Module 16
Diffusion in solids II
Lecture 16
Diffusion in solids II
1 NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | Keywords: Micro‐mechanisms of diffusion, effect of composition on diffusivity, Matano interface, Kirkendall effect Introduction: In the previous module we have learnt about the laws that governs diffusion of species in solids where this is the only process of mixing. We have so far looked at the phenomenology of diffusion. We have also learnt that the diffusivity depends not only on the temperature but also on its structure and the path it follows. Diffusion is primarily associated with the movement of atoms. This may be visualized as thermally activated movement of atoms of a given species in a regular array of atoms. Most solids are made of closely packed atoms. However there are a few vacant sites. It is likely that the movement of atoms would be guided by availability of vacant sites. It has been shown if such movements are assumed to be random it is possible to estimate the average diffusion distance as a function of the diffusivity and time. In this module we would learnt a bit more about the micro‐mechanisms of diffusion in solids. Micro‐mechanisms of diffusion in solids: Atoms in solids are arranged in a regular array. Most metals have nearly close packed structure. Figure 1 represents a typical arrangement of atoms in a common crystalline structure (FCC). We have also seen that that a particular temperature a few of the lattice sites may be vacant (fig 1). How would an atom move in such an array? There are several ways this could take place. On the basis of such a movement mechanism of diffusion can classified as (a) interstitial mechanism (b) vacancy (c) site exchange (d) indirect interstitial mechanism (e) ring mechanism. This is illustrated in slide 1. Fig 1: A sketch showing how the atoms are arranged in a particular plane of a close packed structure. Note that a few sites are vacant. Apart from the vacant sites there are sites between atoms known as interstitial sites. One of these is occupied by an atom of another kind (red color). Diffusion in crystalline solids: mechanisms
Diffusion: Thermally activated motion of atoms
5
1
2
2. Vacancy
3. Site exchange
4. Indirect interstitial
3
2 1. Interstitial
4
5. Ring
Slide 1: This illustrates how atoms of a given species could diffuse through a regular array in a crystalline solid. Light blue filled circles denote regular sites in the lattice. Red circle denotes an atom occupying an interstitial site. Color codes have been used to denote atoms of different kinds (species). Note that there are a few vacant sites. Atoms having much smaller diameter can be accommodated in interstitial sites (like carbon atoms in iron lattice). There are 5 different ways atoms could diffuse. The arrows denote the direction of motion.
NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | In the sketch in slide 1 there is a red atom in an interstitial site (marked 1). It could move along the arrow to the next interstitial site. While moving it has to push the next pair of atoms in the lattice by some distance so that it could pass through the gap. This can happen only in case of an atom having much smaller size than the atoms of the parent lattice. For example iron at room temperature has body centered cubic structure. Carbon has much smaller atomic diameter. It can therefore occupy an interstitial site in iron lattice. Vacancy assisted diffusion is illustrated in the sketch in slide 1 with the help of a grey atom. The arrow (marked 2) shows its direction of motion. As it moves to the vacant site this space gets occupied where as a vacant space is created at the place previously occupied by the grey atom. Atoms could exchange their positions resulting in an overall movement. This is illustrated in slide 1 by two atoms having two different colors marked with curve arrows (number 3). As a result of the exchange the white atoms takes the position previously occupied by the blue atom and vice versa. An atom occupying an interstitial site may push the neighboring atom occupying the main lattice site to the next interstitial site. This is known as indirect interstitial mechanism. This is illustrated in the sketch in slide 1 by a pair of atoms one in an interstitial (marked with green) and the other an atom in a normal lattice site (marked blue). Note that the blue atom moves to the next vacant interstitial site and the green atom takes up the site that was previously