第三章 固体材料中的扩散

第三章固体材料中的扩散

Chapter3 The Diffusion in Solid Materials

本章基本问题：

1. 菲克第一定律的含义和各参数的量纲。

2．能根据一些较简单的扩散问题中的初始条件和边界条件。运用菲克第二定律求解。3．柯肯达耳效应的起因，以及标记面漂移方向与扩散偶中两组元扩散系数大小的关系。4．互扩散系数的图解方法。

5．“下坡扩散”和“上坡扩散”的热力学因子判别条件。

6．扩散的几种机制，着重是间隙机制和空位机制。

7．间隙原子扩散比置换原子扩散容易的原因。

8．计算和求解扩散系数及扩散激活能的方法。

9．影响扩散的主要因素。

Questions for chapter 3

1. What is the the meaning of Fick’s first law?

2. How to solve the problems by Fick’s second law?

3. What is the the Kirkendall effect?

4. How to explain diffusion coefficient schematically?

5. What is the diffusion driving force;

6. What are diffusion mechanisms, expecially interstitial and vacancy mechanisms

7. What is the reason that interstitial diffusion is easier than substitutional diffusion?

8 What are the methods to compute diffusion coefficient and diffusion activation energy?

9. What are main factors affecting diffusion?

The field of diffusion studies in metals is of great practical, as well as theoretical importance. By diffusion one means the movements of atoms within a solution. In general, our interests lie in those atomic movements that occur in solid solutions. This chapter will be devoted in particular to the study of diffusion in substitutional solid solutions and atomic movements in interstitial solid solutions.

Diffusion is a process of mass transport that involoves the movement of one atomic species into another.

3-1扩散方程

Sec.3.1 Diffusion Equations

1 菲克第一定律Fick’s First Law

Diffusion can be modeled as the jumping of atoms from one plane to another.

（1）第一定律描述：单位时间内通过垂直于扩散方向的某一单位面积截面的扩散物质流量（扩散通量J flax or the rate of diffusion）与浓度梯度(concentration gradient)成正比。

The rate of diffusion is proportional to the concentration gradient.

（2）表达式：J=-D(dc/dx)。（C－溶质原子浓度；D-扩散系数Diffusivity or diffusion coefficient。）

（3）适用条件：稳态扩散，dc/dt=0。浓度及浓度梯度不随时间改变。

Fick’s first law assumes that the concentration gradient is independent of time.

2菲克第二定律Fick’s Second Law

一般：?C/?t=?(D?C/?x)/ ?x

二维：

（1）表达式特殊：?C/?t=D?2C/?x2

三维：?C/?t=D(?2/?x2+?2/?y2+?2/?z2)C

稳态扩散：?C/?t=0，?J/?x=0。

（2）适用条件：

非稳态扩散：?C/?t≠0，?J/?x≠0（?C/?t=－?J/?x）。

Assume that diffusivity, D is independent of C, the rate of change in concentration with time, ?C/?t is proportional to the rate at which the concentration gradient changes with distance in a given direction, ?2C/?x2

3扩散第二定律的应用

（1）误差函数解

适用条件：无限长棒和半无限长棒。

表达式：C=C1－(C1-C2)erf(x/2√Dt) (半无限长棒)。

在渗碳条件下：C:x,t处的浓度；C1:表面含碳量;C2:钢的原始含碳量。

（2）正弦解

C x=Cp-A0sin(πx/λ)

Cp:平均成分；A0：振幅Cmax- Cp；λ:枝晶间距的一半。

对于均匀化退火，若要求枝晶中心成分偏析振幅降低到1/100,则：

[C(λ/2,t)- Cp]/( Cmax- Cp)=exp(-π2Dt/λ2)=1/100。

Experimental work has shown that the atoms in face-centered cubic, body-centered cubic, and hexagonal metals move about in the crystal lattice as a result of vacancy motion. Let it now be assumed that the jumps are entirely random; that is, the probability of jumping is the same for all of the atoms surrounding a given vacancy. This statement implies that the jump rate does not depend on the concentration.

