Measurements for Cu and Si diffusivities in
Al–Cu–Si alloys by diffusion couples
Dingfei Zhang
a,∗, John E. Morral
b, Harold D. Brody
c aCollege of Materials Science and Engineering, Chongqing University, Chongqing 400045, ChinabDepartment of Materials Science and Engineering, The Ohio State University, OH 43210, USA cInstitute of Materials Science, University of Connecticut, CT 06268, USA
Received 29 July 2006; received in revised form 13 September 2006; accepted 24 October 2006
Abstract
This paper deals with diffusivity measurements for Cu and Si atoms in Al–Cu–Si alloys by diffusion couples. With Cu and Si concen-tration profiles in the diffusion couples treated by different heating temperatures and times, their diffusivities including DCuCu, DSiSi, DCuSi
and DSiCu in Al–Cu–Si ternary alloys could be calculated and their D0 and Q also could be obtained by their diffusivities under different
temperatures.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Diffusivity; Al–Cu–Si alloys; Diffusion couples
1. Introduction
Being able to predict diffusion behavior is important for the design of heat treatment, especially when particle solution or growth process is involved. The most important parameter for diffusion behavior is diffusivity. Some diffusivities can be obtained from references or handbooks[1–3], and the others have to be obtained from experiments.
In order to simulate the solution processes of Al–Cu–Si alloys and obtain the optimal processing parameters, it is necessary to get diffusivities of Al–Cu–Si alloys including DCuCu, DCuSi, DSiCu and DSiSi besides good software pack-ages. One of the best ways to get these data is diffusion couple experiment[4–8]. The process of diffusion, within a diffusion couple, occurs through a mass transfer of elements from one alloy (one side of the couple) to the other. Lots of informa-tion for diffusivity and phase transformainforma-tion can be obtained from the concentration profiles measured from diffusion cou-ples. Thus, it is possible for us to calculate diffusion rate, predict phase diagram, find the relationship between the micro-hardness and the concentration of an alloying element and so on.
∗Corresponding author. Tel.: +86 23 65112491.
E-mail address:[email protected](D. Zhang).
2. Background
2.1. Fundamental diffusion equations
2.1.1. Diffusion equations in binary and multi-component systems
For binary diffusion in single phase region, diffusion is gov-erned by Fick’s first law and second law,
J = −D∂c ∂x (1) ∂c ∂t = D ∂2c ∂x2 (2)
where J is the flux of the diffusion component, D its diffusivity and c is its concentration; D is supposed to be constant in Eq. (2).
For multi-component diffusion, Eqs.(1)and(2)become, in matrix notation, [J] = −[D]∂[c] ∂x (3) ∂[c] ∂t = [D] ∂2[c] ∂x2 (4)
where [J] and [c] represent (n− 1) ×1 column matrices, or vectors. Asciis constant, only (n− 1) independent variables 0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
are considered. The diffusivity [D] is a square matrix of order (n− 1). From kinetic and thermodynamic laws, [D] must have real positive eigenvalues and can be diagonalized[9].
2.1.2. Ternary and single phase diffusion problem
The diffusivity, in the binary case, is a single scalar quan-tity. In multi-component diffusion problems, the diffusivity is a (n− 1) × (n − 1) matrix. The matrix form of the multi-component diffusivity reflects the influence that each solute element has on the diffusivity of all of the other solute elements on the alloy.
The diffusivity matrix of a ternary alloy has the form, [D] = D11 D12 D21 D22 (5) The first subscript indicates which solute element is diffusing. The second subscript indicates the solute element whose con-centration gradient the diffusing element is moving through.
The ternary and single phase form of Fick’s second law can be written, ∂c1 ∂t = D11∂ 2c 1 ∂x2 + D12 ∂2c 2 ∂x2 (6) ∂c2 ∂t = D21 ∂2c 1 ∂x2 + D22 ∂2c 2 ∂x2 (7)
Unlike for the binary case of Fick’s second law, these two dif-ferential equations cannot be solved directly. Each of the ternary equations depends on two different solute concentration, c1and c2. To make the ternary equations easier to solve, they are re-written for an n-component system using matrix notation. ∂c ∂t = [D] ∂2c ∂2x (8) The next objective is to diagonalize the diffusivity matrix, [D], transforming the system of linear equations in such a way as to have each linear equation be dependent on only one concentra-tion variable (either c1or c2). The derivation and result can be found on paper[10].
