Diffusion in Solid State

Transcription

1 (Diffusion in Solid State): - Diffusion in Solid State Diffusion Mechanism It is known that at high temperature, the Amplitude of vibrations of atoms leads to expansion, while keeping the (Frequency of Vibration) constant in general form

2 Diffusion by voids Nv Na exp Hf / RT The percentage of voids in metal lines of F. C. C to % at high temperatures Wo, is the vibrations between the void and solvent A very far from the solute B W1, is the vibrations between the void and solvent A away, the closest distance from the void and solute B W, are the frequencies between the void and solute B W3, are the frequencies between the void and solvent atoms A, where A is the closest distance

3 from the void. W4, are the frequencies between the void and solvent A away from the solute B. The relationship between these vibrations known (Correlation factor) (f). f Wo W 1 + 7( ) W W4 Wo W 1 + W + 7( W 4 3 W 4 Wo ) W 3 f 7 W1 + W3 7 W1 + W + W 3 Diffusion by Interstitial atoms Q / RT D νa exp Vibration frequency,, Q the activation energy necessary for diffusion, ΔHm ν enthalpy

4 Fick,s first Law dc J D dx Expressed in this law the following equation: Where each of: J, is called flux. C, concentration of solute in the solvent D, diffusion coefficient To derive the law, we assume the existence of two surfaces adjacent the lattice A and B (each an alloy of copper and zinc), where the concentration of zinc in A, more than in the B,

5 The sum of the number of atoms of zinc, which can jump over the energy barrier per second, is: This is a net movement of atoms of zinc, down to include the focus from A to B. And therefore: Where J is, flux. The A, sectional area, from the definition of C: C A na A. a C J ν a[ C A CB ] b C A CB dc a dx B nb A. a وبما ان : () J ν ( n 6 And CA, CB is composition of zinc in both A and B. νa dc J. :. νa D b dc J D dx b dx The negative sign indicates that the flux direction is the direction of decreasing n. In comparison with equation (1) we get: - A n B 1 ). A Fick,s second Law

6 Δv A. Δx Where Δv is the size of the piece. The number of atoms inside the volume at a certain moment nδv n. A. Δx dj J A J A A. x dx dc J A J A. A. x dt dc dj dt dx dc J D dx dc dc D dt dx Solution of Fick,s second Law dc dt x u Dt dc du dc u. ( ) du dt du t dc dx dc du. du dx 1 dc. Dt du dc dx dc du du ) dx dc 1. du 4Dt dc d( ) du du ( dc dc u. du du

7 dc du y dc du :But dy udu y ln y - u + cte y Be u B is a constant dc y du Be u u اذن بما ان u C B e du + A 0 A is anther constant. erf (u) u e 0 u du u x Dt C A + Bref ( x ) Dt erf (u) function erf ( u) π u e 0 u du

8 π u e 0 u du. 0 erf (u) u + u e 0 u du 0 u 0 and π erf ( ) ( ) 1 π π erf ( ) ( ) 1 π erf (-u) -erf (u) Q-1

9 Q-

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12 Q-3 Q-4

13 Q-5 Q-6

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15 D D e Q / RT o (15).. "D": Diffusion coefficient Where Q represents the activation energy. The Do- diffusion constant, and be equal to ( ν a ). Table some of the values of D, Do, Q, for some metals and alloys. 3 x Q جول / مول D o متر / ثانیة D متر / ثانیة o 00 م x x 118 7, x 0, x 0,7 4, x x,6 9, x 0, x 0.5 منصھر Which can be written as the following: متر / ثانیة o 500 م , ,5 عملیة الانتشار المذیب D في المذاب Cu في Cu Fe (α ) في Fe Fe (α ) في C Fe (γ ) في C Fe (γ ) في Mn Fe (γ ) في Ni Cu في Zn Al في Cu ln D ln D o Q 1. R T k If a line chart decreed between ln D and inverted absolute temperature as in Figure () will get a straight line equal to his inclination, and since R is the gas constant, we will have a Q value easily. Since the intersection with the vertical coordinates, represents a ln Do, logarithmic tables, and we can get the value of Do.

16 معامل الانتشار الذاتي D سم / ثانیة 11- x 1.9 x 5 9 x x 5 7- x,68 Log D / كالفن /T k x 1.4 x 1.5 x 1.11 x 1 x درجة الحرارة T o K كالفن Figure () shows the relationship between the diffusion coefficient and the absolute temperature in some metals and alloys.

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18 Q-9 Q-8

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20 types of transactions diffusion: - 1-Self-diffution It is intended that the diffusion of radioactive metal dissolved in the same non-radioactive metal, such as Ni * in Ni, and Ag * in Ag. in general,the self-diffusion equation as follows: D * self D * 0 self e * Q / RT 1- grain boundaries Diffusion The diffusion in fine g,b is greater than large g,b. Equation can be written as follows: D g. b D og. b e Qg. b / RT 1- grain boundaries Diffusion In general, the value of activation energy for the grain boundaries diffusion is half of the value of the Self-diffusion.

21 1 Q g. b Q From this we can deduce that the diffusion coefficient in the g.b will be larger than the diffusion coefficient of Self-diffusion. self D > D g. b self 3 - surface diffusion: - Which is diffusion along the surface of crystals and is usually faster than the g.b diffusion and the Self-diffusion We note that: Ds D s o Q / RT s Q < Q s < Qg. b self s > Dg, b D > D self

22 Phenomenon of Krkindal: -

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24 pair diffusible of( copper nickel) alloy, was heated at 900 o c for a period of 40 hours,. If one of alloys was containing 40% a% nickel in place is located to the right of the surface interval, and the other alloy was containing 50% b% nickel in place is located to the left of the interface,find A The diffusion equation of copper nickel alloy B - the diffusion coefficient ( note that at a point about 0. cm from the boundary is 4.5% )? C - the time required to get the same concentration, but just 0.4 cm from the boundary? A - The first condition, at the beginning of the experiment time will be zero, and the concentration, is % b when X is negative., So the diffusion equation would be: x b% A + Berf ~ D 0 b% A + Berf ( ) b% A B The second condition, at the beginning of the experiment also be time zero, and the concentration is% a when X is positive, so the diffusion equation would be:

25 a% A + Berf + x D 0 a% A + Berf ( ) a % A + B a% + b% A a% b% B ~ N Ni a % + b% + a% b% N Ni 45% - 5% erf erf x ~ x ~ D. t D. t B - Upon passage of 40 hours, the concentration became 4.5%) at distance of 0. cm from the boundary. So: 4.5% 45% 5% erf x ~ D. t erf x D. t erf ( ) 0.5 D x cm D.563x sec

26 C - Since the concentration is fixed, 4.5% 45% 5% erf 4.5% 45% 5% erf x 1 ~ D. t x ~ 1 D. t x 1 t 1 1 erf ~ D. t D. t x t x 1 erf x ~ اذا :