An Analytical Mechanism of the Vacancy Diffusion Process of an Atomic Chain and Its Effects on the Creep Properties

An Analytical Mechanism of the Vacancy Diffusion Process of an Atomic Chain

and Its Effects on the Creep Properties

Tomonori Watanabe

Graduate School of Science & Technology, Chiba University, Chiba 263-8522, Japan

In order to reveal the microscopical mechanism which causes the plastic deformation induced by the vacancy diffusion process, we model and analyze mathematically the motion of the atom with the vacancy in an atomic chain. We describe the dynamics of the atom with the vacancy by a kind of the diffusion equation which includes the essential effects on the plastic deformation, namely, the thermal effect, the interactions of atoms, the friction of the environment and the external force. Strippingoff the fluctuation of the prime motion by the perturbation expansion, we can get the situation where the essential motion of the atom with the vacancy is represented by the propagation of the kink wave which responds to the external force and the temperature, though the system is described by the diffusion process. Then deriving the strain of the atomic chain, we show the strain against time behaves viscoelastically like the typical response, for instance, that of the Voigt model. The properties of the temperature and the applied stress coincide with the well-known results of the primary creep phenomenon.

(Received July 22, 2002; Accepted October 30, 2002)

Keywords: vacancy diffusion process, atomic chain, creep properties, mathematical analysis, perturbation expansion, Voigt model

1. Introduction

Diffusion process plays a decisive role in understanding the various phenomena in material science. Vacancy me-chanism diffusion is one of the diffusion process, which makes the atoms migrate by jumping into near-neighbor vacancies. This gives us the rich picture to understand the micro-mechanics of materials.1)

Recently, the micro-mechanics has been greatly discussed, specifically, to develop the nano-scale electrical devices.2)In fact, the nano-scale metallic contact formed by the several atomic chains is created and examined mainly in terms of the quantum behavior.3)Therefore, it is important to analyze the mechanical properties of the atomic chain for the develop-ment and practical use of the nano-scale metallic contact and for understandingand applyingthe mechanical properties of the materials microscopically.

In the present paper, we discuss the dynamics of the atom with the vacancy in order to understand the mechanical behavior of nano-scale metallic contact. The main purpose of this paper is to reveal the mechanism which causes the plastic deformation induced by the diffusion process. Indeed, several approaches have been performed to understand the mechan-ical properties of nano-scale metallic contact. For instance, by runningthe molecular dynamics simulations, we can see the strain response of the atomic chain under the static tensile force,2) and a detailed description of the evolution of the atomic structure of metallic nanocontacts and of the mechanical deformation processes that take place duringan elongation process.4) By usingthe idealized atomic chain model (which includes the effect of the vacancy and the external force only), the dynamics of the atom in the atomic chain is analytically shown5) and the interestingproperties (for example, discommensurations, devil’s staircases and so forth) of the idealized atomic model are discussed.6)In this study, compared with these approaches, we analyze the simple andpracticalatomic chain model which includes the vacancy, the thermal effect, the interactions of the atoms, the friction of the environment and the external force as the

essential effects on the plastic deformation. We treat the model analytically and mathematically (not numerically), applyingthe theory of soliton system.7) Then we expect to interpret apparently how the microscopical diffusion process causes the mechanical behavior of the materials.

2. Atomic Chain Model

In this section, we model the motion of the atom in the atomic chain and derive the equation which represents the dynamics of the atom with the vacancy in the atomic chain. We consider the nano-scale metallic contact which is made by stretchingthe material in one direction. An example is shown in Fig. 1. In this figure, a sphere represents one atom. A neck has an ordered structure, for instance, like a perfect fcc structure. Namely, this means the nano-scale metallic contact consists of several atomic chain layers. We focus on one of the atomic chain layers at equal spacing a in the metallic nanocontact, that is, one arrangement of the atoms spacingbin the atomic layers in Fig. 2. (More detail of this

Fig. 1 An example of a metallic nanocontact. A sphere represents one atom. A neck has an ordered structure, for instance, like a perfect fcc structure.

