Allometric scaling law in conductive polymer

Polymer 45 (2004) 9067–9070 www.elsevier.com/locate/polymer

Allometric scaling law in conductive polymer Ji-Huan Hea,b,* a

College of Science, Donghua University, 1882 Yan’an Xilu Road, Shanghai 200051, People’s Republic of China b Key Lab of Textile Technology, Ministry of Education, Shanghai, People’s Republic of China Received 30 July 2004; received in revised form 9 September 2004; accepted 12 October 2004

Abstract A mathematical model describing the resistance of conductive polymer is proposed, which is naturally different from that for metal conductors. An allometric scaling law between the resistance and the section area is obtained. In the derivation, He Chengtian’s interpolation, which has millennia history, is applied. q 2004 Elsevier Ltd. All rights reserved. Keywords: Resistance; Polymer; Fiber

1. Introduction After the discovery of conducting polymers in the later 1970s, much progress has been made, and many applications are found in various engineering. Various textile fibers in yarn forms were treated with aniline at different oxidative polymerization conditions to improve their electrical conductivities [1,2], electrospinning can also produce conductive fibers by some special treatment [3]. Though many experiments have been conducted [2], a complete theoretical analysis is not yet dealt with. In theoretical analysis or in engineering design, the calculation of resistance is still based on traditional formulation for metal resistors, which can be written in the form RZk

L ; A

(1)

where R is the resistance of a conductor, A is the section area, L its length and k is the resistance coefficient. Actually, Eq. (1) is valid only for metal conductors where there are plenty of electrons in the conductor. However, in conductive polymer, the current is not caused by electrons, therefore, Eq. (1) should be modified in order to accurately * Address: College of Science, Donghua University, 1882 Yan’an Xilu Road, Shanghai 200051, People’s Republic of China. Tel.: C86 21 62379917; fax: C86 21 62373137. E-mail address: [email protected]. 0032-3861/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymer.2004.10.024

describe the polymer conduction. Though many experiment observations (e.g. see Appendix in Ref. [2]) show the deviation of Eq. (1) when applied to conductive polymer, our understanding of this fundamental phenomenon is scarce and primitive, and its accurate mechanism remains ambiguous, this makes it very difficult to deduce a mathematical model describing the ambiguous mechanism. In this paper, we will establish an allometric scaling law for the conductive polymer by He Chengtian’s interpolation [4].

2. Scaling in nature Before we deduce allometric scaling laws for conductive polymer, we begin with an introduction of allometry in nature. Scaling and dimensional analysis actually started with Newton [5], and allometry exists everywhere in our daily life and scientific activity. The most fruitful achievement is the allometric scaling relationship relating metabolic rate (B) to whole-body mass (M) [6–10] B wM b ;

(2)

where b is the scaling exponent. He [11] tried to explain the biological allometry by

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taking into account the dimension of an organ ðDCN=6Þ=ðDC1Þ ; Borgan wMorgan

Rw (3)

where Borgan is the metabolic rate of the organ, Morgan is its mass, D is the dimension of the organ construction, N is the cell’s degree of freedom of motion in the considered organ, for example DZ2, NZ0 for a leaf, and DZ3, NZ3 for heart. Generally, the form of an allometric relation can be expressed as: Y wX b ;

(4)

where Y and X are two measures describing an event, such as the weight of the antlers X and total weight of a deer Y. The exponent is critical important, and it is relevant to space dimension, therefore, Eq. (4) can be re-written in the form Y wX D=ðDC1Þ ;

In ancient China, there were many interpolation formulae [14,15], among others, we will use hereby He Chengtian’s interpolation [4] to fix approximately value of b. Consider the following inequality: a c ! x! ; b d

am C cn ; bm C dn

a am C cn c ! ! : b bm C dn d

(14)

Applying He Chengtian’s interpolation, in view of Eq. (11), the value of b can be written in the form bZ

n n=m D Z Z ; m C n n=m C 1 D C 1

(15)

where m and n are integers, DZn/m. Therefore, for non-metal material, Eq. (7) should be modified as Rw

1 AD=ðDC1Þ

:

(16)

Hereby D can be considered as the fractal dimension of the section. In order to verify the allometric scaling (16), we consider the surface convection current which is caused by moving charged particles distributed along the surface of the charged jet as that in electrospinning [16–18]. The surface convection current can be written in the form [3,16,17] (17)

where Is is the surface convection current, s surface density of the charge, u the velocity, r the section radius. The resistance for surface convection can be expressed as [17] Rs Z