2.2. Diffusivity measurement
Normally, two diffusion couples are needed to measure the diffusivity matrix of a ternary alloy, but with the constant diffu-sivity approach used here, it is possible to measure [D] using one diffusion couple only. However, the accuracy of [D] increases as more diffusion couples are used. The calculation done in this work is based on a combination of the following two different methods as described by Son and Morral[9]: the square root diffusivity analysis.
It has been known [11]that the amount of solute passing through the initial interface of a single phase couple in time t is given by the equation,
[S] = − t π[r][c 0 ] (9)
in which [r] is the square root diffusivity matrix[12], defined as [D] = [r][r], and [c0] is the initial concentration difference between the right and left side of the couple, also called the composition vector[10]. For a ternary system, the above matrix equation gives two linear equations, one for each element, S1 and S2of [S].
Furthermore, it was shown by Thompson and Morral[13] that the square root diffusivity equations could be modified as: [c0
]= 2√πt[r][∇c] (10)
where [c] is the gradient of concentration at the plane x = 0, at time t. The above matrix equation gives two linear equations also, one for each element,c01andc02, of [c0].
The two equations given by Eq.(9) and the two equations given by Eq.(10) can be solved to find the four elements of the square root diffusivity matrix. The quantities S1, S2,c01, c0
2 andc1 andc2 can be measured from electron probe microanalysis data of one single diffusion couple.
3. Experimental procedure
3.1. Sample preparation
3.1.1. Selections of alloys and temperatures
A ternary single phase system in Al–Cu–Si alloy is selected for diffusion couple experiment, the compositions of two dif-fusion couples are Al–0.4%Si–3.0%Cu versus Al–0.4%Si and Al–1.5%Cu–0.8%Si versus Al–1.5%Cu (to measure Cu and Si diffusivities, respectively, in Al–0.4%Si–1.5%Cu alloy). All the compositions locate in the␣ single phase region. Three heating temperatures were selected as 480, 510 and 540◦C.
3.1.2. Alloying, casting and heat treating
The eutectic alloys of Al–Cu (33.2% Cu) and Al–Si (12.5% Si) were formed first from pure component metals, then final alloys were made from the eutectic alloys and pure aluminum. The liquid alloys after alloying were solidified in a cold copper mold, the fast cooling rate could effectively reduce the segrega-tion of copper and silicon.
All the samples, after solidification, were heat treated for 72 h at 500◦C. Real single phase samples with homogeneous composition were made after the heat treatment.
3.1.3. Polishing
After homogenization, the alloys were machined into cylin-der samples of Ø 6 mm× 3 mm. The polishing process is to make the two surfaces of every samples parallel mutually and the surface used for diffusion flat and mirror-like.
The first polishing step was to grind samples with grinding papers from 320, 400, 600 to 800 grit, the second was done on a Struers MD Chem polishing cloth, impregnated with Leco alpha alumina from 3.0, 1.0 to 0.05m.
3.1.4. Diffusion couple construction
As soon as the polishing work was done, the diffusion cou-ples were made by inserting four alloy wafers into a specially
Fig. 1. The concentration profiles of Cu and Si in Al–0.4%Si–3.0%Cu vs. Al–0.4%Si couple after 98 h heating at 480◦C. designed diffusion couple clamp in order to keep the diffusion
surfaces fresh (no or less oxidization). The clamp consisted of a stainless steel tube, whose bore had reamed to a diameter of 6± 0.05 mm. Each end of the cylinder bore was tapped to receive a stainless steel set screw, allowing the tube to be sealed from both ends.
