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polarizable polar molecules with linear reaction dynamics in a weak alternating field
Chemical Physics Letters 727 (2019) 66–71
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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Research paper
Dielectric relaxation of interacting/polarizable polar molecules with linear reaction dynamics in a weak alternating field
T
Tao Honga, , Zhengming Tanga, Yonghong Zhoua, Huacheng Zhub, Kama Huangb ⁎
a b
School of Electronic Information Engineering, China West Normal University, Nanchong 637002, China College of Electronics and Information Engineering, Sichuan University, Chengdu 610065, China
HIGHLIGHTS
dielectric relaxation of interacting/polarizable reacting polar system is studied. • The self-consistent field approximation is applied to the calculation. • The polarization of both systems is strongly dependent on the concentration of polar molecules. • The • The anisotropic polarizability gives rise to the complex behavior of the third harmonic. ARTICLE INFO
ABSTRACT
Keywords: Self-consistent field approximation Reaction-diffusion Dielectric relaxation
The dielectric relaxation of polar-molecule reaction with linear reaction dynamics is studied via the reactiondiffusion equation in the spherical coordinate (the modified Smoluchowski-Debye equation) based on the selfconsistent field approximation. The polarization of two kinds of reaction systems, the interacting system and the polarizable system, is derived. In the interacting system, the interaction of the polar molecules is dependent on the concentration of the molecules, resulting in the complex interacting behavior of the polarization process. In the polarizable system, the anisotropic polarizability gives rise to the complex behavior of the third harmonic.
1. Introduction Since the microwave-assisted organic synthesis was introduced by Gedye in 1986 [1], microwave heating has attracted interest in chemical engineering due to the advantages of fast heating and better yields [2–5]. However, hot spots and thermal runaway, as the two main drawbacks of microwave heating, may destroy the samples and prevent the large-scale application of microwaves to chemical engineering, since the interaction between microwaves and chemical reactions, especially liquid-phase reactions, is still unclear. Nowadays, researches on the dielectric properties of chemical reactions are mostly investigated through experiments rather than theories [6–8]. Based on Einstein’s study of the rotational Brownian motion of polar molecules [9], Debye proposed an essential theory to describe the orientation polarization of noninteracting polar molecules in a weak alternating electric field [10]. The probability distribution function of the molecular concentration can be described by the Smoluchowski-Debye equation in the spherical coordinate [11,12]
⁎
( , t) = LFP [ ( , t )], t
(1)
where LFP is the Fokker-Planck operator and is the angle the symmetry axis of polar molecule makes with the direction of the applied electric field. As the interaction of dipole moments is apparent in the dielectric relaxation of polar liquids, Onsager [13] and Kirkwood [14] improved the Debye theory by considering the static internal field. Besides, the dynamic dielectric properties of interacting polar molecules were then studied by assuming a special type of memory [15] and a forced rotational diffusion model [16,17]. Meanwhile, the influence of polarizable molecules, which can produce induced dipole moments in the presence of electric field, was usually described in anisotropic polarization which can be found in Kerr-effect theories [18–21]. On the other hand, in chemical reaction, the distribution of molecules is not only dependent on the effect of the thermal motion and external force but also the influence of the chemical reaction process. Therefore, the distribution function of the molecular concentration in chemical reaction is usually expressed by the reaction-diffusion equation in the Cartesian coordinate [22–24]
Corresponding author. E-mail address: [email protected] (T. Hong).
https://doi.org/10.1016/j.cplett.2019.04.053 Received 25 February 2019; Received in revised form 16 April 2019; Accepted 17 April 2019 Available online 19 April 2019 0009-2614/ © 2019 Elsevier B.V. All rights reserved.
Chemical Physics Letters 727 (2019) 66–71
T. Hong, et al.
(x , t ) = f [ (x , t )] + LFP [ (x , t )], t
where NA is Avogadro’s number and c (t ) is the concentration of polar molecules. Now, in order to separate the variables and t , the function of probability distribution function is expanded by the standard basis of Legendre polynomials [31]
(2)
where f is the function based on the chemical reaction dynamics. The transfer of walkers is always investigated theoretically through the equation. However, the reaction-diffusion equation was firstly solved in the spherical coordinate to research the dielectric property of polarmolecule reaction by our group recently [25]. Two vectors, the rotational diffusion vector and the component concentration vector, were proposed to describe the dielectric property. Both the interacting process and anisotropic polarization process are ignored, while both of them may exist in the chemical reaction process. In this paper, the dielectric properties of two kinds of molecules with linear reaction dynamics, including interacting polar molecules and polarizable polar molecules, are studied in the presence of a weak alternating electric field, respectively. In Sections 2 and 3, the formulas for the dielectric response in the interacting system and polarizable system of reacting polar molecules are derived through the reactiondiffusion equation in spherical coordinate, respectively. In the last section, the dielectric response in both systems is analyzed.
( , t) = n=0
P (t ) = NA µc (t )
D
sin
sin
( , t)
+
1 V ( , t) ( , t) kB T
An (t ) = t
t
)chem = k ,
A 0 (t ) = t
µF (t ) cos
=
µ [E (t )
0
2
An (t ) = t
0
2
( , t )sin d
,
(11)
(12)
[k + n (n + 1) Dr ] An (t ) + [u (t )
A1 (t )]
n (n + 1) Dr [An 2n + 1
1 (t )
An + 1 (t )].
