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Physica B 403 (2008) 3214– 3216

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/physb

The current of a particle along a microtubule in microscopic plasma

Li Wei a, Chen Jun-fang a,�, Wang Teng b, Lai Xiu-Qiong a

a School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou, Guangdong 510006, Chinab School of Computer, South China Normal University, 510006 Guangzhou, China

a r t i c l e i n f o

Article history:

Received 3 March 2008

Received in revised form

1 April 2008

Accepted 5 April 2008

PACS:

04.25.�8

05.60.�k

52.25. Fi

Keywords:

Current

Plasma

Transport

26/$ - see front matter & 2008 Elsevier B.V. A

016/j.physb.2008.04.006

esponding author. Tel.: +86 20 85565160.

ail address: tolwwt@163.com (C. Jun-fang).

a b s t r a c t

Transport of a particle along the axis of a microtubule in a plasma-enhanced chemical vapor deposition

(PECVD) system is investigated. The current, respectively, as a function of the temperature, the magnetic

field and the external force is obtained. The value and direction of the current may be controlled by

changing the above parameters.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

Because the plasma technologies have been more and morewidely used for material processing, it is necessary to study thefundamental physical processes inside a plasma system. Transportphenomenon is important in plasma systems. Many researcheshave been conducted in this field in the past decades and moststudies have been focused on the macroscopic local equilibriumplasma [1]. Modern kinetic theory is based on the Boltzmanequation, and different approximate analytical solutions fornonsteady transport processes are studied [2,3]. More progresseshave been achieved by the application of statistical means [4]. Inrecent years, transport and diffusion of nonlinear systems havebeen studied using approximate methods; some computationsand simulations for transport and distribution of plasma havebeen made [5–9]. It is necessary for us to analyze the transportphenomenon along the surface of the substrate in a plasma-enhanced chemical vapor deposition (PECVD) system in order toobtain a more suitable film.

In this work, our aim is to find how the change of someenvironmental parameters can induce a net current of particlesalong the surface of a microtubule in a plasma system. Our modelwas proposed to describe the movement of a particle along a

ll rights reserved.

microtubule in a plasma system. The phenomenon of currentreversal will hold our interest in some application of PECVD.

2. The model

Here, we consider the simplest imaginable system containingonly some crucial elements. We assume an active particle movingalong the surface of a microtubule in a plasma system, which isacted upon by a zero-time-averaged external force caused by theroles of other particles in this system. We assume that there existsonly a homogeneous changing magnetic field in the system forsimplifying the model. So, the motion of this particle can bedescribed by the Langevin equation along the axis [10]:

dx

dt¼�ðB� FðtÞÞ

gþ

ffiffiffiffiffiffiffiffiffiffikT=g

qxðtÞ (1)

where x is the coordinate, g is the space-dependent frictioncoefficient of the plasma, B is the positive force along the x

direction caused by a magnetic field, k is the Boltzman constant, T

is the temperature of this system, x(t) is the Gaussian white noisewith zero mean and correlation function: /x(t)x(t0)S ¼ 2d(t�t0),/yS denotes an ensemble average over the distribution of thenoise. d(t) is the Dirac delta function. F(t) is an external force along

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-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

0.5 1 1.5 2 2.5 3B

0

J

F0=0.5

Fig. 2. Current J vs. B at F0 ¼ 0.5, kT ¼ 1, a ¼ 1 and e ¼ 0.

-0.1

-0.2

-0.3

-0.4

-0.5

0B=3

J

L. Wei et al. / Physica B 403 (2008) 3214–3216 3215

the x direction, which is the energy source, and satisfies

FðtÞ ¼

1� �

1þ �F0; ntptpntþ 0:5ð1þ �Þ

�F0; ntþ 0:5ð1þ �Þptpðnþ 1Þt

8<: (2)

where t is the period of the unbiased force, F0 is the magnitude ofF(t) and e is temporal asymmetric parameter with �1pep1. Theshape of the microtubule is described by its radius o(x) ¼ a, wherea is a constant.

The equation of motion along the axis of the microtubule canbe equivalently given by the Fokker–Planck equation [11]

qPðx; tÞ

qt¼

qqx

1

gkT

qPðx; tÞ

qxþ ðB� FðtÞÞPðx; tÞ

� �

¼ �qjðx; tÞ

qx(3)

where j(x, t) is the probability current density and P(x, t) is theprobability density for the particle of the plasma at position x andat time t. It satisfies the normalization condition

R 2p0 Pðx; tÞdx ¼ 1

and the periodicity P(x, t) ¼ P(x+2p,t).If the period of F(t) is longer than any other time scale of the

system, there exists a quasi-steady state. This equation can besolved for the probability current j. By following the method inRefs. [11–13], we can obtain the current

jðFðtÞÞ ¼

kTð1� expðB� FðtÞÞ=ðkTÞÞR 2p0 expð�Bþ FðtÞxÞ=ðkTÞdx

R xþ2px ð1þ o0ðyÞ2Þ1=2 expðB� FðtÞyÞ=ðkTÞÞdy

(4)

For the given external force, the average current is given by

J ¼1

t

Z t

0jðFðtÞÞdt

¼1

2ð1þ �Þj

1� �

1þ �F0

� �þ ð1� �Þjð�F0Þ

� �(5)

0.50 1 1.5 2F0

2.5 3

Fig. 3. Current J vs. F0 at B ¼ 3, kT ¼ 1, a ¼ 1 and e ¼ 0.

