} changes to electric field energy.

• These stages are repeated. What this equation indicates is transition of the electromagnetic wave energy, not a chemical reaction; thus, to consider the effect on the chemical reaction, one uses the following quantities:

• P _{X_prop} indicates the amount of energy present in the field acting on the reaction;

• P _{X_loss} indicates the amount of energy lost to the field affecting the reaction;

• A combination of these.

Whether P _{X_prop} is involved as a field that affects the reaction or P _{X_loss} is involved as a result of affecting the reaction is not clearly determined now. In any case, however, it should be energy terms which affect the chemical reaction.

3.3 Interpretation of the Arrhenius equations

In the reaction kinetics, the activation energy theory described by Arrhenius (Eq. (70)), which originated from gas molecular kinetics, is discussed:

$$k={\mathit{Ae}}^{-\frac{E_a}{\mathit{RT}}}.$$ E70

Strictly speaking, it cannot be applied theoretically except for the secondary reaction of two gas molecules, but it is useful as an empirical formula and can be used in a solution system. Here, k, A, E _a, and R are the reaction rate constant, A-factor, activation energy [J/mol], and molar gas constant [J/Kmol]. E _a is assumed to be a constant that does not change with the reaction temperature.

The Arrhenius equation can be interpreted as follows. The horizontal axis in Figure 10 shows the molecular velocity v in an arbitrary reaction coordinate system. The vertical axis φ _v represents the existence probability of the molecule with a normal distribution. In Figure 10 , it is assumed that a molecule that thermally and isotropically exchanges kinetic energy at a temperature T has a normal velocity distribution according to Eq. (71):

Figure 10.

Relationship between molecular velocity v and existence probability φ _E. Line 1: small distribution ∼ Line 3: large distribution.

$${\varphi }_v={\mathit{Ae}}^{-\frac{{\mathit{mv}}_x^2}{2\mathit{RT}}}.$$ E71

Here, the mass per mol is m [kg/mol], the kinetic energy of the molecule is E _v = mv²/2 [J/ mol], and the horizontal axis is energy, E. If the range of E is 0 ≤ E < +∞, this frequency distribution φ _E is canonical, as shown in Figure 11 .

Figure 11.

Reaction coordinate and energy distribution at temperature T.

$${\varphi }_E={\mathit{Ae}}^{-\frac{E}{\mathit{RT}}}$$ E72

The progress of the reaction is assumed to occur when the motion of the original molecule coincides with the positive direction of the reaction coordinate system and its kinetic energy exceeds E _a. The reaction coordinate is shown on the left-hand side of Figure 11 . Here, the substrate, transition state, and product are Sb, Tr, and Pd, respectively. The integrals of the region E _a < E < ∞ and of the whole region 0 ≤ E < ∞ are expressed by Eqs. (73) and (74):

$${\int }_{E_a}^{\infty }{\mathit{Ae}}^{-\frac{E}{\mathit{RT}}}\mathit{dE}={\mathit{\text{ARTe}}}^{-\frac{E_a}{\mathit{RT}}};$$ E73

$${\int }_0^{\infty }{\mathit{Ae}}^{-\frac{E}{\mathit{RT}}}\mathit{dE}=\mathit{ART}.$$ E74

Therefore, the ratio of the region exceeding E _a to the entire region is represented by Eq. (75), indicating that A is irrelevant:

$$\frac{{\int }_{E_a}^{\infty }{\mathit{Ae}}^{-\frac{E}{\mathit{RT}}}\mathit{dE}}{{\int }_0^{\infty }{\mathit{Ae}}^{-\frac{E}{\mathit{RT}}}\mathit{dE}}=e^{-\frac{E_a}{\mathit{RT}}}.$$ E75

The value obtained by Eq. (75) is the Boltzmann factor. In the Arrhenius equation, the reaction rate is proportional to the Boltzmann factor because of the occupation ratio of the high energy state.

Here, the temperature T of the heat bath is increased to T + ΔT. In gas molecule kinetics, the kinetic energy of a molecule increases with temperature, and this is expressed by (m _B v ²)/2 = 3k _B T/2. Here, m _B and k _B are the mass of one molecule and the Boltzmann constant, respectively. Therefore, a temperature increase of ΔT is the same as one molecule receiving k _B ΔT/2 of energy as m _B v _x ²/2, m _B v _y ²/2, and m _B v _z ²/2, respectively. Considering this at 1 mol, since R = N _A k _B (where N _A is Avogadro’s number), the kinetic energy of the system is distributed to the x, y, and z components by RΔT/2, respectively.