(13)
By setting n = 1 and neglecting the higher order of the expansion coefficients, we have
A1 (t ) = t
(5)
( , t )cos sin d
(10)
Now, the infinite hierarchy for the expansion coefficients can be simplified as
2 A1 (t )] Dr A0 (t ). 3
[k + 2Dr ] A1 (t ) + [u (t )
(14)
This equation may be integrated by assuming the system to be equilibrium at t = 0
A1 (t ) =
2Dr c (t ) 3
t 0
exp(
t t
(2Dr +
2Dr c (t1)) dt1) u (t ) dt . 3
(15)
According to Eqs. (9), (12) and (15), the polarization of the interacting systems of reacting polar molecules can be expressed as
P (t ) =
(6)
2Dr NA µ c (t ) 3
t 0
exp(
t t
(2Dr +
2Dr c (t1)) dt1) u (t ) dt . 3
(16)
It is obvious that the polarization cannot be expressed by the two vectors [25,28] which are the rotational diffusion vector and the component concentration vector due to the existence of the term representing the interacting of polar molecules. Meanwhile, the interacting of the polar molecules is strongly dependent on the number of the polar molecules, while the concentration of polar molecules is decreasing with the reacting process, leading to the complexity of Eq. (16) which is shown as c (t1) .
where F (t ) is the total effective field, µ is the dipole moment, E (t ) is the electric field and P (t ) is the polarization. The polarization of the reacting polar molecules can be expressed by the function of probability distribution function ( , t ) [25]
P (t ) = NA µc (t )
kA 0 (t ).
A0 (t ) = c (t ).
,
where Us is an effective single dipole free energy and Vint represents the effect of interaction torques. The effect of interaction torques is usually proportional to the constant number of polar molecules per unit volume in matter [29,30]. However, in the isomerization chemical reaction process, the concentration of polar molecules is varying with time, leading to the decreasing number of polar molecules. We may calculate the potential V ( , t ) through the total field of a typical dipole [17]
V ( , t) =
An + 1 (t )],
The equation is just the same form of the linear reaction dynamics, hence we may get
where k is the rate constant. The potential V ( , t ) can be described as
4 P (t )]cos , 3
1 (t )
4 NA µ2
µE (t )
(4)
V ( , t ) = Us ( , t ) + Vint ( , t ).
c (t ) A1 (t ) n (n + 1) ] Dr [An A 0 (t ) 2n + 1
where u (t ) = k T and = 3k T . It is obvious that the analytical exB B pression of each expansion coefficient can not be achieved without any simplification of the hierarchy. In practical situation, the electric field is 1. Hence, we only need to get the usually weak enough so that u (t ) analytical expressions of the first two expansion coefficients. By setting n = 0 , Eq. (10) can be reduced to
where Dr is the rotational diffusion coefficient, kB is the Boltzmann constant, T is the absolute temperature, v is the effect of the chemical reaction rate and V ( , t ) is the potential. The effect of the chemical reaction rate can be expressed by the chemical reaction dynamics [25]
(
(9)
[k + n (n + 1) Dr ] An (t ) + [u (t )
(3)
v=
A1 (t ) . A 0 (t )
Hence, by combining Eqs. (3), (4), (6), (8) and (9), we can get the infinite hierarchy of differential recurrence relations for the expansion coefficients of Legendre polynomials
In this section, we consider a system consisting of independent groups of interacting polar molecules with linear reaction dynamics based on the assumption of self-consistent field approximation [16,17]. In order to take the chemical reaction process into consideration, a kind of isomerization reaction is discussed. In this kind of reaction, the polar molecules will undergo removals at a constant per capita rate and then become nonpolar. Thus, the investigation may be started with the reaction-diffusion equation (the modified Smoluchowski-Debye equation) for the probability distribution function of the molecular concentration in the spherical coordinate [25–28].
v+
(8)
By taking Eq. (8) into Eq. (7), and using the orthogonality of Legendre polynomials, we obtain
2. Dielectric response in interacting system of reacting polar molecules
( , t) = t
2n + 1 An (t ) Pn (cos ). 2
(7) 67
Chemical Physics Letters 727 (2019) 66–71
T. Hong, et al.
3. Dielectric response in polarizable system of reacting polar molecules
4. Results and discussion In Sections 2 and 3, we have derived the general analytical expressions Eqs. (16) and (23), for the polarization of both the interacting system and the polarizable system of reacting polar molecules, respectively. The results show that the polarization process is complicated in both systems and it is challenging to analyze the polarization in the frequency domain due to the existence of the reaction process. In order to compare the results with the former one [25,28], the electric field is considered as a time-harmonic field
In this section, we consider independent groups of polarizable polar molecules with linear reaction dynamics via anisotropic rotational diffusion [19,20]. The polar molecules will undergo the same reaction process as the one in Section 2. Hence, the reaction-diffusion equation (the modified Smoluchowski-Debye equation) for the probability distribution function of the molecular concentration can be also described by Eq. (3). However, the potential V ( , t ) in this kind of system can be given by
V ( , t ) = Us ( , t ) + Viso ( , t ) =
µE (t ) cos
1 2
E 2 (t )cos2
,
where E0 is the amplitude of the electric field and is the angular frequency of the electric field. In addition, both systems are assumed to be equilibrium at t = 0 and the reaction process begins at this moment. By expanding the exponential in the integral of Eq. (16) with Taylor’s series, we have
(17)
where Viso is the effect of anisotropic polarizability and is the difference between the principal electric polarizabilities, parallel and perpendicular to the symmetry axis, respectively. Similar to Section 2, the variables and t can be separated in the probability distribution function of the molecular concentration in the spherical coordinate by combining Eqs. (3), (4), (8) and (17), and the infinite hierarchy of differential recurrence relations for the expansion coefficients of Legendre polynomials in this system can be expressed by
An (t ) = t
k + n (n + 1) Dr (1 n (n + 1) + Dr u (t )[An 2n + 1 An+ 1 (t )] + Dr (t )
P (t ) =
P (t ) =
1) n (n + 1) An 1)(2n + 1)
kA 0 (t ).