Special Issue on Diffusion in Materials and Its Application —Recent Development—

[image:1.595.361.492.585.749.2]

description is discussed in the papers.2,4)) We consider the dynamics of the atom in the atomic chain in Fig. 3. Supposingone space exists as the vacancy which is made by missingthe particle in the atomic chain (Fig.3(a)), we consider the motion of the ith atom i¼1;2;. . .;N (N: the total number of the atoms in the chain) which is located at the nearest-neighbor of the vacancy with the interactions of the other atoms. We assume that the interaction derived from the nearest-neighbor atoms of the same chain is described by the harmonic oscillation and that the interactions derived from the atoms of the other chains are represented by the periodical potential force. Frenkel-Kontorova model5)is the typical and well-known model which is based on the above assumption. It has been used for the various analysis and provides the

interestingresults.6)

In this study, for the more practical analysis, we consider, addingto the above effects, the thermal effect, the friction of the environment and the external force as the essential effects on the mechanical properties. More precisely, we consider the followingsituation. The ith atom exists with the interactions of the atoms and the thermal force. Then it changes the displacement ui from the equilibrium point by the external forceF against the friction of the environment. We show this behavior of theith atom in Figs.3(a)–(c). Then, we will find that the change of the displacement ui can be represented by the propagation of the kink type wave (Here we mention the nature of the external forceF). The external forceF should be given naturally by the interaction among atoms which originates from the macroscopic force added to the material surface. Indeed, recently, the force and con-ductance have been measured experimentally usingthe scanningtunnelingmicroscope supplemented by a force sensor.8,9)However, in this study, we do not consider how the external forceF is concretely related with the macroscopic force, because we would like to discuss not the external force

F but the dynamics driven by the external force F. We assume that the external force F works on each atom uniformly (the direction is equal to the x direction in Fig. 3. In this figure, we explicitly show the force which works on theith atom for convenience).

From the above observation of the atomic chain, we can obtain the Langevin equation as the equation of the motion which describes the dynamics of the ith atom with the vacancy;

0¼ duiðtÞ dt þ

K

b2 uiþ1ðtÞ 2uiðtÞ þui1ðtÞ

@Ui

@ui

þFþiðtÞ;

ð1Þ

where;K;Uiandiare the frictional coefficient, the spring coefficient, the potential energy ofith atom and the thermal fluctuation. The properties of the thermal effect are given by Gaussian white noise:

hiðtÞiav¼0;

hiðtÞjðt0Þiav¼2kBTijðtt0Þ;

ð2Þ

where h i_av means the ensemble average. kB and T are

Boltzmann constant and temperature. Here, we mention that in the left hand side of eq. (1) we suppose the inertia is sufficiently small because we consider the motion is very slow against time due to the large friction of the environ-ment.10)

In eq. (1), we give the periodical potential Ui

ðA=pÞð1cospuiÞ, ðp2=bÞ with the Peierls force A. Supposingthe lattice spacingb is sufficiently small, by the limitingprocedure: uðib;tÞ !uðx;tÞ as b! þ0 with

uðib;tÞ uiðtÞandxib, we obtain the partial differential equation:

0¼ @uðx;tÞ

@t þK

@2_uðx;tÞ

@x2 Asinpuðx;tÞ þFþðx;tÞ; ð3Þ

where we have

i+1

a

b

F

i

i-1

Fig. 2 Atomic chain layers in the metallic nanocontact.

t

A

b

u

i

V

F

u

i

a

i

x

b

u

i

(a)

F

u

i

(b)

F

u

i

(c)

[image:2.595.74.266.72.204.2] [image:2.595.68.267.255.583.2]

hðx;tÞiav¼0;

hðx;tÞðx0;t0Þiav¼2kBTðxx0Þðtt0Þ:

ð4Þ

In this study, we discuss the mechanical properties of the atomic chain by analyzingeq. (3).

3. Dynamics of the Atom Induced by the Vacancy

Diffusion Process

The dynamics of the atom in the atomic chain with the vacancy is described by eq. (3). Here we can find eq. (3) is a kind of the diffusion equations which generally work to uniformize the field, for example, like the Fick’ Law. However, referringto the former section (Figs.3(a)–(c)), we should consider that the atom with the vacancy propagates like the kink wave in response to the external forceF. In this section, we show how the kink wave is derived from the diffusion eq. (3), by applyingthe analysis of the singular perturbation method.11)

Now supposingthat eq. (3) has the travelingwave solution, we consider the transformation:

¼xþ Z t

0

vðt0Þdt0; ¼t:

We expand eq. (3) as

uðx;tÞ ¼u0ð Þ þu1ð ; Þ þ2u2ð ; Þ þOð3Þ;

vðtÞ ¼v0þv1ðÞ þ2v2ðÞ þOð3Þ;

where is the perturbation parameter and Oð Þ means Landau’s symbol for the higher order terms. Then we obtain the followingequations inOð0ÞandOð1Þwith respect tou;