3.1.5. Heating cycles
Once the diffusion clamps have been constructed, they must be heated as soon as possible. After the heating, the diffusion clamps were removed and quenched into water as quickly as possible. It was found from the micro-hardness analysis of the samples under 510◦C× 144 h and 540◦C× 96 h that the diffu-sion areas were large enough for measurement and calculation, so the heating time under 480◦C was reduced to 98 h.
3.2. Electron probe analyses
Following the thermal cycles of the diffusion experiments, the diffusion clamps were sectioned to expose the diffusion couples. Electron probe analyses and micro-hardness checks were made after the diffusion couples were polished carefully.
The microprobe data included weight percentage, atomic per-centage of copper, silicon and aluminum for every measuring point, the distance between two adjacent points was 15m. The measuring range depends on where the concentration of ele-ment measured keeps unchangeable. Micro-hardness profiles were measured before microprobe measurement to check the
approximate diffusion distances and whether diffusion process in diffusion couple worked well.
4. Results
4.1. Concentration profiles
All the Cu and Si concentration profiles under these heating temperatures and times could be obtained by the electron probe measuring data, two of the measured concentration profiles are shown in Figs. 1 and 2. Then based on Eqs.(9) and(10), the diffusivities of DCuCu, DSiSi, DCuSiand DSiCucould be calculated from these profiles.
4.2. DCuCuand DSiSicalculation
It was found from Fig. 1 that great copper concentra-tion change resulted in little silicon concentraconcentra-tion change in the couple, which meant that copper concentration change in 0–3.0 wt.% had little effect on the silicon activity in the Al–3.0%Cu–0.4%Si versus Al–0.4%Si couple, while it was also proved from Fig. 2 that silicon concentration change in 0–0.8 wt.% had little effect on the copper activity in the Al–0.8%Si–1.5%Cu versus Al–1.5%Si couple. Thus, DCuSiand DSiCu should be so much smaller than DCuCu and DSiSi that their values could be regarded approximately to zero in the calculation. The calculation results were listed inTable 1.
Table 1
Calculation results of DCuCuand DSiSi(×10−14m2/s)
Heat treating Element SR SL (SR+ SL)/2 C0(wt.%) D1 C (wt.%/cm) D2 Dave
480◦C (98 h) Copper 141.0 168.2 154.6 −3.01 2.35 −101.4 2.00 2.18 Silicon 54.9 49.7 52.3 −0.67 5.46 −16.1 3.88 4.67 510◦C (144 h) Copper 275.3 306.9 291.1 −3.00 5.70 −57.6 4.12 4.91 Silicon – – – −0.76 − −11.4 8.91 8.91 540◦C (96 h) Copper 262.7 344.5 288.6 −3.15 8.46 −54.0 7.59 8.03 Silicon 75.9 89.4 82.7 −0.61 16.8 −6.3 21.2 19.0
SR: the amount of solute diffused to the lower concentration side, wt.%m. Its value equals to the area as shown inFig. 1. SL: the amount of solute lost by diffusing to the lower concentration side, wt.%m. Its value equals to the area as shown inFig. 1. SLshould be equal to SRif the interface of a diffusion couple is just located on the 0m. D1: Diffusivity calculated from(9). D2: Diffusivity calculated from(10). Dave: The average of D1and D2.
Fig. 3. ln (D)–1/T curves of Cu and Si.
All the data inTable 1looks reasonable except the data of silicon at 510◦C the profile of which is shown inFig. 2b, it was found that a very thin oxidizing film in its interface made its concentration profile break in the interface, so only its gradient of concentration was used to calculate its approximate diffusivity by Eq.(10). D1should be equal to D2if there are no experimental or measuring error, comparatively the error of D1 is less than that of D2during measuring as the accuracy of area measuring is higher than that of slope measuring.
5. Discussion
5.1. Calculation of Q and D0of Cu and Si
The relationship equation between diffusivity and tempera-ture has the form:
D = D0 exp − Q RT (11) where Q is the activated energy of diffusion (kJ/mole), T the temperature (K), R is 8.31 J/mole K and D0is the constant (m2/s). Having drawn ln (D) and 1/T curves with above data of dif-fusivity and temperature (Fig. 3), we could obtain the D0and Q of copper and silicon (Table 2).