(t ) 2 ) A1 (t ) + Dr u (t ) A 0 (t ). 5 3
P (t ) =
0
t
(2Dr
2Dr (t1)) dt1) u (t ) dt . 5
2Dr NA µ c (t ) 3
t 0
exp(
t t
(2Dr
2Dr (t1)) dt1) u (t ) dt . 5
2Dr NA µ e 3
2Dr t *u (t )
]+
4Dr2 NA µ 9k
c 2 (t ) [(e
2Dr t
(2Dr k ) t
e
)
(27)
4Dr2 NA µ 9
2Dr t ,
2Dr NA µu0 c (t ) 3
2 + (2D
(e
2Dr t
e
(2Dr k ) t
)]
(28)
r
k)
1+
2Dr c (t ) 3k 2 + 4D 2 r
[ sin( t ) + 2Dr cos( t )
2Dr e
2Dr t ]
,
2 [ sin( t ) + (2Dr
k ) cos( t )
(2Dr
k) e
2Dr t ]
(29) µE
where u 0 = k T0 . B If the reaction dynamics is ignored and the integration range is extended to the infinity, Eq. (16) can be reduced to the steady response obtained by Déjardin [17] and Deshmukh [32], viz.,
(21)
P (t ) = NA c0 u0 µ (22)
3+ 1 2 1) + 9
(3 +
2 2
cos t +
(3 +
3 t 2 + 9
1)
2 2
sin t , (30)
Hence the polarization of the polarizable systems of reacting polar molecules can be given by
P (t ) =
[e
,
2Dr c (t ) 3k
If the system is assumed to be equilibrium at t = 0 , we can get t
(25)
is the rotational diffusion vector and * represents the convolution. Hence, the polarization of the interacting system may be expressed as two vectors which are similar to those in our previous works [25,28]. However, in the component concentration vector of Eq. (26), the rate constant concerned with the chemical reaction process, which should be included in the component concentration vector, is shown. By taking Eq. (24) into Eq. (26), we get
By setting n = 1 and neglecting the higher order of the expansion coefficients, we have
exp(
2Dr NA µ c (t ) 3
(t ) = [
(20)
t
2Dr c (t1) dt1) u (t ) dt . 3
is the component concentration vector,
(19)
A0 (t ) = c (t ).
2Dr c (t ) 3
t
C (t ) = [c (t ), c 2 (t )]
(18)
Eq. (19) is the same form of Eq. (11), hence we can also get
A1 (t ) =
t
(1
where
E2 (t )
k + 2Dr (1
2Dr (t t )
(26)
2 (t )
where (t ) = k T . It can be seen that the expansion of the poB larizable system is different from the one of the interacting system as shown in Eq. (10) due to the effect of the anisotropic polarizability and the interacting of polar molecules. And the analytical expression of each expansion coefficient cannot be achieved without any simplification either. But the polarization can be obtained by deriving the analytical expressions of the first two expansion coefficients which is similar to Section 2. Eq. (18) can be simplified by setting n = 0
A1 (t ) = t
e
= C (t )[ (t )*u (t )]
n (n + 1)(n + 2) An + 2 (t ) , (2n + 1)(2n + 3)
A 0 (t ) = t
t 0
*u (t ) ]
1 (t )
(n (2n
2Dr NA µ c (t ) 3
If the integration range is extended to the infinity, the polarization of the interacting system can be expressed by
(t ) ) An (t ) 1)(2n + 3)
(2n
(24)
E (t ) = E0 cos( t ),
where c0 is the initial concentration, 1 = c0 and = Similarly, by expanding the exponential in the integral of Eq. (23) with Taylor’s series, we have 1 . 2Dr
(23)
P (t ) =
It can be seen that the polarization cannot be expressed by the two vectors either due to the existence of the term representing the anisotropic polarizability of polar molecules.
2Dr NA µ c (t ) 3
t 0
e
2Dr (t t )
(1 +
t t
2Dr (t1) dt1) u (t ) dt 5
(31)
The polarization of polarizable system may not be simplified due to the higher order of the electric field. By combining Eqs. (24) and (31), 68
Chemical Physics Letters 727 (2019) 66–71
T. Hong, et al.
Fig. 1. The probability distribution function (normalized by c(0)) of the polar-molecule concentration in the interacting system.
we can obtain the polarization of the polarizable system
P (t ) =
+
2Dr NA µu0 c (t ) 3
{
80Dr3 + 20Dr 2 + 12Dr3 0 10(4Dr2 + 2)2
+
20Dr2 + 5 3 + 4Dr2 0 5(4Dr2 + 2)2
+
8Dr2 0 10(4Dr2 + 2)(4Dr2 + 9 2)
Dr 0 10
sin( t ) + sin(3 t )
Dr 0 2
cos( t )
4Dr3 0 3Dr 0 2 10(4Dr2 + 2)(4Dr2 + 9 2) 2Dr
e 4Dr2 + 2
are vanishing with the reaction process. On the other hand, in the time harmonic electric field, the polarization is oscillating and reducing with the reaction process which is similar to the process of damped oscillation. In both situations, the amplitude of the polarization is strongly dependent on the interacting of polar molecules. The stronger the interacting is, the smaller amplitude of polarization can be found. The probability distribution function of the polar-molecule concentration in the polarizable system is shown at various time in the cases u 0 = 0.1, k = 0.1, Dr = 2, 0 = - 2, 0 or 2 and = 0 or 1 in Fig. 3. Similar results can be found that the mean value of the probability distribution function decreases with increasing time due to the reaction process. If the parallel polarizability is stronger than the perpendicular one, the more heterogeneous distribution of molecules can be found in the presence of the static electric field. On the other hand, the frequency of the electric field will also affect the value of the probability distribution function. In Fig. 4, the polarization of the reacting polar molecules in the polarizable system is shown in the cases u 0 = 0.1, k = 0.1, Dr = 2, 2, 0 or 2 and = 0 or 1. The higher peak of polarization can be 0 = observed with the larger parallel polarizability at the same moment, since the polar molecules in the direction of the electric filed are more than those in the other directions which can be seen in Fig. 3. In the time harmonic electric field, the polarization is oscillating and reducing with the reaction process. However, the oscillation in the polarizable system is different from the one in the interacting system due to the existence of the third harmonic electric field. In both systems, the polarization is strongly dependent on the concentration of the polar molecules. Hence, the polarization of both systems is composed of different frequency components, leading to the challenge to the analysis in the frequency domain. If the interaction or
2Dr t (1
cos(3 t )
+
Dr 0 t 5
,
sin(2 t ))
2 Dr 0 2D t 4Dr2 e r 5 (4Dr2 + 2)2
Dr 0 2D t 3 e r ( 2 20 4Dr + 9 2
+
1 ) 4Dr2 + 2
}
(32)
E02
where 0 = k T . B In Fig. 1, the probability distribution function of the polar-molecule concentration in the interacting system is shown at various time in the cases u 0 = 0.1, k = 0.1, Dr = 2, c (0) = 0, 0.5 or 2 and = 0 or 1. Here c (0) is the initial concentration of polar molecules. We can find that the mean value of the probability distribution function in the interacting system decreases with time due to the reaction process. Meanwhile, the conclusion can be made that the stronger the interacting is, the more homogeneous distribution of molecules can be found in the presence of the static electric field. While in the harmonic electric field, the value of the probability distribution function is affected by the frequency of the electric field, and the different probability distributions will be consistent with the reaction process in different interacting situations. In Fig. 2, the polarization of the reacting polar molecules in the interacting system is shown in the cases u 0 = 0.1, k = 0.1, Dr = 2, c (0) = 0, 0.5 or 2 and = 0 or 1. As shown in Fig. 2(a), in the presence of the static electric field, the polarization reaches the maximum peak rapidly and then decreases with time, since the polar molecules 69