When any one direction is taken as the reaction coordinate, the energy distribution is the same in any state from Sb to Pd in the reaction coordinate system. Therefore, as expected, E _a does not change, even if the temperature is raised. This is shown on the left-hand side of Figure 12 . On the other hand, an increase in T corresponds to a decrease in the kurtosis of the Boltzmann distribution in the original molecule and an increase in the tail. As a result, the occupancy rate of molecules having an energy exceeding E _a increases, and the reaction rate k increases, as shown at right in Figure 12 .

Figure 12.

Reaction coordinate and energy distribution at temperature T + ΔT.

Next, consider the supply of microwave energy instead of ΔT as an external energy supply. The Arrhenius equation presupposes that it is in thermal equilibrium, which indicates that energy can be exchanged quickly with an external heat bath. Therefore, if the rate at which the system releases heat to the outside matches the heating rate by microwaves, a constant temperature state can be maintained, and T can be regarded as constant. This means that there can be reaction systems with the same T and different microwave application intensities. Microwave energy is calculated as energy per unit time. On the other hand, the reaction kinetics discussed earlier calculate the change in the molar concentration of molecules as the reaction rate. Therefore, in order to align the units, the electromagnetic wave energy supplied to the volume per mole with respect to the concentration M [mol/m³] of the target reaction molecule is E _add [J/mol]. E _add is proportional to the oscillated energy E _MW [J/mol], which is the product of power per mole and irradiation time, but not all energy works effectively.

Herein, the effective efficiency is defined by α and γ, and E _add = αRT = γE _MW. This means that αRT is added to the thermal potential RT that the field already has. Assuming that this added amount of external energy is incorporated into the canonical distribution in the same manner as RΔT, the potential distribution is derived in Eq. (76) by replacing RT in Eq. (72) with (1 + α)RT:

$${\varphi }_E=A_1{e}^{-\frac{E}{\left(1+\alpha \right)\mathit{RT}}}=A_1{e}^{-\frac{E}{\mathit{RT}+E_{\mathit{add}}}}.$$ E76

The coefficient A ₁ in Eq. (76) was introduced to consider the possibility that the A-factor changes depending on the presence or absence of external energy.

From the right-hand side of Figure 13 , integration of the region E _a < E < ∞, integration of the whole region 0 ≤ E < ∞, and the ratio between both are obtained as Eqs. (77), (78), and (79):

Figure 13.

Reaction coordinate and energy distribution at temperature T under microwave irradiation.

$${\int }_{E_a}^{\infty }{A}_1{e}^{-\frac{E}{\left(1+\alpha \right)\mathit{RT}}}\mathit{dE}=\left(1+\alpha \right)A_1{\mathit{RTe}}^{-\frac{E_a}{\left(1+\alpha \right)\mathit{RT}}};$$ E77

$${\int }_0^{\infty }{A}_1{e}^{-\frac{E}{\left(1+\alpha \right)\mathit{RT}}}\mathit{dE}=\left(1+\alpha \right)A_1\mathit{RT};$$ E78

$$\frac{{\int }_{E_a}^{\infty }{A}_1{e}^{-\frac{E}{\left(1+\alpha \right)\mathit{RT}}}\mathit{dE}}{{\int }_0^{\infty }{A}_1{e}^{-\frac{E}{\left(1+\alpha \right)\mathit{RT}}}\mathit{dE}}=e^{-\frac{E}{\left(1+\alpha \right)\mathit{RT}}}=e^{-\frac{E}{\mathit{RT}+E_{\mathit{add}}}}.$$ E79

From the aforementioned, it is apparent that the exponential form of the Boltzmann factor is maintained, and this expression is irrelevant to A ₁, just as was Eq. (75). Here, when the reaction rate constant under microwave non-irradiation is k ₀ and that under irradiation is k ₁, the Arrhenius equation is described as follows:

$$k_0=A_0{e}^{-\frac{E_a}{\mathit{RT}}};$$ E80

$$k_1=A_1{e}^{-\frac{E_a}{\mathit{RT}+E_{\mathit{add}}}}.$$ E81

Comparing both equations, k ₀ = k ₁ at E _add = 0. Therefore, A ₀ = A ₁ is derived. Eq. (81) shows that the A-factor does not change under microwave irradiation.

3.4 Introduction of a nonthermal constant

Generalizing Eq. (81) under an applied microwave irradiation yields Eq. (82):

$$k={\mathit{Ae}}^{-\frac{E_a}{\mathit{RT}+\gamma E_{\mathit{MW}}}}.$$ E82

E _MW is an intensive property that can change the applied amount from the outside. Therefore, changing k by a variable independent of temperature T is not thermal, that is, a nonthermal effect. Therefore, the coefficient γ in Eq. (82) is a nonthermal constant. Here, if γ is a positive value, the group has the same T, but the variance of the canonical distribution widens, and the occupation ratio of E _a or higher is increased and k ₀ < k ₁.