occupied by the blue atom. Diffusion can also be visualized as exchange of positions amongst a set of atoms. This is illustrated as the 5th mechanism in the sketch in slide 1. A set of four atoms marked with number 5 denotes the exchange of position may take place. Each of these mechanisms would have specific thermal barrier. This can be estimated on the basis of the crystal structure. Most metals have nearly close packed structure. In such cases the most likely mechanism appears to be either interstitial for atoms like C, N, H in iron lattice or vacancy assisted in cases of self diffusion or diffusion of substitutional atoms like Mn in iron lattice. Interstitial diffusion: Carbon atoms are much smaller than that of iron. Therefore in a Fe‐C alloy carbons atom occupy interstitial sites. The sketch in slide 2 shows the possible location of carbon atoms. However the solubility of carbon atom in Fe is limited. Only a very small fraction of these sites are likely to be occupied. 3 NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | Slide 2: The sketch shows the possible locations of carbon atoms in  iron having BCC structure. It also illustrates Consider: diffusion of C atom in  iron (BCC)
Interstitials already present. Diffusion involves
how to estimate the number of interstitial sites in a plane. migration
Consider plane 1. Apparently it has 5 sites. However these may be shared by the planes in the neighboring unit cell as No. interstitial in plane 1
= 4x1/2+1x1 =3
well. Those at the centers of the edges are shared by two Plane 1
No. interstitial in plane 2
whereas the one at the center belongs exclusively to this. = 4x1/2+4x1/4 =3
Thus the number comes out to be 3. In plane 2 there are 4 Plane 2
corner sites. These are shared by the extension of the J = J12-J21 = net flux from 1-2
same plane to 4 neighboring unit cells. Thus this too has the same number of sites. The above illustration shows that every plane in the lattice has identical number of interstitial sites. Only a few of these are likely to be occupied by carbon atoms. This is determined by the concentration of carbon in respective planes. The solubility of carbon in  iron (ferrite) is very small (~0.02%). Therefore it is reasonable to assume that all the nearest neighboring interstitial sites around a carbon atom are vacant. On an average each interstitial site has 6 nearest interstitial sites of these 4 are in the same plane and one each on the two neighboring planes (one in the front and the other at the back). The probability of an atom in plane 1 to move to plane 2 is therefore 1/6. The distance between the two planes is a/2 (where a = the lattice parameter). This denotes the average jump distance of a carbon atom. The net flux of carbon atoms from plane 1 to plane 2 can therefore be estimated from the ; where J12 difference in the jump frequency between the two planes. This is given by
denotes flux from plane 1 to plane 2 and J21 denotes flux of carbon atoms from plane 2 to plane 1. The flux from the plane 1 to plane 2 is equal to the frequency of jump x probability of a successful jump from plane 1 to plane 2 x the concentration of carbon atom in this plane. The calculation steps are given in slide 3. Slide 3: Illustrates how the diffusivity of an interstitial atom can be related to the average Interstitial diffusion
jump frequency and lattice parameter. The J12 = concentration x jump frequency x (1/6)
sketch on the right shows a plot of free energy a/2
G
as a function of the location of the interstitial J = (/6) (n1-n2)
G
atom. The arrays A, B & C give the location of a
Area conc. = n1 = c (a/2)
the interstitial atom during initial, intermediate    dc  a     a 
and the final stage of the jump. The positions A A
C
J   c  c       
6   dx  2     2 
& C are the positions of thermodynamic B
a 2 dc
a 2
equilibrium (G=0), whereas the intermediate 
D
24 dx
24
state B has the maximum energy. For diffusion to occur this barrier must be overcome. Atomistic Mechanism
4 The jump frequency is given by . The number of carbon per unit area of plane 1 and plane 2 are n1 & n2. The flux (J) in terms of number of atoms moving from plane 1 to plane 2 is given by
. If c is the concentration of carbon (per unit volume) in plane 1, the number of carbon atoms in plane 1 is NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | . The concentration of carbon in plane 2 is given by
given by
carbon in terms of concentration / unit volume is given by