Chemical Physics Letters 727 (2019) 66–71
T. Hong, et al.
Fig. 2. The polarization (normalized by NA µc (0) ) of the reacting polar molecules in the interacting system.
the anisotropic polarizability of polar molecules is ignored, the results in both systems will be the same as those in our previous works. Once the linear reaction dynamics is ignored, the polarization of the interacting or polarizable system can be reduced to the linear one in literature [17] and [19], respectively. We may also notice that the results shown in Eqs. (16) and (23) are based on the self-consistent field approximation, and the probability distribution function factorizes in a product of one-body function so that the probability distribution function in this paper is unity. Therefore, the results in this paper are just qualitative ones and can describe the short-range interactions. Meanwhile, there barely exist experiments
that concern both the chemical reaction process and dielectric relaxation. Still, we have proved that the dielectric relaxation of chemical reaction is dependent on both the molecular concentration, which is determined by the chemical reaction process, and the rotational diffusion property, which mainly relies on the temperature, at a certain frequency. The theoretical results may explain the formulas used in the Ref. [7]. If the stronger (long-range) interaction is discussed, it may be necessary to consider the multi-body function [33,34] or the fractional relaxation equation [35,36].
Fig. 3. The probability distribution function (normalized by c(0)) of the polar-molecule concentration in the polarizable system. 70
Chemical Physics Letters 727 (2019) 66–71
T. Hong, et al.
Fig. 4. The polarization (normalized by NA µc (0) ) of the reacting polar molecules in the polarizable system.
5. Conclusion
[3] A. de la Hoz, A. Diaz-Ortiz, A. Moreno, Chem. Soc. Rev. 34 (2005) 164. [4] C.O. Kappe, Chem. Soc. Rev. 37 (2008) 1127. [5] W. Zhang, B.W. Cue, Green Techniques for Organic Synthesis and Medicinal Chemistry, John Wiley & Sons, Weinheim, Germany, 2018. [6] K. Huang, X. Cao, C. Liu, X.B. Xu, IEEE Trans. Microw. Theory Tech. 51 (2003) 2106. [7] K. Huang, H. Zhu, L. Wu, Bioresource Technol. 131 (2013) 541. [8] H.C. Zhu, J.Q. Lan, L. Wu, T. Gulati, Q. Chen, T. Hong, K.M. Huang, Int. J. Appl. Electrom. 47 (2015) 927. [9] A. Einstein, Investigations on the Theory of the Brownian Movement, Dover, New York, 1956. [10] P. Debye, Polar Molecules, Chem. Catalog. Co., New York, 1929. [11] J.L. Déjardin, Y.P. Kalmykov, Phys. Rev. E 61 (2000) 1211. [12] N. Wei, P.M. Déjardin, Y.P. Kalmykov, W.T. Coffey, Phys. Rev. E 93 (2016). [13] L. Onsager, J. Am. Chem. Soc. 58 (1936) 1486. [14] J.G. Kirkwood, J. Chem. Phys. 7 (1939) 911. [15] T.W. Nee, R. Zwanzig, J. Chem. Phys. 52 (1970) 6353. [16] B.J. Berne, J. Chem. Phys. 62 (1975) 1154. [17] P.M. Déjardin, F. Ladieu, J. Chem. Phys. 140 (2014). [18] H. Benoit, Ann. Phys. 6 (1951) 561. [19] J.L. Déjardin, G. Debiais, A. Ouadjou, J. Chem. Phys. 98 (1993) 8149. [20] J.L. Déjardin, Y.P. Kalmykov, J. Chem. Phys. 112 (2000) 2916. [21] B.X. Li, V. Borshch, S.V. Shiyanovskii, S.B. Liu, O.D. Lavrentovich, Phys. Rev. E 92 (2015). [22] J. Zhu, O.B. Spirina, R.I. Cukier, J. Chem. Phys. 100 (1994) 8109. [23] J. Zhu, R. Ma, Y. Lu, G. Stell, J. Chem. Phys. 123 (2005). [24] K. Dhole, B. Modak, A. Samanta, S.K. Ghosh, Phys. Rev. E 82 (2010). [25] K. Huang, T. Hong, J. Phys. Chem. A 119 (2015) 8898. [26] L. Monchick, J. Chem. Phys. 24 (1956) 381. [27] G. Schwarz, J. Phys. Chem. 71 (1967) 4021. [28] T. Hong, Z. Tang, H. Zhu, J. Chem. Phys. 145 (2016). [29] M. Warchol, W.E. Vaughan, J. Chem. Phys. 71 (1979) 502. [30] P.M. Déjardin, J. Appl. Phys. 110 (2011). [31] Z. Sekkat, J. Wood, W. Knoll, J. Phys. Chem. 99 (1995) 17226. [32] S.D. Deshmukh, P.M. Déjardin, Y.P. Kalmykov, J. Chem. Phys. 147 (2017). [33] P.M. Déjardin, Y. Cornaton, P. Ghesquière, C. Caliot, R. Brouzet, J. Chem. Phys. 148 (2018). [34] P.M. Déjardin, S.V. Titov, Y. Cornaton, Phys. Rev. B 99 (2019). [35] R. Metzler, E. Barkai, J. Klafter, Phys. Rev. Lett. 82 (1999) 3563. [36] B. Dumitru, D. Kai, S. Enrico, Fractional Calculus: Models and Numerical Methods, World Scientific, Singapore, 2012.