0¼ v0 du0ð Þ

d þK d2u0ð Þ

d 2 f u0ð Þ

þF; ð5Þ @u1ð ; Þ

@ þLu1ð ; Þ ¼ v1ðÞ

du0ð Þ

d þð ; Þ; ð6Þ

where

f u0ð Þ

Asinpu0ð Þ;

and the operatorLis given by

L K @

2

@ 2þv0ðÞ

@ @ þf

0_u

0 ð Þ:

We suppose that the operatorL and the adjoint operatorLA

satisfy the complete orthonormal function system;

L bð Þ ¼0; L lð Þ ¼l lð Þ;

LAA_bð Þ ¼0; LAA_lð Þ ¼lAlð Þ;

wherelis the eigenvalue and landAl are the eigenfunc-tions. We choose the eigenfunctions to be normalized so that

A_l l0

¼ðll0Þ; A_b b

¼1; A_b l

¼0¼ A_l b

;

where we shall use the brackets to indicate the inner product:

hujvi Z1

1

uð Þvð Þd :

Then the closure relation must be

ð 0Þ ¼A_bð 0Þ bð Þ þ

Z 1

1

A_lð 0Þ lð Þdl: We expand eq. (5) as

u0ð Þ ¼uð00Þð Þ þ!u

ð1Þ

0 ð Þ þ! 2_uð2Þ

0 ð Þ þOð! 3_Þ;

v0¼vð00Þþ!v

ð1Þ

0 þ! 2_vð2Þ

0 þOð! 3_Þ;

where ! is the perturbation parameter, consideringthat

F¼!Fð1Þ_{. Here we requirev}ð0Þ

0 0because of the balance

against the external force. (The direction of the kink wave propagation should be physically equal to that of the external force.) Then we obtain the followingequations inOð!0Þand

Oð!1Þwith respect tou0; 0¼Kd

2_uð0Þ

0 ð Þ d 2 f u

ð0Þ

0 ð Þ

; ð7Þ

Lð0Þuð₀1Þð Þ ¼ vð₀1Þdu

ð0Þ

0 ð Þ d þF

ð1Þ_;

ð8Þ

where the operatorLð0Þ_{is given by}

Lð0Þ K @

2

@ 2þf

0 _uð0Þ

0

:

Let the eigenvaluesð_l0Þand the eigenfunctions_bð0Þandð_l0Þ

be

Lð0Þð_b0Þð Þ ¼0; Lð0Þ_lð0Þð Þ ¼_lð0Þð_l0Þð Þ; ð_biÞ ð_bjÞ

D E

¼0; ð_liÞ ð_ljÞ

D E

¼0; ði6¼jÞ:

Because the operator is self-adjoint, ð_b0Þ ¼ ð_b0Þ. These relations are useful to the practical calculation.

From eqs. (7) and (8), we evaluate the ensemble averaged values;

uð ; Þ

h i_av¼uð₀0Þð Þ_avþ2hu2ð Þiðav0ÞþOð!; 3_Þ;

vðÞ

h iav¼!v

ð1Þ

0 þ! 2 _v

2ðÞ

h ið_av1ÞþOð!2; 3Þ:

Here we can regard eq. (7) as the specialized equation of the sine-Gordon equation which is a kind of soliton equations and exactly solvable.7) Actually, we obtain the kink type wave solution:

u₀ð0Þð Þ ¼u₀ð0Þð Þ_av

¼4

parctanðe Þ;

ffiffiffiffiffiffiffi

Ap K

r

:

ð9Þ

We have the eigenfunctions;

ð_b0Þ¼ ð_b0Þ¼ ffiffiffiffi

2 r

sech ; ð10Þ

ð0Þ

k ¼

eik _kþitanh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k2_þ2

q ; ð11Þ

ð_k0Þ¼ ð0_kÞ ¼e

ik _kitanh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k2_þ2

q ; ð12Þ

and the eigenvalues:

ð_k0Þ¼K

pðk

From eq. (8), we have the relations;

vð₀1Þ ¼ 1 Nð0Þ

ð0Þ

b ð ÞF

ð1Þ

;

Nð0Þ _¼ ð0Þ

b ð Þ

du

ð0Þ

0 ð Þ d

* +

:

Then we obtain

vð₀1Þ ¼ p 4F

ð1Þ_¼ p

4A ffiffiffiffiffiffiffi

KA p

s

Fð1Þ; ð13Þ

Nð0Þ ¼2

p

ffiffiffiffiffiffi 2

p

: ð14Þ

Now in order to observe the behavior of solution (9), we normalize each variable appropriately, that is,

ðu;x;tÞ7!ðuu~;xx~;tt~Þ where uu~, xx~ and tt~ are the normalized variables. Then we have the normalized displacement functionuu~ðxx~;tt~Þ:

~

u

uðxx~;tt~Þ ¼arctanðexx~þtt~Þ:

The time evolution of this displacement function is shown in Fig.4. We can see that the kink wave is propagated without dissipating. Thus it is concluded that the propagating movement of the atom with the vacancy is derived from the diffusion eq. (3) by strippingoff the fluctuations from the essential motion. Moreover, the ensemble averaged velocity of the travelingwave, which will be minutely shown by eq. (17) in the next section, depends on the external forceFand the temperature T. Namely, the atom with the vacancy responds to the external force and the temperature. It migrates faster with increasing the external force and the temperature, though it derives form the diffusion process. This behavior can be qualitatively reasonable6,12) and will lead one to understandingthe mechanism of the dynamics of the atom with the vacancy induced by the diffusion process.

4. Derivation of the Strain

In this section, we derive the strain of the atomic chain. Applyingto the results of the former section, we can calculate the higher order terms;

u2ð Þ

h ið_av0Þ¼sech

4K

tanh

4 1þ sech2

3

; ð15Þ

v2 h ið_av1Þ ¼

p2_Fð1Þ _1þJ

2

32K2 ; ð16Þ

where

J¼2

Z 1

0

dv

sinh2ðvÞ 1 1 ð1þv2_Þ1=2

¼0:1494094 . . .:

Thus we have the ensemble averaged displacement:

uð ; Þ

h i_av¼uð₀0Þð Þ_avþ2hu2ð Þiðav0ÞþOð!; 3

Þ

¼ 4

parctanðe Þ þ

2 _u 2ð Þ

h ið_av0ÞþOð!; 3Þ;

and the ensemble averaged velocityhvðÞiav:

vðÞ

h iav¼!vð 1Þ

0 þ! 2 _v

2ðÞ

h ið_av1ÞþOð!2; 3Þ

¼!pF

ð1Þ

4 1þ 1þJ

2

p

8K kBT

0 B B @ 1 C C

AþOð!2; 3Þ:

ð17Þ

From these results, since we are interested in the essential properties of the dynamics, we shall define the strain of the atomic chain by

"ðx;tÞ uðx;tÞ

a

’2b

aarctan expððxþv0tÞÞ

;

ð18Þ

where the velocityv0is given by the lowest term ofhvðÞiav;

v0

pF

4 1þ 1þJ

2

p

8K kBT

0 B B @ 1 C C A:

5. Mechanical Properties under the Constant Applied Force

In this section, we discuss the mechanical properties of the atomic chain under the constant applied stress, — the creep response —, by analyzingstrain (18).

From the derivation of strain (18) by timet, we obtain the relation:

@"ðx;tÞ

@t ¼ v0b

a sin a

b "ðx;tÞ

: ð19Þ

Here consideringthe initial strain and the practical strain under the constant applied force are respectively represented by"ð0Þ_ðx

0Þand"ð1Þðx0;tÞwith the constantx0, we describe the

strain"by

"ðx0;tÞ ¼"ð0Þðx0Þ þ"ð1Þðx0;tÞ:

Usingthis strain into relation (19) and linearizingit, we have the differential equation of"ð1Þ_ðx

0;tÞ:

-10 -5 10 0 4 6 8 10 0 0.5 1 1.5 -10 -5 0 5

Normalized

Place

,

Time

,

Normalized

Normalized 2

Displacement

, u

~

x

~

t

~

[image:4.595.67.268.76.210.2] [image:4.595.49.288.262.403.2]

d"ð1Þ_ðx

0;tÞ dt ¼

v0b

a sin a

b "

ð0Þ_ðx

0Þ

þv0cos

a b "

ð0Þ

ðx0Þ

"ð1Þðx0;tÞ: ð20Þ

This equation is the first order ordinary differential equation of"ð1Þ_{. Moreover, lettingthe initial strain be}

1 2

b a < "

ð0Þ_ðx

0Þ<

b

a; ð21Þ

we can find eq. (20) agrees with the Voigt model12)which describes the creep viscoelasticity;

d"VðtÞ

dt ¼ V

G

"VðtÞ; ð22Þ

where "V andV are the strain and the stress of the Voigt model, andGandare constants. This result indicates that the atomic chain under the constant applied force behaves viscoelastically like the solution of the Voigt model.