. Therefore the flux of . On comparison of this with the equation representing Fick’s first law the expression of diffusivity of carbon atom in iron lattice comes out to be: (1) In this expression ‘a’ denotes lattice parameter. If the same is expressed in terms of the jump distance (a) it is given by . Now let us examine how can we express  the jump frequency. This represents the number of successful jumps an atom makes in unit time. This should be a strong function of temperature. Let D denotes the number of attempts made by an atom to jump from plane 1 to plane 2. For a successful jump the activation barrier (hill) must be crossed. In this case let Gm denotes the activation barrier (see slide 3.) Therefore the expression for diffusivity of an interstitial atom can be given by: (2) In the above equation k is Boltzmann constant and D is Debye frequency (1013 /s). Diffusion of substitutional atom: We have just seen how the process of diffusion of an interstitial atom can be represented in terms of its random movements (jumps). The same can be extended to the diffusion of atoms that occupy the normal lattice sites. The main difference lies in the estimation of the number sites available for an atom to move to. In case of interstitial sites the number of sites is too many and only a small (negligible) fraction of it is occupied. In the case of the diffusion of atoms in normal sites it is not so. If Z denotes the co‐ordination number and cv is the concentration of vacancy the number of vacant sites is Zcv. Therefore diffusion here would depend on the concentration of vacancies. This is illustrated in slide 4. Diffusion of atoms in normal lattice site
Atoms can move only if
the nearest site is vacant
z = coordination number
cv = vacant sites (in fraction)
Self diffusion of iron: DFe BCC
z=8
Probability that nearest site is vacant = z cv
5 D
2 2
 G 
zcv D exp   m 

6
6
 kT 
Slide 4: Illustrates the relation between diffusion of atoms & the number of vacancies. The sketch on the left shows an atom (red dot) that can move if the next site is vacant. Or else it has to wait till the next site falls vacant.  denotes frequency at which a vacant site can change its position. 1/ denotes relaxation time (). It is a measure of the average time a vacancy spends in one location. The probability that an atom can jump from plane 1 to plane 2 is 1/6. Thus the diffusivity (D) is given by
, where  is the smallest jump step. NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | It is easiest to visualize the process of diffusion in terms of probability of an atom moving from one plane to the next atomic plane in a simple cubic lattice. An atom has 6 nearest neighbor four of these lie in a plane (say plane 1). Any movement within this plane does not amount to a movement in the direction of interest. There is only one site in plane 2 which would represent diffusion along a direction say x (see fig 2). x z y Fig 2: A sketch showing the way atoms are arranged in a simple cubic lattice. Let us consider diffusion along the direction x. The atom shown in red can move to any of the six sites. However if it moves to the site shown with dotted circle, it amounts to diffusion along x axis. This is one amongst 6 sites. The jump distance is one atomic distance (a). There is another condition that needs to be satisfied for diffusion to take place. The site in the next plane should be vacant. The probability that it is vacant is Zcv. Therefore diffusivity of a substitutional atom is given by (3) The fraction of sites that are vacant (cv) is given by Boltzmann statistic. If the free energy associated with the creation of a vacancy is Gf it is given by
. On its substitution in equation 3 you get: (4) It is well known that Gibb’s free energy (G) is given by G = H –TS where H is enthalpy and S is entropy. Therefore both Gm & Gf would have two components. On substitution of this and isolating the temperature dependent and temperature independent terms one gets the following general expression for diffusivity: (5) The temperature independent part is given by (6) Therefore the general expression for diffusivity is given by: 6 (7) In the above equation q represent activation energy of diffusion. Its dimension is J/atom. Often for most engineering application it is represented as J/mole. The most commonly used expression for diffusivity of a given species is as follows: (8) NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | In equation 8, R is universal gas constant in J/mole/°K and Q