Table 2
Calculation of D0and Q
Elements Equation of ln (D) and 1/T D0(m2/s) Q (kJ/mole) Copper ln (D) =−13.44 − 13507/T 1.45× 10−6 112.3 Silicon ln (D) =−11.67 − 14361/T 8.57× 10−6 119.4
5.2. Estimation of DCuSiand DSiCu
By definition, the diffusivity, [D], is related to its square root, [r], by:
[D] = [r][r] (12)
Eq.(12)is valid for any number of components and the function Dijversus rijcan be obtained in a straightforward way by matrix
multiplication. As Al–Cu–Si alloy system, each coefficient of [D] can be described in terms of [r] coefficients by:
DCuCu= r2CuCu+ rCuSirSiCu (13)
DCuSi= rCuSi(rCuCu+ rSiSi) (14)
DSiCu= rSiCu(rCuCu+ rSiSi) (15)
DSiSi= rSiSi2 + rCuSirSiCu (16)
and the [r] can be calculated by equations below, SCu SSi = − t π rCuCu rCuSi rSiCu rSiSi c0 Cu c0 Si (17) c0 Cu c0 Si = −2√πt rCuCu rCuSi rSiCu rSiSi ∇cCu ∇cSi (18) Because it was found in our experiments that the copper’s concentration change resulted from silicon’s diffusion or the silicon’s from the copper’s near the interface of diffusion couple is too small to be measured accurately, only the approximate calculation results for DCuSi and DSiCu could be obtained as Table 3
Approximate calculation results of DCuSiand DSiCu
Diffusion temperatures (◦C) rSiCu(m/s1/2) rCuSi(m/s1/2) DSiCu(m2/s) DCuSi(m2/s)
480 −7.61 × 10−10 −1.53 × 10−8 −2.77 × 10−16 −5.57 × 10−15
shown inTable 3. The corresponding concentration data near the interface of the diffusion couple heated at 540◦C is too random to obtain approximate DCuSiand DSiCu, only calculation results at 480 and 510◦C are listed inTable 3.
6. Summary
Diffusion couple experiment is an effective method to mea-sure diffusivities. Ternary diffusivities can be meamea-sured with one diffusion couple only, using constant [D] models, as shown in Eqs. (9) and (10). Good diffusivity data of DCuCu, DSiSi and their D0, Q in Al–Cu–Si alloys have been obtained by proper diffusion couple experiments, diffusivities of DCuSiand DSiCuin Al–Cu–Si alloys could only be estimated in this work because their values are so small and their concentration data are com-paratively random.
References
[1] Landolt-B¨ornstein, Numerical Data and Function Relationships in Science and Technology, New Series III/26, Springer-Verlag, Berlin, 1998. [2] J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford, 1975. [3] J. Hatch, Aluminum: Properties and Physical Metallurgy, American Society
for Metals, Metals Park, Ohio, USA, 1984.
[4] D. Zhang, J. Peng, T. Liu, Mater. Sci. Eng. A 425 (6) (2006) 78. [5] V. Jindal, V.C. Srivastava, A. Das, et al., Mater. Lett. 60 (6) (2006) 1758. [6] L.R. Ram-Mohan, M.A. Dayananda, Acta Mater. 54 (5) (2006) 2325. [7] C.E. Campbell, J.-C. Zhao, M.F. Henry, Mater. Sci. Eng. A 407 (10) (2005)
135.
[8] H. Yang, J.E. Morral, Y. Wang, Acta Mater. 53 (8) (2005) 3775. [9] Y.-H. Son, J.E. Morral, Metallurgica 28 (1993) 833.
[10] P.K. Gupta, A.R. Cooper Jr., Physica 54 (1971) 148. [11] M.S. Thompson, J.E. Morral, Acta Metall. 34 (1986) 339. [12] J.E. Morral, Scr. Met. 18 (1984) 1251.