Fig. 3.1 Hypothetical single crystal with a concentration gradient

Fig.3.2 Atomistic view of section of the hypothetical crystal of Fig. 3.1

Fig.3.1 represents a single crystal bar composed of a solid solution of A and B atoms in which the composition of the solute varies continuously along the length of the bar, but is uniform over the cross-section. For the sake of simplifying the argument, the crystal structure of the bar is assumed to be simple cubic with a <100> direction along the axis of the bar. It is further assumed that the concentration is greatest at the right end of the bar and least at the left end, and that the macroscopic concentration gradient dn A /dx applies on an atomic scale so that the difference in composition between two adjacent transverse atomic planes is:

dx dn a A )( 3-1 where a is the interatomic, or lattice spacing (see Fig.3.2). Let the mean time of stay of an atom in a lattice side be τ. The average frequency with which the atoms jump is therefore 1/τ. In the simple cubic lattice pictured in Fig.3.2, any given atom, such as that indicated by the symbol x , can jump in six different directions: right or left, up or down, or into or out of the plane of the paper. The exchange of A atoms between

two adjacent transverse atomic planes, such as those designated X and Y in Fig.3.2, will now be considered. Of the six possible jumps that an A atom can make in either of these planes, only one will carry it over to the other indicated plane, so that the average frequency with which an A atom jumps from X to Y is 1/6τ. The number of these atoms that will jump per second from plane X to plane Y equals the total number of the atoms in plane X times the average frequency with which an atom jumps from plane X to plane Y. The number of solute atoms in plane X equals the number of solute atoms per unit volume (the concentration n A ) times the volume of the atoms in plane X, (Aa ), so that flux of solute atoms from plane X to plane Y is

)(61a n J A y X τ

=→ 3-2 Where y X J → = flux of solute atoms from plane X to plane Y per unit cross-section

τ= mean time of stay of a solute atom at a lattice site

A n = number of A atoms per unit volume

a = lattice constant of crystal

The concentration of A atoms in plane Y may be written:

dx

dn a n n A A Y A )()(+= 3.3 Where A n is the concentration at plane X , and a is the lattice constant, or distance between planes X and Y . The rate at which A atoms move from plane Y to X is thus

τ6)(a dx dn a n J A A X Y ?????

?+=→ 3.4 Where J Y →X represents the flux of A atoms from plane Y to plane X. Because the flux of solute atoms from right to left is not the same as that from left to right, there is a net flux (designated by the symbol J ) which can be expressed mathematically as follows:

ττ6)()(6a dx dn a n n a J J J A A A X Y Y X ?????

?+-=

-=→→ 3.5 Or

dx dn a J A ?-=τ62 3.6 Since the cross-sectional area was chosen to be a unit area. Notice that in Eq.3-6, the flux (J ) of A atoms is negative when the concentration gradient is positive (concentration of A atoms increases from left to right in Fig.3.2). This result is general for diffusion in an ideal solution; the diffusion flux is down the concentration gradient. Notice that if one considers the flow of B atoms instead of A atoms, the net flux will

be from left to right, in agreement with a decreasing concentration of the B component as one moves from left to right. Again, the flux (in this case of B atoms) is down the concentration gradient.

Let us now make the substitution

τ

62

a D = 3.7 in the equation for the net flux, which gives:

dx

dn D J A -= 3.8 This equation is identical with that first proposed by Adolf Fick in 1855 on theoretical grounds for diffusion in solutions. In this equation, called Fick’s first law , J is the flux, or quantity per second, of diffusing matter passing normally through a unit area under the action of a concentration gradient dn A /dx . The factor D is known as the diffusivity , or the diffusion coefficient .

3-2 扩散的微观机制

Sec. 3.2 Mechanisms of Diffusion

The mechanism of diffusion determines the energy barrier that must be overcome (i.e., the activation energy Q) for the process to occur. Since energy is supplied thermally, the higher the temperature , the greater the probability that large numbers of atoms will have sufficient energy to overcome the energy barrier and the more rapid will be the diffusion process.

The lattice geometry also affects the diffusion coefficient through the preexponential constant.

Interstitial confussion machanism: when a small interstitial atom or ion is present in the crystal structure, the atom or ion moves from one interstitial site to another. No vacancies are required for this mechanism.

