mass transport in solid
a. diffusion drived by dμᵢ b. electromigration by dϕ c. thermomigration by dT d. capillary by dγ e. stress migration, e.g. creep
J=c∙υ=c∙MF, M= υ/F mobility; Jₑ=ρE=(neμ)E=nμF, μ: mobility
mobility and diffusivity
J=cM∙(-dμ) ↔ J=-Ddc Ficks 1st law
i.e. μᵢ=μᵢ,0+kTlnXᵢ=μᵢ,0+kTlncᵢ<ω>, <ω>: average atomic volume, → dμᵢ=(kT/cᵢ)dcᵢ
Jᵢ=cᵢMᵢ∙(-dμ)=-MᵢkTdcᵢ ↔ Jᵢ=-Dᵢdcᵢ ⸫ MᵢkT=Dᵢ, Mᵢ=Dᵢ/kT einstein relation
Ficks 2nd law non-steady diffusion equation
∂c(x,t)/∂t=D(∂²c(x,t)/∂x²) I.C.: c(x,0)=c₀, B.C.: c(0,t)=cₛ, c(∞,t)=c₀
using Laplace tansform PDE to ODE, ℒ[f(t)]=∫₀∞e⁻ₛₜf(t)dt ⸫ℒ[c(x,t)]=c(x,s)
→ ℒ[∂c(x,t)/∂t]=ℒ[D(∂²c(x,t)/∂x²)] ⸪ℒ[f(t)]=sℒ[f(t)]-f(0)
→ s∙c(x,s)-c(x,0)=D(∂²c(x,s)/∂x²) → ∂²c(x,s)/∂x²-(s/D)∙c(x,s)=-c₀/D.....ODE of c(x,s)
B.C. → Laplace tansform
ℒ[c(0,t)]=∫₀∞e⁻ₛₜcₛdt=-(cₛ/s)e⁻ₛₜ|₀∞= cₛ/s, ℒ[c(∞,t)]=∫₀∞e⁻ₛₜc₀dt=-(c₀/s)e⁻ₛₜ|₀∞= c₀/s 代入cₚ求A, B
A=0, B=(cₛ-c₀)/s ⸫ c(x,s)=[(cₛ-c₀)/s]exp[-(s/D)½x]+c₀/s
c(x,t)=ℒ⁻¹[c(x,s)]=ℒ⁻¹[[(cₛ-c₀)/s]exp[-(s/D)½x]+c₀/s]=(cₛ-c₀)ℒ⁻¹[1/s∙exp[-(s/D)½x]]+c₀ ℒ⁻¹[1/s]
i.e. ℒ⁻¹[1/s∙exp[-(s/D)½x]]=erfc[x/2(Dt)½]
擴散的目的是為了降低系統的自由能, 不管是濃度梯度∂c/∂x的up或down過程,其驅動力都是自由能的下降
擴散種類: interstitial diffusion, substitional diffusion, vacancy diffusion, inter-diffusion
interstitial diffusion depend on “strain energy barrier”, 同時interstitial solute conc. << octahedral #
e.g. dilute interstitial solid solution(simple cubic lattice), CN=6
1. homogenization treatment to reduce the segregation in casting, 可以用無限多組的sin波合成組成元素的濃度分布,例如:t=0的濃度 c(x,0)=c+β₀sin(πx/l), t=∞的濃度c(x,∞)= c
c(x,t)=c+β₀sin(πx/l)exp(-t/τ), τ=l²/π²D relaxation time, β=β₀exp(-t/τ) if t= τ → β/β₀=1/e=0.368
因此均質化程度由個別的sin函數中的τ決定decay rate,長波長(l比較大)的τ值極為關鍵
2. 滲碳: 增加鋼表面的碳濃度,達到耐磨耗的特性
∂c/∂t=D(∂²c/∂x²) I.C.: c(x,0)=c₀, B.C.: c(0,t)=cₛ, c(∞,t)=c₀
solution: cₓ=cₛ-(cₛ-c₀)erf[x/2(Dt)½], if x=(Dt)½ erf(0.5)≈0.5 cₓ=(Dt)½=½(cₛ+c₀)
3. 脫碳: 降低脆性
∂c/∂t=D(∂²c/∂x²) I.C.: c(x,0)=c₀, B.C.: c(0,t)=0, c(∞,t)=c₀
, B.C.: c(0,s)=0, c(∞,s)=c₀/s → A=0, B=-c₀/s
c(x,t)=ℒ⁻¹[c(x,s)]=ℒ⁻¹[(-c₀/s)exp[-(s/D)½x]+c₀/s]=-c₀(1-erf[x/2(Dt)½])+c₀
⸫c(x,t)=c₀∙erf[x/2(Dt)½]
4. 焊接接合的退火處理 ∂c/∂t=D(∂²c/∂x²) I.C.: c(x,0)=c₁ if x≥0 and c(x,0)=c₂ if x≤0, B.C.: c(0,t)=½(c₁+c₂), c(±∞,t)=½(c₁+c₂)
substitional diffusion: 與interstitial diffusion 不同的,除了solute原子跳躍外,還需要"空位”的存在; self-diffusion用同位素A*測量
Dᴀ*=Dᴀ=⅙α²Γ, Γ:jump frequency=υzXᵥexp(-ΔGₘ/RT), Xᵥ:vacant probability=exp(-ΔGᵥ/RT)
→ Dᴀ=⅙α²υzexp(-ΔGᵥ/RT)exp(-ΔGₘ/RT)
對照interstitial diffusion, Dᵢ=⅙α²ω=⅙α²υzexp(-ΔGₘ/RT)
vacancy diffusion, Dᵥ=⅙α²Γᵥ 可以類比interstitial diffusion. Dᵥ=Dᴀ/Xᵥ ⸫ Dᵥ>>Dᴀ