In this paper, the dielectric polarization of two polar-molecule systems with linear reaction dynamics, including the interacting system and the polarizable system, is derived. The polarization of both systems is quite different from the polarization in conventional matter due to the existence of the chemical reaction process. In the interacting system, the decreasing number of polar molecules leads to the weaker interaction of polar molecules and gives rise to the more homogeneous distribution of polar molecules. The polarization of this kind of system then becomes similar to the process of damped oscillation, which can be expressed by the component concentration vector and the rotational diffusion vector. In the polarizable system, the polarization process becomes similar to the process of two kinds of damped oscillations, since the third harmonic exists due to the anisotropic polarizability. Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 61731013) and Sichuan Science and Technology Program (Grant No. 2018FZ0008). References [1] R. Gedye, F. Smith, K. Westaway, H. Ali, L. Baldisera, L. Laberge, J. Rousell, Tetrahedron Lett. 27 (1986) 279. [2] D.M.P. Mingos, D.R. Baghurst, Chem. Soc. Rev. 20 (1991) 1.
71
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Research paper
Dielectric relaxation of interacting/polarizable polar molecules with linear reaction dynamics in a weak alternating field
T
Tao Honga, , Zhengming Tanga, Yonghong Zhoua, Huacheng Zhub, Kama Huangb ⁎
a b
School of Electronic Information Engineering, China West Normal University, Nanchong 637002, China College of Electronics and Information Engineering, Sichuan University, Chengdu 610065, China
HIGHLIGHTS
dielectric relaxation of interacting/polarizable reacting polar system is studied. • The self-consistent field approximation is applied to the calculation. • The polarization of both systems is strongly dependent on the concentration of polar molecules. • The • The anisotropic polarizability gives rise to the complex behavior of the third harmonic. ARTICLE INFO
ABSTRACT
Keywords: Self-consistent field approximation Reaction-diffusion Dielectric relaxation
The dielectric relaxation of polar-molecule reaction with linear reaction dynamics is studied via the reactiondiffusion equation in the spherical coordinate (the modified Smoluchowski-Debye equation) based on the selfconsistent field approximation. The polarization of two kinds of reaction systems, the interacting system and the polarizable system, is derived. In the interacting system, the interaction of the polar molecules is dependent on the concentration of the molecules, resulting in the complex interacting behavior of the polarization process. In the polarizable system, the anisotropic polarizability gives rise to the complex behavior of the third harmonic.
1. Introduction Since the microwave-assisted organic synthesis was introduced by Gedye in 1986 [1], microwave heating has attracted interest in chemical engineering due to the advantages of fast heating and better yields [2–5]. However, hot spots and thermal runaway, as the two main drawbacks of microwave heating, may destroy the samples and prevent the large-scale application of microwaves to chemical engineering, since the interaction between microwaves and chemical reactions, especially liquid-phase reactions, is still unclear. Nowadays, researches on the dielectric properties of chemical reactions are mostly investigated through experiments rather than theories [6–8]. Based on Einstein’s study of the rotational Brownian motion of polar molecules [9], Debye proposed an essential theory to describe the orientation polarization of noninteracting polar molecules in a weak alternating electric field [10]. The probability distribution function of the molecular concentration can be described by the Smoluchowski-Debye equation in the spherical coordinate [11,12]
⁎
( , t) = LFP [ ( , t )], t
(1)
where LFP is the Fokker-Planck operator and is the angle the symmetry axis of polar molecule makes with the direction of the applied electric field. As the interaction of dipole moments is apparent in the dielectric relaxation of polar liquids, Onsager [13] and Kirkwood [14] improved the Debye theory by considering the static internal field. Besides, the dynamic dielectric properties of interacting polar molecules were then studied by assuming a special type of memory [15] and a forced rotational diffusion model [16,17]. Meanwhile, the influence of polarizable molecules, which can produce induced dipole moments in the presence of electric field, was usually described in anisotropic polarization which can be found in Kerr-effect theories [18–21]. On the other hand, in chemical reaction, the distribution of molecules is not only dependent on the effect of the thermal motion and external force but also the influence of the chemical reaction process. Therefore, the distribution function of the molecular concentration in chemical reaction is usually expressed by the reaction-diffusion equation in the Cartesian coordinate [22–24]
Corresponding author. E-mail address: [email protected] (T. Hong).
https://doi.org/10.1016/j.cplett.2019.04.053 Received 25 February 2019; Received in revised form 16 April 2019; Accepted 17 April 2019 Available online 19 April 2019 0009-2614/ © 2019 Elsevier B.V. All rights reserved.