is activation energy in J/mole. Experimental determination of self diffusivity: We have seen that the most common mechanism of diffusion in solids (metals) is through the movement of vacancies. Even in pure metal atoms keep changing their positions with vacant sites. The concentration of vacancy in metals is a function of temperature. If we quench a piece of metal from a high temperature it would have higher concentration of vacancies. This is because fast cooling does not allow enough time for the excess vacancies to diffuse out. The presence of excess vacancies makes it unstable. This would try to diffuse out with time. If the concentration of vacancies can be monitored at a given temperature as a function of time it should be possible to estimate its diffusivity. One of the common methods is to measure the change in resistivity of metals as a function of time at a given temperature. Figure 3 shows how it would change with time at a given temperature (T1). If the temperature is suddenly increased to T2 there will be a change in resistivity. T1 (d/dt)T2 
(d/dt)T1 T2 Fig 3: A sketch showing the change in resistivity as the function of time at a given temperature T1. The slope of the curve gives the rate of change of resistivity at T1. If at given time the temperature is raised to T2 there is an instantaneous change in the rate of change of resistivity. It can be estimated from the plot as shown. Time The rate of change of resistivity is proportional to the rate of change of the concentration of vacancy. This in turn is inversely proportional to the relaxation time () for vacancies to diffuse out. Therefore using the data on the rate of change of resistivity at two different temperatures it is possible to get an estimate for the activation energy for the diffusion of vacancies or self diffusion coefficient can be estimated. The calculation steps are as follows: ∝
∝ ∝
(9) (10) Equation 10 thus gives an estimate of the activation energy for self diffusion or that for the diffusion of vacancy in a matrix. Effect of concentration on diffusivity: 7 While introducing the concept of solid state diffusion we did assume that the diffusivity is independent of composition. The diffusivity of a species depends on the number of jumps an atom makes at any instant and the number atoms of the same kind present at a given location. Therefore it is more likely that it should be a function of composition. In a diffusion couple the concentration profile is a function of distance. Hence the more general form of Fick’s second law should be as follows: NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | (11) (12) The slide 5 illustrates the likely concentration profile across a diffusion couple made of two alloys A & B at three different times (0, t, ∞). The solution of the equation is much more complex. Note that equation 11 is partial differential equation in one dimension. A more general form of this equation in 3D would be an extension of the same with additional terms. Let us look at the essential features and procedure for solving equation 11. Let us introduce a new variable given by √
(13) Slide 5: Illustrates how the concentration profile of a species in two alloys A & B would vary with time. The initial concentration of the species in A is c0 and that in B is c. At time = 0 the concentration profile is a step function denoted by t0. At infinitely long time the concentration of the species should be uniform all through. This is denoted by the profile t. At an intermediate time the profile is given by t. The more general form of Fick’s second law should be represented as shown. Effect of composition on D: diffusivity
A
B
concentration
c
t0
t
c   c

D
t x x
tα D
  D c
c0
0
c: concentration of B
If you substitute equation 13 in equation 11, it gets converted into an ordinary differential equation. The derivation is given in slide 6. Effect of composition on D: diffusivity
If

x
t
Fig 6: Illustrates how by introducing a new variable  that is equal x over square root of t, it is possible to transform the partial differential equation into an ordinary differential equation between two variables c and  at a given time t. This suggests if the concentration of a given species is known as a function of distance it can be solved numerically to estimate diffusivity for different concentration. c   c
 D
t x x
c c 
1 x c
c c  c 1


&


t  t
2 t t 
x  x  t

1 x c
1    c 1 

D

2 t t 
t    t 
Partial diff eqn is transformed to ordinary diff eqn.