间隙－间隙；

（1）间隙机制 平衡位置－间隙－间隙：较困难；

间隙－篡位－结点位置。

（间隙固溶体中间隙原子的扩散机制。）

Vacancy exchange mechanism of diffusion : in self-diffusion and diffusion involving substitutional atoms, an atom leaves its lattice site to fill a nearby vacancy( thus creating a new vacancy at the original lattice site). As diffusion continues, we have countercurrent flows of atoms and vacancies, called vacancy diffusion.

方式：原子跃迁到与之相邻的空位；

（2）空位机制 条件：原子近旁存在空位。

（金属和置换固溶体中原子的扩散。）

直接换位direct interchange diffusion mechanism

（3）换位机制

环形换位Zener ring mechanism for diffusion

（所需能量较高。）

In general, the activation energies for vacancy-assisted diffusion Q v are higher than those for interstitial diffusion Q i. The reason is that the former mechanism requires energy to both form a vacancy and move an atom into the vacancy, while in the latter case energy is needed only to move the interstitial atom into the interstitial site.

3-3 扩散的驱动力

Sec. 3.3 The Driving Force for Diffusion

（1）扩散的驱动力

对于多元体系，设n为组元i的原子数，则在等温等压条件下，组元i原子的自由能可用化学位表示：

μi=?G/?n i

扩散的驱动力为化学位梯度，即

F=-?μi /?x

负号表示扩散驱动力指向化学位降低的方向。

（2）扩散的热力学因子

组元i的扩散系数可表示为

D i=KTB i(1+? lnγi/? lnxi)

其中，(1+? lnγi/? lnxi)称为热力学因子。当(1+? lnγi/? lnxi)<0时，D I<0,发生上坡扩散。

（3）上坡扩散

概念：原子由低浓度处向高浓度处迁移的扩散。

驱动力：化学位梯度。

其它引起上坡扩散的因素：

弹性应力的作用－大直径原子跑向点阵的受拉部分，小直径原子跑向点阵的受压部分。

晶界的内吸附：某些原子易富集在晶界上。

电场作用：大电场作用可使原子按一定方向扩散。

3-4 影响扩散系数的因素

Sec. 3.4 Factors Affecting Diffusion Coefficient

1 温度Temperature

D=D0exp(-Q/RT)

可以看出，温度越高，扩散系数越大。

The kinetics of process of diffusion are strongly dependent on temperature. The diffusion coefficient D is related to temperature by an Arrhenus-type equation.

2 原子键力和晶体结构

Dependence on bonding and crystal structure

原子键力越强，扩散激活能越高；致密度低的结构中扩散系数大（举例：渗碳选择在奥氏体区进行）；在对称性低的结构中，可出现明显的扩散各向异性。

A number of factors influence the activation energy for diffusion, and hence, the rate of diffusion. Interstitial diffusion, with a low-activation energy, usually occurs much faster than vacacy, or substitutional diffusion. Activation

3 固溶体类型和组元浓度的影响

Dependence on types of diffusion and concentration of diffusing species and composition of matrix

The diffusion coefficient depends not only on temperature, but also on the concentration of diffusing species and composition of matrix.

间隙扩散机制的扩散激活能低于置换型扩散；提高组元浓度可提高扩散系数。

4 晶体缺陷的影响

Dependence on crystal defects

（缺陷能量较高，扩散激活能小）

空位是空位扩散机制的必要条件；

位错是空隙管道，低温下对扩散起重要促进作用；

界面扩散－（短路扩散）：原子界面处的快速扩散。

如对银：Q表面=Q晶界/2=Q晶内/3

5 第三组元的影响

Dependence on components

如在钢中加入合金元素对碳在 中扩散的影响。

强碳化物形成元素，如W, Mo, Cr,与碳亲和力大，能显著组织碳的扩散；

弱碳化物形成元素，如Mn，对碳的扩散影响不大；

固溶元素，如Co, Ni, 提高碳的扩散系数；Si降低碳的扩散系数。

3-5 反应扩散

Sec. 3.5 Diffusion in Non-isomorphic Alloy Systems

（1）反应扩散：有新相生成的扩散过程。

（2）相分布规律：二元扩散偶中不存在两相区，只能形成不同的单相区；

三元扩散偶中可以存在两相区，不能形成三相区。

3-6 其它固体材料中的扩散（略）

Sec. 3.6 Mechanisms of Diffusion in other materials