3. Results and discussion

The main results are shown in Figs. 1–4.

6

5

4

3

0

-1

-2

0 0.5 1 1.5 2 2.5 3T

2

1

7

J

x10-3

B=0.1F0=0.2

Fig. 1. Current J vs. T at B ¼ 0.1, F0 ¼ 0.2, a ¼ 1 and e ¼ 0.

Fig. 1 shows the current J as a function of T at B ¼ 0.1, F0 ¼ 0.2,a ¼ 1 and e ¼ 0. The current J obtains the maximum value whenthe value of T is very small. J decreases with the increasing valueof T. The behavior of J as a function of T is due to the synergeticeffect of the thermal fluctuations, the magnetic field and theexternal force. When T-0, J is mainly controlled by B and F0.When T-N, the thermal noise is very large, and there is littlemeaning for this system. This indicates that thermal fluctuationscan facilitate the energy conversion of this system by reversingthe direction of the current J. Fig. 1 shows that on increasing thetemperature the current J may reverse its direction with someproper values of B and F0.

Fig. 2 shows the current J as a function of B at F0 ¼ 0.5, kT ¼ 1,a ¼ 1 and e ¼ 0. The behavior of J as a function of B is due to thesynergetic effect of the thermal fluctuations, the magnetic fieldand the external force. When B-0, J is mainly controlled by T andF0. When B-N, the magnetic field is very large, and there is littlemeaning for this system. This indicates that the magnetic field canfacilitate the energy conversion of this system by reversing thedirection of the current J. Fig. 2 shows that the current J decreaseswith the increasing value of B. The current J, on increasing B, mayreverse its direction with some proper value of F0.

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0.1

0

-0.1

-0.2

-0.3

-0.4

-0.5

-0.8

-0.82

-0.84

-0.86

-0.9

-0.92

-0.94

-0.96

-0.88

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1ε

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1ε

kT=0.5

B=3

F0=0.6

kT=2

B=3F0=1

JJ

Fig. 4. Current J vs. e for different values of T, F0 and B at a ¼ 1.

L. Wei et al. / Physica B 403 (2008) 3214–32163216

Fig. 3 shows the current J as a function of F0 at B ¼ 3, kT ¼ 1,a ¼ 1 and e ¼ 0. The behavior of J as a function of F0 is due to thesynergetic effect of the thermal fluctuations, the magnetic fieldand the external force. When F0-0, J is mainly controlled by T

and B. When F0-N, the external force is very large, and there islittle meaning for this system. This indicates that external forcecan facilitate the energy conversion of this system by reversingthe direction of the current J. Fig. 3 shows that there are optimizedvalues of F0 at which the current J takes its maximum or

minimum values. The current J on increasing F0 may reverse itsdirection with some proper value of B.

Fig. 4 shows the current J as a function of e, for different valuesof F0, B and T. The behavior of J as a function of e is due to thesynergetic effect of the thermal fluctuations, the magnetic fieldand the external force. The asymmetry of the unbiased externalforce can induce a current when the shape of the tube issymmetric. From Fig. 4, we can find that the current J may takeits maximum or minimum values with optimized values of e. Thecurrent J may reverse its direction with the change of e.

4. Conclusions

In this work, we studied the transport of a particle along thesurface of a microtubule in a plasma system. The movementequation can be described by the Langevin equation andFokker–Planck equation. In our model, the thermal fluctuations,the magnetic field and the external force are three driving factorsfor obtaining the net probability current. When the three drivingfactors compete with each other, the current J may reverse itsdirection upon increasing some parameter. The current J isobtained, respectively, as the function of T, B, F0 and e. Weconclude that it is possible to achieve the suitable value anddirection of the current J by suitably tuning the parameter valuesof the PECVD system.

Acknowledgments

This work was sponsored in part by the National NaturalScience Foundation of China [10575039], the Chinese SpecializedResearch Fund for the Doctoral Program of Higher Education[2004057408], the Key Project of Science Research Fund ofGuangdong (China) [05100534] and by the Science ProjectFoundation of Guangzhou (China) [2005Z3-D2031].

References

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