8 The Fick’s equation in terms of the new variable  can therefore be written as follows: (14) On integrating equation 14 you get the following equation: NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | (15) On substitution of equation 13 in equation 15 you get the following equation: (16) Look at the concentration profile in slide 5. The slope .Therefore (17) c Fig 7: Illustrates how to locate the interface from where the distance x should be measured to find the integral in equation 17. It is so located that the two areas on either sides of the interface are equal. This is known as the Matano interface. 0 c c0 x Figure 7 explains how to locate the interface from where x should be measured to evaluate the integral in equation 17. Note that the area between the limit c0 and c is zero. This is because on the left of the interface x is negative and that on the right it is positive. c c c
Elemental area c0 x Fig 8: Illustrates how to estimate the integral in equation 17. The vertical dashed line at the centre is the Matano interface. The dark shaded portion is the elemental area xdc. Extend it fill up the region marked by a set of short vertical lines to the total area till the desired concentration c. The gradient of the inclined dashed line denotes the slope of the concentration profile in the right hand side of equation 17. Now that the integral in equation has been obtained, the slope dc/dx has been determined and time t is known therefore the diffusivity can be estimated. It can be done for all values of concentration c. Kirkendall effect: 9 Consider a diffusion couple made of two metals A & B. Assume that A & B are completely miscible in each other. If kept for an infinitely long time at high temperature the two would mix completely to form a homogeneous solid solution. Let there be a set of markers at the interface. These are made of metal wires having little solubility in both A and B. If it is kept in a furnace at a temperature high enough for A atoms to diffuse into B and B atoms to diffuse into A. What would happen if the diffusivity of A (DA) is greater than that of B (DB)? Obviously more number of A atoms would diffuse into B or cross the marker NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | than those of B atoms. Consequently the location of the marker would shift. The phenomenon is known as Kirkendall effect. The direction of movement will just be the opposite if DB > DA. If the number of A atoms crossing the markers is greater than the number of B atoms moving over to A by crossing the markers it might leave a trail of vacancies in A. Some of these may coalesce to form voids. The detection of Kirkendall voids in metals suggests that the mechanism of diffusion in metals is controlled by vacancies. Marker Fig 9: Illustrates what happens in diffusion couple when the DA > DB. The circles denote markers. At time t = 0 these are at the interface of A & B. After sometime the markers move towards A by a distance = x. A
B
A
B
Experimental measurements have shown that metals having lower melting points have higher diffusivity at the same temperature. For example diffusivity of copper at a given temperature is higher than that of nickel (DCu > DNi). Therefore in a diffusion couple made of copper and nickel; the formation of voids or pores is expected within copper near the interface. Slide 7 suggests a simple experiment to demonstrate Kirkendall effect. Kirkendall effect
Cu
 brass
More of Zn diffuses
into Cu than Cu into
brass.
cZn
Mo markers move closer
Slide 7: A piece of  brass and copper base alloy having 30% Zn is inserted in a block of copper with Mo markers at the interface as shown in the top sketch. The next figure shows a section of the above sketch. Note the distance between the markers at time t = 0 and the concentration of Zn as a function of distance. Diffusivity of Zn is higher than that of Cu. More number of Zn atoms from brass would cross the markers than the number of Cu atoms. Therefore the area occupied by brass would shrink. The distance between the markers would decrease.
Darken equation: Kirkendall effect shows that in a diffusion couple made of two metals the movement of the two species may not be the same. The flux of A and B atoms / unit area across the interface are given by:
and respectively; where nA and nB are the number of A and B atoms per unit volume at a given location. Location of marker at time t x 10 Fig 10: A sketch showing the movement of markers in a diffusion couple between time t and t+t. The distance is x. The velocity is given by
∆
∆
. If the cross section area of the couple =1, v also denotes the volume of the matter that has passed through Location of marker at time t+t the marker in the opposite direction. NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | The volume of matter that has crossed the marker is given by
. Equating this with v and keeping in mind that the directions of net flux and v are opposite the following expression is obtained: (18) and atom fraction B =