Chemical Physics Letters 727 (2019) 66–71
T. Hong, et al.
(x , t ) = f [ (x , t )] + LFP [ (x , t )], t
where NA is Avogadro’s number and c (t ) is the concentration of polar molecules. Now, in order to separate the variables and t , the function of probability distribution function is expanded by the standard basis of Legendre polynomials [31]
(2)
where f is the function based on the chemical reaction dynamics. The transfer of walkers is always investigated theoretically through the equation. However, the reaction-diffusion equation was firstly solved in the spherical coordinate to research the dielectric property of polarmolecule reaction by our group recently [25]. Two vectors, the rotational diffusion vector and the component concentration vector, were proposed to describe the dielectric property. Both the interacting process and anisotropic polarization process are ignored, while both of them may exist in the chemical reaction process. In this paper, the dielectric properties of two kinds of molecules with linear reaction dynamics, including interacting polar molecules and polarizable polar molecules, are studied in the presence of a weak alternating electric field, respectively. In Sections 2 and 3, the formulas for the dielectric response in the interacting system and polarizable system of reacting polar molecules are derived through the reactiondiffusion equation in spherical coordinate, respectively. In the last section, the dielectric response in both systems is analyzed.
( , t) = n=0
P (t ) = NA µc (t )
D
sin
sin
( , t)
+
1 V ( , t) ( , t) kB T
An (t ) = t
t
)chem = k ,
A 0 (t ) = t
µF (t ) cos
=
µ [E (t )
0
2
An (t ) = t
0
2
( , t )sin d
,
(11)
(12)
[k + n (n + 1) Dr ] An (t ) + [u (t )
A1 (t )]
n (n + 1) Dr [An 2n + 1
1 (t )
An + 1 (t )].
(13)
By setting n = 1 and neglecting the higher order of the expansion coefficients, we have
A1 (t ) = t
(5)
( , t )cos sin d
(10)
Now, the infinite hierarchy for the expansion coefficients can be simplified as
2 A1 (t )] Dr A0 (t ). 3
[k + 2Dr ] A1 (t ) + [u (t )
(14)
This equation may be integrated by assuming the system to be equilibrium at t = 0
A1 (t ) =
2Dr c (t ) 3
t 0
exp(
t t
(2Dr +
2Dr c (t1)) dt1) u (t ) dt . 3
(15)
According to Eqs. (9), (12) and (15), the polarization of the interacting systems of reacting polar molecules can be expressed as
P (t ) =
(6)
2Dr NA µ c (t ) 3
t 0
exp(
t t
(2Dr +
2Dr c (t1)) dt1) u (t ) dt . 3
(16)
It is obvious that the polarization cannot be expressed by the two vectors [25,28] which are the rotational diffusion vector and the component concentration vector due to the existence of the term representing the interacting of polar molecules. Meanwhile, the interacting of the polar molecules is strongly dependent on the number of the polar molecules, while the concentration of polar molecules is decreasing with the reacting process, leading to the complexity of Eq. (16) which is shown as c (t1) .
where F (t ) is the total effective field, µ is the dipole moment, E (t ) is the electric field and P (t ) is the polarization. The polarization of the reacting polar molecules can be expressed by the function of probability distribution function ( , t ) [25]
P (t ) = NA µc (t )
kA 0 (t ).
A0 (t ) = c (t ).
,
where Us is an effective single dipole free energy and Vint represents the effect of interaction torques. The effect of interaction torques is usually proportional to the constant number of polar molecules per unit volume in matter [29,30]. However, in the isomerization chemical reaction process, the concentration of polar molecules is varying with time, leading to the decreasing number of polar molecules. We may calculate the potential V ( , t ) through the total field of a typical dipole [17]
V ( , t) =
An + 1 (t )],
The equation is just the same form of the linear reaction dynamics, hence we may get
where k is the rate constant. The potential V ( , t ) can be described as
4 P (t )]cos , 3
1 (t )
4 NA µ2
µE (t )
(4)
V ( , t ) = Us ( , t ) + Vint ( , t ).
c (t ) A1 (t ) n (n + 1) ] Dr [An A 0 (t ) 2n + 1
where u (t ) = k T and = 3k T . It is obvious that the analytical exB B pression of each expansion coefficient can not be achieved without any simplification of the hierarchy. In practical situation, the electric field is 1. Hence, we only need to get the usually weak enough so that u (t ) analytical expressions of the first two expansion coefficients. By setting n = 0 , Eq. (10) can be reduced to
where Dr is the rotational diffusion coefficient, kB is the Boltzmann constant, T is the absolute temperature, v is the effect of the chemical reaction rate and V ( , t ) is the potential. The effect of the chemical reaction rate can be expressed by the chemical reaction dynamics [25]
(
(9)
[k + n (n + 1) Dr ] An (t ) + [u (t )
(3)
v=
A1 (t ) . A 0 (t )
Hence, by combining Eqs. (3), (4), (6), (8) and (9), we can get the infinite hierarchy of differential recurrence relations for the expansion coefficients of Legendre polynomials
In this section, we consider a system consisting of independent groups of interacting polar molecules with linear reaction dynamics based on the assumption of self-consistent field approximation [16,17]. In order to take the chemical reaction process into consideration, a kind of isomerization reaction is discussed. In this kind of reaction, the polar molecules will undergo removals at a constant per capita rate and then become nonpolar. Thus, the investigation may be started with the reaction-diffusion equation (the modified Smoluchowski-Debye equation) for the probability distribution function of the molecular concentration in the spherical coordinate [25–28].