(9)

(13)

where m and n are weighting factors. It is easy to prove that

(8)

or

For conductive polymer, we assume that

(12)

Is Z 2prsu; (7)

For an insulator, the resistance is independent of its section area, in scaling form, we write

1 : A0=1

(11)

xZ

The resistance of a conductive metal scales as:

Rw

0 1 ! b! 1 1

(6)

3. Allometric scaling law for conductive polymer

R wA0 ;

where b is constant relative to conductive character of the polymer The conductive behavior of conductive polymer lies between that for metal and that for insulator, so the value of b lies between 0 and 1, i.e.

where a, b, c, and d are integers. According to He Chengtian’s interpolation, the value of x can be approximately identified as follows:

where D is dimension of the discussed problem and k is an integral. The best known example is the simple pendulum, with the period TwR1/2, where R is the length of pendulum, T is its period, and here DZ1, kZ1. The Kepler’s 3rd law can be expressed in the form TwR3/2, where R is the distance from the planet to the Sun, T is the period, DZ1, kZ3. Another example is the drainage basin of a river. In Ref. [12], it is shown that the mass of the river and the area of the drainage basin follow a scaling law with an exponent of 2/3, this is because that the drainage basin is distributed on a plane, leading to DZ2. An example of a one-dimensional system (DZ1) that exhibits scaling is given in Ref. [13] where the flow through a tube is investigated. It is found that the amount of water in the tube and its length are related by a scaling law with exponent 1/2 [13].

1 Rw ; A

(10)

(5)

or Y wX k=ðDC1Þ ;

1 ; Ab

E E E 1 Z 1=2 Z w 1=2 ; 1=2 Is 2prsu 2p suA A

(18)

where Rs is the resistance for surface convection, E is the voltage applied. Due to the charged particles are distributed along the

J.-H. He / Polymer 45 (2004) 9067–9070

R Zk

L0:99 : c1:02 A0:64

9069

(25)

For non-woven conductive material, R Zk

L0:99 : c1:01 A0:64

(26)

Our prediction agrees very well with Westbroek et al.’s experimental data.

4. Conclusion Fig. 1. Resistance for surface convection sales as Rwr

.

K1

surface of the section as illustrated in Fig. 1, its dimension DZ1, so our prediction (16) agrees well with Eq. (18). For conductive textile DZ2, we have Rw

1 : A2=3

(19)

If we want to design an electrochemical cell constructed of two electrodes, which are made of knitted, woven or nonwoven conductive textile material, the allometry leads to the following formulations: R wL;

(20)

R wcK1 ;

(21)

where L the distance between the electrodes and c the concentration of the electrolyte solution. Therefore, we can easily obtain the following mathematical formulation for resistance calculation: RZk

L ; cA2=3

(22)

where k is a constant. Westbroek et al. [19] developed an electrochemical cell to test the resistance of knitted, woven and non-woven conductive textile material. This cell is constructed of two electrodes positioned in a planar manner against each other using a series of PVC plates. For each experiment performed at different electrolyte concentrations (10K5, 10K4, 10K3, 10K2, 10K1 mol lK1), different electrode surface areas(19.6, 78.5, 176,314, 491 mm2), and different distances between the electrodes (27, 91, 112 mm), fitting the experiment data, Westbroek et al. obtained [19] RZk

L : cA0:65

(23)

They also obtained the following similar results [19]: For knitted conductive material, RZk

L0:98 : cA0:62

For woven conductive material,

(24)

It would not be an exaggeration to say that practically most complex coupled physical problems can not be cast into an exact mathematical description. Various approximate tools are suggested in open literature, among others, the ancient Chinese interpolation, called He Chengtian’s interpolation [4], plays a critical role in our derivation. In practical design, it is highly desirable to get an insight into the problem in a mathematical formulation, which can help us understand the basic properties and specific features of the considered cases. The proposed theoretical model is of vital importance for the industry, especially the specialists in design, manufacturing and using conductive polymer.

Acknowledgements The author thanks two unknown reviewers for their careful reading and helpful comments. The work is supported by grant 10372021 from National Natural Science Foundation of China.

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