Eq. (82) does not change E _a and A. Therefore, the reaction route is not changed. This is a reaction rate equation that can be applied, even when the microwave irradiation is 0, and the irradiation intensity of the microwave is variable. In this case, the population maintains a canonical distribution. If there is a nonthermal effect, it must be expressed by a reaction equation with a microwave intensity independent of temperature as a variable. Eq. (82) obtained by deduction meets this condition.

3.5 Working principle of a nonthermal constant

Eqs. (81) or (82) can be transformed into Eq. (83). E _a and A can be obtained without microwave irradiation. If E _a and A are not changed by microwave irradiation, E _add, that is, γE_MW can be obtained from k at T. Assuming that the volume of the irradiation target is same and E_MW is proportional to the irradiation power, γ can be calculated from Eq. (83):

$$\mathit{\alpha RT}=E_{\mathit{add}}=\gamma E_{\mathit{MW}}=\frac{E_a}{ln\left(A\right)-ln\left(k\right)}-\mathit{RT}.$$ E83

If the effect of microwave irradiation is expressed by Eq. (81), the ratio r between Eqs. (81) and (80) indicates the efficiency of increase by microwave irradiation (Eq. (84)):

$$r=\frac{{\mathit{Ae}}^{-\frac{E_a}{\left(1+\alpha \right)\mathit{RT}}}}{{\mathit{Ae}}^{-\frac{E_a}{\mathit{RT}}}}=e^{\left(\frac{\alpha }{1+\alpha }\right)\frac{E_a}{\mathit{RT}}}.$$ E84

When α is 0, there is no irradiation, so r = 1. The upper limit of r is considered when α is infinite. Therefore, the upper limit r _max of r is expressed by Eq. (85):

$$r_{\mathit{ma}x}=\underset{a\to \infty }{lim}{e}^{\left(\frac{\alpha }{1+\alpha }\right)\frac{E_a}{\mathit{RT}}}=e^{\frac{E_a}{\mathit{RT}}}.$$ E85

This equation suggests that in a system having an effective α, the effect of microwave irradiation appears more markedly as the reaction temperature T is lower and E _a is higher.

4. Conclusions

Microwave chemistry was described with a focus on the behavior of dielectrics. We discussed the relationship between physical properties such as dielectric constant, energy propagation in the material, and energy loss in the material. By discussing energy, if there is a special effect other than heat in microwave chemical reactions, we have derived a priori by mathematical formulas factors that are required. The “Arrhenius equation under microwave irradiation” proposed in this section is one model. In this equation, a microwave energy term is added.

In order to confirm the results of this deduction inductively, correct measurements are required. A simple method is a surface temperature measurement with a radiation thermometer, but the fact that there is a temperature difference from the inside has been investigated in various experiments, and unless this problem is solved, it will lead to incorrect measurement. The temperature and its distribution should be validated with multiple methods, such as internal radiation measurements using an optical fiber, a fiber-grating method that measures the local volume changes of the optical fiber, and temperature dependence of the lifetime of the fluorescent material at the tip [30, 31, 32].

As presented in this paper, understanding microwave effects algebraically in terms of energy theory is different from considering energy distribution geometrically. For example, the existence of nonequilibrium local heating, which is influenced by the structure of the irradiated object, has been reported [33]. This phenomenon, which can be explained as a peculiar heating method occurring in microwave heating, is due to the geometric intensity of the electromagnetic field distribution. In this example, the unique heat distribution structure gives “geometric” microwave effects rather than “algebraic” microwave effects.

Microwaves in the GHz band have a decimator wavelength. If the size of the reaction vessel is smaller than the wavelength, the result may vary greatly depending on the place of installation and the shape of the vessel, based on the microwave interference. Therefore, in chemical synthesis assisted by microwave heating, it is necessary to consider the shape of the electromagnetic field distribution in the apparatus. Otherwise, other conditions may change at the same time that the temperature, substrate, solvent, and scale are changed. If this unintended influence is ignored, it becomes difficult to clarify general trends in the condition search, or the result will lead to excessively good or bad evaluations. Thus, microwave chemistry seems to be complicated in terms of chemistry and electromagnetic field analysis. However, it can be expected that more efficient reaction control will be possible by fully utilizing the control as an external field and utilizing it highly. In addition to the algebraic interpretation based on energetics, if the structure of the irradiated object is geometrically controlled as a metamaterial, further microwave effects will be manifested. We believe that microwave chemistry will be a useful technique that can be used to manipulate chemical synthesis by applying external energy to a simple heating reaction.