v+
(8)
By taking Eq. (8) into Eq. (7), and using the orthogonality of Legendre polynomials, we obtain
2. Dielectric response in interacting system of reacting polar molecules
( , t) = t
2n + 1 An (t ) Pn (cos ). 2
(7) 67
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3. Dielectric response in polarizable system of reacting polar molecules
4. Results and discussion In Sections 2 and 3, we have derived the general analytical expressions Eqs. (16) and (23), for the polarization of both the interacting system and the polarizable system of reacting polar molecules, respectively. The results show that the polarization process is complicated in both systems and it is challenging to analyze the polarization in the frequency domain due to the existence of the reaction process. In order to compare the results with the former one [25,28], the electric field is considered as a time-harmonic field
In this section, we consider independent groups of polarizable polar molecules with linear reaction dynamics via anisotropic rotational diffusion [19,20]. The polar molecules will undergo the same reaction process as the one in Section 2. Hence, the reaction-diffusion equation (the modified Smoluchowski-Debye equation) for the probability distribution function of the molecular concentration can be also described by Eq. (3). However, the potential V ( , t ) in this kind of system can be given by
V ( , t ) = Us ( , t ) + Viso ( , t ) =
µE (t ) cos
1 2
E 2 (t )cos2
,
where E0 is the amplitude of the electric field and is the angular frequency of the electric field. In addition, both systems are assumed to be equilibrium at t = 0 and the reaction process begins at this moment. By expanding the exponential in the integral of Eq. (16) with Taylor’s series, we have
(17)
where Viso is the effect of anisotropic polarizability and is the difference between the principal electric polarizabilities, parallel and perpendicular to the symmetry axis, respectively. Similar to Section 2, the variables and t can be separated in the probability distribution function of the molecular concentration in the spherical coordinate by combining Eqs. (3), (4), (8) and (17), and the infinite hierarchy of differential recurrence relations for the expansion coefficients of Legendre polynomials in this system can be expressed by
An (t ) = t
k + n (n + 1) Dr (1 n (n + 1) + Dr u (t )[An 2n + 1 An+ 1 (t )] + Dr (t )
P (t ) =
P (t ) =
1) n (n + 1) An 1)(2n + 1)
kA 0 (t ).
(t ) 2 ) A1 (t ) + Dr u (t ) A 0 (t ). 5 3
P (t ) =
0
t
(2Dr
2Dr (t1)) dt1) u (t ) dt . 5
2Dr NA µ c (t ) 3
t 0
exp(
t t
(2Dr
2Dr (t1)) dt1) u (t ) dt . 5
2Dr NA µ e 3
2Dr t *u (t )
]+
4Dr2 NA µ 9k
c 2 (t ) [(e
2Dr t
(2Dr k ) t
e
)
(27)
4Dr2 NA µ 9
2Dr t ,
2Dr NA µu0 c (t ) 3
2 + (2D
(e
2Dr t
e
(2Dr k ) t
)]
(28)
r
k)
1+
2Dr c (t ) 3k 2 + 4D 2 r
[ sin( t ) + 2Dr cos( t )
2Dr e
2Dr t ]
,
2 [ sin( t ) + (2Dr
k ) cos( t )
(2Dr
k) e
2Dr t ]
(29) µE
where u 0 = k T0 . B If the reaction dynamics is ignored and the integration range is extended to the infinity, Eq. (16) can be reduced to the steady response obtained by Déjardin [17] and Deshmukh [32], viz.,
(21)
P (t ) = NA c0 u0 µ (22)
3+ 1 2 1) + 9
(3 +
2 2
cos t +
(3 +
3 t 2 + 9
1)
2 2
sin t , (30)
Hence the polarization of the polarizable systems of reacting polar molecules can be given by
P (t ) =
[e
,
2Dr c (t ) 3k
If the system is assumed to be equilibrium at t = 0 , we can get t
(25)
is the rotational diffusion vector and * represents the convolution. Hence, the polarization of the interacting system may be expressed as two vectors which are similar to those in our previous works [25,28]. However, in the component concentration vector of Eq. (26), the rate constant concerned with the chemical reaction process, which should be included in the component concentration vector, is shown. By taking Eq. (24) into Eq. (26), we get
By setting n = 1 and neglecting the higher order of the expansion coefficients, we have
exp(
2Dr NA µ c (t ) 3
(t ) = [
(20)
t
2Dr c (t1) dt1) u (t ) dt . 3
is the component concentration vector,
(19)
A0 (t ) = c (t ).
2Dr c (t ) 3
t
C (t ) = [c (t ), c 2 (t )]
(18)
Eq. (19) is the same form of Eq. (11), hence we can also get
A1 (t ) =
t
(1
where
E2 (t )
k + 2Dr (1
2Dr (t t )
(26)
2 (t )
where (t ) = k T . It can be seen that the expansion of the poB larizable system is different from the one of the interacting system as shown in Eq. (10) due to the effect of the anisotropic polarizability and the interacting of polar molecules. And the analytical expression of each expansion coefficient cannot be achieved without any simplification either. But the polarization can be obtained by deriving the analytical expressions of the first two expansion coefficients which is similar to Section 2. Eq. (18) can be simplified by setting n = 0
A1 (t ) = t
e
= C (t )[ (t )*u (t )]
n (n + 1)(n + 2) An + 2 (t ) , (2n + 1)(2n + 3)
A 0 (t ) = t
t 0
*u (t ) ]
1 (t )
(n (2n
2Dr NA µ c (t ) 3
If the integration range is extended to the infinity, the polarization of the interacting system can be expressed by
(t ) ) An (t ) 1)(2n + 3)
(2n
(24)
E (t ) = E0 cos( t ),
where c0 is the initial concentration, 1 = c0 and = Similarly, by expanding the exponential in the integral of Eq. (23) with Taylor’s series, we have 1 . 2Dr
(23)
P (t ) =
It can be seen that the polarization cannot be expressed by the two vectors either due to the existence of the term representing the anisotropic polarizability of polar molecules.
2Dr NA µ c (t ) 3
t 0
e
2Dr (t t )
(1 +
t t
2Dr (t1) dt1) u (t ) dt 5
(31)
The polarization of polarizable system may not be simplified due to the higher order of the electric field. By combining Eqs. (24) and (31), 68
Chemical Physics Letters 727 (2019) 66–71
T. Hong, et al.
Fig. 1. The probability distribution function (normalized by c(0)) of the polar-molecule concentration in the interacting system.
we can obtain the polarization of the polarizable system
P (t ) =
+
2Dr NA µu0 c (t ) 3
{
80Dr3 + 20Dr 2 + 12Dr3 0 10(4Dr2 + 2)2
+
20Dr2 + 5 3 + 4Dr2 0 5(4Dr2 + 2)2
+
8Dr2 0 10(4Dr2 + 2)(4Dr2 + 9 2)
Dr 0 10
sin( t ) + sin(3 t )
Dr 0 2
cos( t )
4Dr3 0 3Dr 0 2 10(4Dr2 + 2)(4Dr2 + 9 2) 2Dr
e 4Dr2 + 2
are vanishing with the reaction process. On the other hand, in the time harmonic electric field, the polarization is oscillating and reducing with the reaction process which is similar to the process of damped oscillation. In both situations, the amplitude of the polarization is strongly dependent on the interacting of polar molecules. The stronger the interacting is, the smaller amplitude of polarization can be found. The probability distribution function of the polar-molecule concentration in the polarizable system is shown at various time in the cases u 0 = 0.1, k = 0.1, Dr = 2, 0 = - 2, 0 or 2 and = 0 or 1 in Fig. 3. Similar results can be found that the mean value of the probability distribution function decreases with increasing time due to the reaction process. If the parallel polarizability is stronger than the perpendicular one, the more heterogeneous distribution of molecules can be found in the presence of the static electric field. On the other hand, the frequency of the electric field will also affect the value of the probability distribution function. In Fig. 4, the polarization of the reacting polar molecules in the polarizable system is shown in the cases u 0 = 0.1, k = 0.1, Dr = 2, 2, 0 or 2 and = 0 or 1. The higher peak of polarization can be 0 = observed with the larger parallel polarizability at the same moment, since the polar molecules in the direction of the electric filed are more than those in the other directions which can be seen in Fig. 3. In the time harmonic electric field, the polarization is oscillating and reducing with the reaction process. However, the oscillation in the polarizable system is different from the one in the interacting system due to the existence of the third harmonic electric field. In both systems, the polarization is strongly dependent on the concentration of the polar molecules. Hence, the polarization of both systems is composed of different frequency components, leading to the challenge to the analysis in the frequency domain. If the interaction or
2Dr t (1
cos(3 t )
+
Dr 0 t 5
,
sin(2 t ))
2 Dr 0 2D t 4Dr2 e r 5 (4Dr2 + 2)2
Dr 0 2D t 3 e r ( 2 20 4Dr + 9 2
+
1 ) 4Dr2 + 2
}
(32)
E02
where 0 = k T . B In Fig. 1, the probability distribution function of the polar-molecule concentration in the interacting system is shown at various time in the cases u 0 = 0.1, k = 0.1, Dr = 2, c (0) = 0, 0.5 or 2 and = 0 or 1. Here c (0) is the initial concentration of polar molecules. We can find that the mean value of the probability distribution function in the interacting system decreases with time due to the reaction process. Meanwhile, the conclusion can be made that the stronger the interacting is, the more homogeneous distribution of molecules can be found in the presence of the static electric field. While in the harmonic electric field, the value of the probability distribution function is affected by the frequency of the electric field, and the different probability distributions will be consistent with the reaction process in different interacting situations. In Fig. 2, the polarization of the reacting polar molecules in the interacting system is shown in the cases u 0 = 0.1, k = 0.1, Dr = 2, c (0) = 0, 0.5 or 2 and = 0 or 1. As shown in Fig. 2(a), in the presence of the static electric field, the polarization reaches the maximum peak rapidly and then decreases with time, since the polar molecules 69
Chemical Physics Letters 727 (2019) 66–71
T. Hong, et al.
Fig. 2. The polarization (normalized by NA µc (0) ) of the reacting polar molecules in the interacting system.
the anisotropic polarizability of polar molecules is ignored, the results in both systems will be the same as those in our previous works. Once the linear reaction dynamics is ignored, the polarization of the interacting or polarizable system can be reduced to the linear one in literature [17] and [19], respectively. We may also notice that the results shown in Eqs. (16) and (23) are based on the self-consistent field approximation, and the probability distribution function factorizes in a product of one-body function so that the probability distribution function in this paper is unity. Therefore, the results in this paper are just qualitative ones and can describe the short-range interactions. Meanwhile, there barely exist experiments
that concern both the chemical reaction process and dielectric relaxation. Still, we have proved that the dielectric relaxation of chemical reaction is dependent on both the molecular concentration, which is determined by the chemical reaction process, and the rotational diffusion property, which mainly relies on the temperature, at a certain frequency. The theoretical results may explain the formulas used in the Ref. [7]. If the stronger (long-range) interaction is discussed, it may be necessary to consider the multi-body function [33,34] or the fractional relaxation equation [35,36].
Fig. 3. The probability distribution function (normalized by c(0)) of the polar-molecule concentration in the polarizable system. 70
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T. Hong, et al.
Fig. 4. The polarization (normalized by NA µc (0) ) of the reacting polar molecules in the polarizable system.
5. Conclusion
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In this paper, the dielectric polarization of two polar-molecule systems with linear reaction dynamics, including the interacting system and the polarizable system, is derived. The polarization of both systems is quite different from the polarization in conventional matter due to the existence of the chemical reaction process. In the interacting system, the decreasing number of polar molecules leads to the weaker interaction of polar molecules and gives rise to the more homogeneous distribution of polar molecules. The polarization of this kind of system then becomes similar to the process of damped oscillation, which can be expressed by the component concentration vector and the rotational diffusion vector. In the polarizable system, the polarization process becomes similar to the process of two kinds of damped oscillations, since the third harmonic exists due to the anisotropic polarizability. Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 61731013) and Sichuan Science and Technology Program (Grant No. 2018FZ0008). References [1] R. Gedye, F. Smith, K. Westaway, H. Ali, L. Baldisera, L. Laberge, J. Rousell, Tetrahedron Lett. 27 (1986) 279. [2] D.M.P. Mingos, D.R. Baghurst, Chem. Soc. Rev. 20 (1991) 1.
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