Furthermore, we define the external stressbyF=b2, (namely, the external stressis proportional to the external forceF), and regard"VðtÞandVas"ð1ÞðtÞandrespectively, (that is "VðtÞ ¼"ð1ÞðtÞ and V ¼). Then, comparingthe equation derived from the atomic chain model (20) with the Voigt model, we have the relations;

GðÞ ¼ a

b

1

tan a

b"

ð0Þ_ðx

0Þ

; ð23Þ

ðTÞ ¼ 2aA

b2_sin a

b "

ð0Þ_ðx

0Þ

ffiffiffiffiffiffiffi_KA

p

s þ

1þJ 2 8 0 B B @

1 C C AkBT 0

B B @

1 C C A

: ð24Þ

These relations indicateGanddepend on the external stress

and the temperature T. This result suggests the initial inclination of the creep curve becomes steep with increasing the external forceF or the temperatureT.

We show strain "ð1Þ _{against the time under the constant}

applied load by usingeq. (18) in Fig.5. In this figure, we use

the values; v0¼0:005h1, "ð0Þ¼0:99 and b=a¼1. For

the comparison, the gray line shows the strain of the Voigt model "VðtÞon the same condition. Two dotted lines (black and gray) correspond with the respective strains on the conditionv0¼0:01h1. This figure shows the strain of the

atomic chain model under the constant applied load behaves as the viscoelastic response like the primary creep phenom-enon. Because the dotted lines mean the higher temperature or applied force, compared with the smooth lines, the strain rate becomes high with increasing the temperature or applied force. This behavior qualitatively agrees with the experi-mental result about the dependence of the primary creep curve on the temperature and the applied force.12)Therefore, it is concluded that the dynamics of the atom with the vacancy in the atomic chain induces the viscoelastic proper-ties like the primary creep phenomenon.

6. Conclusions

In the present paper, we mainly purpose to reveal the mechanism which causes the plastic deformation induced by the vacancy diffusion process.

We model the dynamics of the atom with the vacancy in the atomic chain. We derive a kind of the diffusion equation which describes the motion of the atom with the vacancy includingthe essential effects on the plastic deformation, namely, the thermal effect, the interactions of the atoms, the friction of the environment and the external force. We analyze the obtained diffusion equation mathematically. Strippingoff the fluctuation of the prime motion by means of the perturbation expansion, we can get the situation where the essential motion of the atom with the vacancy is represented by the propagation of the kink wave which responds to the external force and the temperature, though the system is described by the diffusion process.

Then we discuss the viscoelastic properties of the atomic chain by analyzingthe strain of the atomic chain under the constant applied force. The behavior of the creep curve induced from the atomic chain agrees with that of the primary creep phenomenon which is represented experimentally and typically by the Voigt model. The strain rate becomes high with increasingthe temperature or the applied force.

Although it is well-known that the vacancy diffusion process affects the creep properties of the materials, we believe that it is important to show analytically how the vacancy acts on the dynamics of the atomic chain and the other factors (namely, the interactions of the atoms, the friction of the environment and the external force) relate to the diffusion process of the atomic chain, and how such microscopical factors cause the creep properties. We hope that the materials presented here can be useful and valuable for further investigation in related problems, especially, the study and application of nano-scale metallic contact.

Acknowledgments

The author expresses his sincere gratitude to Professor Dr. Tamotsu Majima (Chiba University), Professor Dr. Tetsuji Tokihiro (the University of Tokyo), Professor Dr. Junkichi Satsuma (the University of Tokyo), Professor Dr. Katsuhiro

200

400

600

800 1000

0.002

0.004

0.006

0.008

0.010

Time

, t /

h

Strain

,

-1

Fig. 5 Strain response of the atomic chain under the constant load on the condition that v0¼0:005h1, "ð0Þ¼0:99 and b=a¼1. For the

[image:5.595.54.282.585.727.2]

Nishinari (Ryukoku University) and Professor Dr. Yasuhiro Ohta (Hiroshima University) for stimulatingdiscussions, valuable comments and warm support with magnificent hearts.

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