Since atom fraction A = and 1therefore: (19) On substitution of equation 19 in equation 18: (20) This relates the two diffusivities with the velocity of the marker. Note that if DA = DB then the markers remain stationary. However more often these are not likely to be the same. There will be a definite velocity at which the markers would move. It can be determined experimentally. So far while deriving the second law of diffusion the difference between the diffusivities of the species were ignored. Let us now examine how it gets modified due to Kirkendall effect. This derivation is based on the assumption that no void or pore is formed in the diffusion couple. Slide 8 presents the basic concepts based on which a more general form of an equation that describes how the concentrations of species would evolve with time in a diffusion couple made of two metals A & B. The flux of A atoms at x+x is given by: (21) ∆
Darken equation
A
Marker at time t
Slide 8: A sketch showing the locations of the markers in a diffusion couple after it has been at a given temperature (T) for a duration t and t+t. Over this period the markers have moved through a distance x. On the basis of the assumption that pores do not develop during diffusion the flux of atoms at x and x+x are described by the equations given in this slide. B
x
Marker at time t+t
Velocity of markers = v = x/t
Assumption: no porosity formation due to diffusion
nA
 vnA
x
J
Flux of A at x  x  J A  A x
x
Flux of A at x  J A   DA
11 Note that denotes flux of A atoms crossing the marker at x at time t whereas ∆ is the flux of A atoms crossing the markers at x+x at time t+t. The cross section area of the diffusion couple is unity. Therefore the difference between the two represents the accumulation of A atoms within the volume between the two markers. The volume is equal to x. Thus the rate of change of the number of A atoms within this volume is given by: NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | ∆
(22) This can be converted to atom fraction (or concentration) by dividing the both sides by the total number of atoms within this elemental volume. (23) On substituting the expression for the velocity (equation 20) in equation 23 and noting that
1: Where (24) (25) The diffusivity can be obtained from experimental data using equation 17. In addition by placing markers at the interface it is possible to find its velocity at a given temperature. We now have two equations 20 & 25 therefore intrinsic diffusivities (for individual species A & B) DA & DB can be determined. Summary: In this module we have learnt about the mechanisms of diffusion in solid. The importance of the presence of vacancies has been highlighted. Factors determining diffusivity have been explained. This includes temperature, crystal structure, microstructure and composition. Experimental methods for determining diffusivity, the concept of Matano plane and Kirkendall effect have been introduced. The formation of voids or pores indicates that diffusion in solids is controlled by vacancy. Using atomistic model Darken equations relating diffusivities of individual species in a binary system with the velocity of markers and the effective diffusivity have been derived. Exercise: 12 1. Plot the concentration profile of carbon in iron sample which was kept at 927˚C for 8hours while maintaining 1 % carbon concentration at its surface. Assume that the initial carbon content of iron is negligible. 2. It is often thought that that species having lower activation energy diffuses faster than the one having higher activation energy. Is this always true? 3. Rank the following samples in order of increasing self diffusion coefficients (a) Aluminum single crystal, (b) Polycrystalline aluminum whose average grain size is 5micron (c) Polycrystalline aluminum whose average grain size is 10micron. NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | | Answer: 1. Carbon concentration (profile) is given by D0 Q T D t (Dt)0.5 2.00E‐05 143 1200 1.18E‐11 8 0.000584 √
m/s2 kJ K m/s2 hr 1
√
x, m
c % 0
0.0002
0.0004
1.00 0.81 0.63 0.0006
0.0008
0.47 0.33 0.001
0.0012
0.0014
0.0016
0.0018
0.002
0.23 0.15 0.09 0.05 0.03 0.02 1.2
1
% C
0.8
0.6
0.4
0.2
0
0
0.0005
0.001
0.0015
0.002
0.0025
x, m
2. Assume diffusivities of two species in a given matrix are given by 13 & respectively. Let Q1 > Q2. Let us plot logarithm of diffusivity against reciprocal of temperature. Such plots are known as Arrhenius plot whose slope gives activation energy. This clearly shows that there might 1 be a critical temperature shown by dotted line where both have Log D 2 identical diffusivity. In this case at NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | lower temperatures species 2 has | | higher diffusivity where as at higher temperature species 1 has higher 1
diffusivity. 3. Diffusion through grain boundaries is much faster. Since single crystal does not have any grain boundary its diffusivity is the lowest. Finer the grains provide more number of high diffusivity paths. Therefore diffusivity is expected to be higher. Thus 14 NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering || | | |