Kinetics of Materials

Page 73 — 2 corrections

Eq. 3.115

$$\vec J_1 = -D_1 \left( \nabla c_1 + \frac{\Omega_1 c_1}{kT} \nabla P \right) = -D_1 c_1 \nabla \left( \ln c_1 + \frac{\Omega_1}{kT} P \right)$$

Eq. 3.118

$$\vec F_1 \propto \nabla P \propto \left( \hat u_r \PFD {P}{r} + \... ... \frac{1}{r^2}\left( -\hat u_r \sin \theta + \hat u_\theta \cos \theta \right)$$

Page 76 — 3 corrections

First equation on page

$$A = \pi \left( r^2 + h^2 \right)$$

Eq. 3.125

$$R = \frac{r^2}{2h} \left( 1 + \frac{h^2}{r^2} \right)$$

Eq. 3.127

$$dV = \frac{\pi r^2}{2} \left( 1 + \frac{h^2}{r^2} \right) dh$$

Page 82

Fig. 4.2

 

Page 92

Eq. 4.68

$$v = \Omega_1 \left[ {D_1 - D_2} \right]_m \left( {{\frac{\partial c_1}{\partial x}}} \right)_m$$

Page 97 — 3 corrections

Eq. 4.100

$$D(\theta )= -\frac{\vec J \cdot \nabla c}{\vert\nabla c\vert^2} =... ...ht)^2+D_{33}\left( {{\frac{\partial c}{\partial \hat x_3}}} \right)^2}\right]$$

Eq. 4.101

$$\cos^2 \theta = \left[ \frac{\nabla c \cdot \hat k}{\vert\nabla c... ...\right]^2 = \frac{1}{\vert\nabla c\vert^2} \left( \PFD {c}{\hat x_3} \right)^2$$

Eq. 4.102

$$\sin^2 \theta = \left[ \frac{\nabla c \cdot \left( \hat \imath + ... ...t( \PFD {c}{\hat x_1} \right)^2 + \left( \PFD {c}{\hat x_2} \right)^2 \right]$$

Page 109 — 7 corrections

Eq. 5.39

$$u(x) = \frac{b_o}{2} + \sum_{n=1}^{\infty} \left[ a_n \sin \left( n \pi \frac{x}{L}\right) + b_n \cos \left( n \pi \frac{x}{L} \right) \right]$$

Eq. 5.40

$$a_n = \frac{1}{L} \int_{-L}^{L} u(x) \sin \left( n \pi \frac{x}{L}\right) dx$$

Eq. 5.41

$$b_n = \frac{1}{L} \int_{-L}^{L} u(x) \cos \left( n \pi \frac{x}{L}\right) dx$$

Eq. 5.42

$$u(x) = \sum_{n=1}^{\infty} a_n \sin \left( n \pi \frac{x}{L} \right)$$

Eq. 5.43

$$a_n = \frac{2}{L} \int_{0}^{L} u(x) \sin \left( n \pi \frac{x}{L} \right) dx$$

Eq. 5.44

$$u(x) = \frac{b_o}{2} + \sum_{n=1}^{\infty} b_n \cos\left( n \pi \frac{x}{L} \right)$$

Eq. 5.45

$$b_n = \frac{2}{L} \int_{0}^{L} u(x) \cos \left( n \pi \frac{x}{L} \right) dx$$

Page 112

last 2 lines of Table 5.3

$\frac{e^{-qx}}{p}$	erfc$\left( \frac{x}{\sqrt{4 D t}} \right)$
$\frac{e^{-qx}}{pq}$	$\sqrt{\frac{4 D t}{\pi}} e^{-x^2/(4Dt)} - x$   erfc$\left( \frac{x}{\sqrt{4 D t}} \right)$

Page 114 — 3 corrections

Text at top of page before Eq. 5.71

An estimate of the penetration distance for the error-function solution (Eq. 5.23) is the distance where $c(x,t) = \cnot /8$ , or equivalently, $\erf [x/(2\sqrt{Dt})]$ = $-3/4$ , which corresponds to

Eq. 5.72

$$c^A(x) = \cnot - \left( \cnot - c^{A/B} \right) \frac{x}{L}$$

Eq. 5.73

$$J^A = -D^A \FD {c^A}{x} = \frac{D^A\left( \cnot - c^{A/B} \right)} {L}$$

Page 115

Eq. 5.80

$$0 = \frac{\partial}{\partial r} \left( r D(r) \frac{\partial c}{\partial r} \right)$$

Page 116 — 7 corrections

Eq. 5.83

$$c^\mathrm{in} (r) = \frac{c^\mathrm{i/o} - c^\mathrm{in}} {\ln \l... ...mathrm{in}}\right)} \ln \left( \frac{r}{R^\mathrm{in}} \right) + c^\mathrm{in}$$

Eq. 5.84

$$J^\mathrm{i/o} = -D^\mathrm{in} \frac{c^\mathrm{i/o} - c^\mathrm{... ...{in} + \Delta R/2}{R^\mathrm{in}}\right)} \frac{1}{R^\mathrm{in} + \Delta R/2}$$

Eq. 5.85

$$c^\mathrm{out} (r) = \frac{c^\mathrm{out} - c^\mathrm{i/o}} {\ln... ...right)} \ln \left(\frac{r}{R^\mathrm{in} + \Delta R/2}\right) + c^\mathrm{i/o}$$

Eq. 5.86

$$J^\mathrm{i/o} = -D^\mathrm{out}\frac{c^\mathrm{out} - c^\mathrm{... ...ta R}{R^\mathrm{in} + \Delta R/2}\right)} \frac{1}{R^\mathrm{in} + \Delta R/2}$$

Eq. 5.88

$$\begin{array}{lcl} \alpha^\mathrm{out} \equiv \frac{D^\mathr... ...{R^\mathrm{in} + \Delta R/2}{R^\mathrm{in}}\right)} \end{array}$$

Eq. 5.89

$$I = 2 \pi \left( R^\mathrm{in}+\frac{\Delta R}{2} \right) J^\math... ...} \right)} {\ln \left( \frac{R^\mathrm{in}+\Delta R/2}{R^\mathrm{in}} \right)}$$

Eq. 5.90

$$c^\mathrm{i/o} - c^\mathrm{in} = \frac{\alpha^\mathrm{out} \left( c^\mathrm{out}-c^\mathrm{in} \right)}{\alpha^\mathrm{out} + \alpha^\mathrm{in}}$$

Page 118

Eq. 5.94

$$\PFD { ^{\star}\!c}{t} = -\nabla \cdot \vec J = \PFD {}{x} \left( ^{\star}\!D\PFD { ^{\star}\!c}{x} - \ave {v_A} c \right)$$

Page 121

Eq. 5.118

$$\begin{split} \int_{-{a}/{2}}^{{a}/{2}} & e^{-( x - \chi)^2/... ...erf \left( \frac{x-a/2}{\sqrt{4Dt}} \right) \right] \end{split}$$

Page 122 — 4 corrections

First equation on page

$$\begin{split} c(x,y,z,t) = \frac{c_o}{8} & \times \left[ \... ...\left( \frac{z-a/2}{\sqrt{4 D t}} \right) \right] \end{split}$$

Eq. 5.120

$$\PFD {T}{t} = \hat \kappa_\parallel \left( \PSD {T}{\hat x_1} + \PSD {T}{\hat x_2} \right) + \hat \kappa_\perp \PSD {T}{\hat x_3}$$

Eq. 5.122

$$\PFD {T}{t} = \left( \hat \kappa_\parallel^2 \hat \kappa_\perp \right)^{1/3} \left[ \PSD {T}{\xi_1} + \PSD {T}{\xi_2} + \PSD {T}{\xi_3} \right]$$

Eq. 5.124

$$T \left( \hat x_1, \hat x_2, \hat x_3, t \right)= \frac{\alpha}{t... ...\kappa_\parallel t} + \frac{\hat x_3^2}{4 \hat \kappa_\perp t} \right) \right]$$

Page 123

Eq. 5.126

$$T \left( x_1^\prime, x_2^\prime, x_3^\prime = 0, t \right) = \fr... ...l t} + \frac{3}{16 \hat \kappa_\perp t} \right) {x_2^\prime}^2\right] \right\}$$

Page 124

Eq. 5.127

$$T \left( \hat x_1, \hat x_2 = 0, \hat x_3, t \right) = \frac{\alp... ...kappa_\parallel t} + \frac{{\hat x_3}^2}{4 \hat \kappa_\perp t} \right)\right]$$

Page 126 — 2 corrections

Eq. 5.137

$$T = T_\circ \sin \left(\pi \frac{x}{L}\right) e^{-\pi^2 \kappa t/L^2}$$

Eq. 5.139

$$T_\infty = \frac{1}{L} \int_0^L T_\circ \sin \left( \pi \frac{x}{L} \right) dx = \frac{2}{\pi} T_\circ$$

Page 127 — 5 corrections

Eq. 5.146

$$-D\left( \PFD {\hat c}{x}\right)_{x=0} = -\alpha \hat c(0,p)$$

Eq. 5.147

$$a_2 = -\frac{\alpha \cnot }{p\sqrt{D}\left( \sqrt{p} + \frac{\alpha}{\sqrt D} \right)}$$

Eq. 5.148

$$\hat c = -\frac{h \cnot }{p \left( h+\sqrt{p/D}\right)} e^{-\sqrt{{p}/{D}} x } + \frac{\cnot }{p}$$

Eq. 5.149

$$c(x,t) = \cnot \left[ \erf \left( \frac{x}{2\sqrt{Dt}} \right) + e^{hx + h^2 Dt} \erfc \left( \frac{x}{2\sqrt{Dt}} +h\sqrt{Dt} \right) \right]$$

Eq. 5.150

$$c(0,t) = \cnot e^{-h^2/(Dt)} \erfc \left( h \sqrt{Dt} \right)$$

Page 128 — 3 corrections

Eq. 5.152

$$c(0,t) = \cnot \left( 1+h^2 Dt \right) \left( 1-\frac{2}{\sqrt{\pi}} h \sqrt{Dt} \right) \cdots$$

Eq. 5.153

$$\left[ \PFD {\hat c^I}{x} \right]_{x=0} = \left[ \PFD {\hat c^{II}}{x} \right]_{x=\infty} = 0$$

Eq. 5.154

$$\left[ \PFD {\hat c^I}{x} \right]_{x=L} = \left[ \PFD {\hat c^{II}}{x} \right]_{x=L}$$

Page 129 — 2 corrections

Eq. 5.157

$$\begin{split} c^I(x,t) & = \frac{c_0}{2} \left[ \erf \left( ... ...- \erf \left( \frac{x-L}{2\sqrt{Dt}}\right) \right] \end{split}$$

Eq. 5.158

$$c(x,t) = \frac{c_0}{2} \left[ \erf \left( \frac{x+L}{2\sqrt{Dt}}\right) - \erf \left( \frac{x-L}{2\sqrt{Dt}}\right) \right]$$

Page 137 — 2 corrections

Eq. 6.45

$$\begin{split} \partt {\left[ \begin{array}{l} \breve{c}_1 ... ..._2\\ \vdots\\ \breve{c}_{N-1} \end{array}\right]} \end{split}$$

Eq. 6.47

$$\begin{split} \frac{\partial \breve{c}_1 }{\partial t} & = \... ...ial t} & = \lambda_{N-1} \nabla^2 \breve{c}_{N-1} \end{split}$$

Page 141

Eq. 6.54

$$\begin{split} c_{\alpha} = & a_{1} + a_{2} \erf \left(... ... \erf \left( \frac{x}{\sqrt{4\lambda_{+}Dt}}\right) \end{split}$$

Page 142 — 2 corrections

Eq. 6.56

$$\begin{split} c_{1} = & \frac{D_{11}-D_{22}-\Delta}{2D_{2... ... \erf \left( \frac{x}{\sqrt{4\lambda_{+}Dt}}\right) \end{split}$$

Eq. 6.60

$$\begin{split} c_{2} = & \frac{c_{2}^{R} + c_{2}^{L}}{2} \... ...,\erf \left( \frac{x}{\sqrt{4\lambda_{+}t}}\right) \end{split}$$

Page 143 — 2 corrections

Eq. 6.61

$$\begin{split} c_{\alpha} = & \frac{a_{\alpha}}{(4\pi\lambd... ...^{3/2}} \exp\left(-\frac{r^2}{4\lambda_{+}t}\right) \end{split}$$

Eq. 6.65

$$\begin{split} c_{2} = & \frac{2 D_{21}N_{1}+(\Delta+D_{22}-D... ...^{3/2}} \exp\left(-\frac{r^2}{4\lambda_{-}t}\right) \end{split}$$

Page 159

Eq. 7.56

$$\left[ \PFD {\ln \Gamma^\prime}{P} \right]_T = \left[ \PFD {\ln \nu}{P} \right]_T - \frac{1}{kT}\left[ \PFD {G^m}{P} \right]_T$$

Page 161

Eq. 7.61

$$\begin{split} \langle R^2 \rangle & = \frac{\int_0^\infty \i... ...int_0^\infty c(x_1,x_2,x_3,t) dx_1 dx_2 dx_3} \end{split}$$

Page 193 — 2 corrections

Eq. 8.114

$$\frac{dc^{[100]}}{dt} = -12 \Gamma' \left( c^{[100]} - \frac{c^\mathrm{tot}}{3} \right)$$

Eq. 8.115

$$c^{[100]}(t) -\frac{c^\mathrm{tot}}{3} = \left[ c^{[100]}(o) -\frac{c^\mathrm{tot}}{3}\right] e^{-12 \Gamma' t}$$

Page 195 — 2 corrections

Eq. 8.118

$$J = J(\rightarrow) - J(\leftarrow) = c A e^{-(U_{\circ}+\Delta U/2)/(kT)} -\left(c + \FD {c}{x} a\right) A e^{-(U_{\circ}-\Delta U/2)/(kT)}$$

Eq. 8.119

$$J = -A e^{-U_{\circ}/(kT)} \left[ a \FD {c}{x}+ \frac{c\Delta U}{kT} \right]$$

Page 196

Eq. 8.123

$$\begin{split} p_6 & = 1 \left( \frac{1}{6} \right) + 0 \left... ...\right)^3 + 2 \left( \frac{1}{6} \right)^4 = 0.0015 \end{split}$$

Page 197

First equation on page

$$\begin{split} \ave {\cos \theta} & = 0.1960 \left( -1 \right... ... + 0.0015 \left( 1 \right)\\ \nonumber & = -0.2293 \end{split}$$

Page 198 — 2 corrections

Eq. 8.130

$$-J^{\mathrm{net}} = D_{11} \FD {n}{x} = \left( \frac{3a^2}{4} \Ga... ...a^2}{4} \Gamma_\circ^\prime + \frac{a^2}{2} \Gamma_x^\prime \right) \FD {n}{x}$$

Eq. 8.138

$$\ave {r_3^2} = (1-p) \left( \frac{c}{2}\right)^2$$

Page 202 — 5 corrections

Eq. 8.154

$$\left[ V_\mathrm{Zr}^{\prime \prime \prime \prime }\right] = \fra... ...{\bullet} \right] + \frac{1}{2} \left[ V_\mathrm{O}^{\bullet \bullet} \right]$$

Eq. 8.155

$$\left[ V_\mathrm{Zr}^{\prime \prime \prime \prime }\right] \left(... ...} \left[ \mathrm{Ta}_\mathrm{Zr}^{\bullet} \right] \right)^2 = e^{-G_S^f/(kT)}$$

Equation following Eq. 8.155

$$\left[ V_\mathrm{Zr}^{\prime \prime \prime \prime }\right] = \fra... ...{\bullet \bullet} \right] \gg \left[ \mathrm{Ta}_\mathrm{Zr}^{\bullet} \right]$$

Eq. 8.156

$$\left[ V_\mathrm{Zr}^{\prime \prime \prime \prime }\right] = \left( \frac{1}{4}\right)^{1/3} e^{-G_S^f/(3kT)}$$

Equation following Eq. 8.156

$$\left[ V_\mathrm{Zr}^{\prime \prime \prime \prime }\right] = \frac{1}{4} \left[ \mathrm{Ta}_\mathrm{Zr}^{\bullet} \right]$$

Page 204

Eq. 8.161

$$4[V_{\mathrm{O}}^{\bullet \bullet}] \left( [V_{\mathrm{O}}^{\bull... ...thrm{M}}^{\prime \prime}]\right)^2 P_{\mathrm{O}_2}^{1/2} = e^{-\Delta G/(kT)}$$

Page 206

Eq. 8.177

$${\frac{dc_1}{dt}}=-6\Gamma'\left(c_1-\frac{c^{\text{tot}}}{3}\right)$$

Page 207

Eq. 8.178

$$\left[c_1(t) -\frac{c^{\text{tot}}}{3}\right]= \left[c_1(0) -\frac{c^{\text{tot}}}{3}\right]=e^{-6\Gamma't}$$

Page 208 — 5 corrections

Eq. 8.180

$$\begin{split} \FD {c_{1}}{t} = & -2\left(\Gamma'_{1 \rightar... ...) c_{1} + 2 \Gamma'_{3 \rightarrow 2}c^{\text{tot}} \end{split}$$

Eq. 8.181

$$\begin{array}{cc} \Gamma'_{1\rightarrow2} = \Gamma'\left( 1 ... ...\left( 1 + \frac{U_{1 \rightarrow 3}}{2kT} \right) \end{array}$$

Eq. 8.183

$$c_{1}(t) = \left( \frac{c^{\text{tot}}}{3} - c_{1}^{\text{eq}} \... ...{c^{\text{tot}}}{3} - c_{2}^{\text{eq}} \right) e^{-k' t} + c_{1}^{\text{eq}}$$

Eq. 8.184

$$\begin{split} k' & = 6 \Gamma'\\ c_{1}^{\text{eq}} & = \fra... ... 2}}{3kT} + \frac{U_{1 \rightarrow 3}}{3kT}\right) \end{split}$$

Eq. 8.185

$$\begin{split} p_{1} = & \frac{e^{-U_{1}/(kT)}}{Z} = \frac{e^... ... 2}}{3kT} + \frac{U_{1 \rightarrow 3}}{3kT}\right) \end{split}$$

Page 226 — 4 corrections

Eq. 9.22

$$0={\frac{\partial ^2c^B_2 (y_1,t_1)}{\partial y_1^2}}-\frac{2c^B_... ...rtial x_1} \; \erf \left( {{\frac{x_1}{2t_1^{1/2}}}} \right)} \right]_{x_1=0}$$

Eq. 9.23

$$c^B_2(y_1,t_1)=\exp \left[ -\left( \frac{4}{\pi k^2t_1} \right)^{1/4} y_1 \right]$$

Eq. 9.24

$$c^{XL}_2(x_1,y_1,t_1)=\frac{1}{k}\exp \left[ {-\left( {\frac{4}{{... ...} \right] \left[ {1-\erf \left( {{\frac{x_1}{2t_1^{1/2}}}} \right)} \right]$$

Eq. 9.25

$${\frac{d\ln \Delta N}{dy}}=-\left[ {{\frac{4D^{XL}_2}{\pi t}}} \right]^{1/4} \left[ {\frac{1}{{k\delta D^B_2}}} \right]^{1/2}$$

Page 228

Eq. 9.31

$$J=-\left[ {\frac{\pi \delta ^2}{4} ^\star \! D^D {\frac{\part... ...t] \left[ {\frac{1}{\delta }\frac{\theta}{b}(\sin \phi +\cos \phi )} \right]$$

Page 276 — 6 corrections

First equation on page

$$\hat n = \frac{\hat \zeta \times \vec b}{\left\vert \hat \zeta \times \vec b \right\vert }$$

Eq. 11.42

$$\vec f_\mu =-\frac{\hat \zeta \times \vec b}{\left\vert \hat \zeta \times \vec b \right\vert} \PFD {G}{y}$$

Text following Eq. 11.42

Since the climb distance that results from the destruction of $\delta \! N_V$ vacancies is now $\delta y=(\Omega /\left\vert {\hat \zeta \times \vec b} \right\vert)\, \delta \! N_V$ , using Eqs. 3.64 and 3.66,

Eq. 11.43

$$\vec f_\mu = -\frac{\hat \zeta \times \vec b}{\Omega} \PFD {G}{N_... ...\zeta \times \vec b}{\Omega} kT \ln \left( \frac{X_V}{X_V^\mathrm{eq}} \right)$$

Eq. 11.44

$$df=2\mu b^2\sin \left(\frac{d\theta}{2}\right)\approx \mu b^2 d\theta = \frac{\mu b^2 ds}{R}$$

Eq. 11.45

$$\left\vert {\vec f_\kappa } \right\vert={\frac{df}{ds}}=\frac{\mu b^2}{R}$$

Page 277 — 3 corrections

Text following Eq. 11.49

Comparing the last result with Eq. 11.10, we see that the curvature given by $\kappa =a/(a^2+p^2)$ is constant everywhere and that the principal normal vector at any point on the helix is pointed toward the helix axis and is perpendicular to it. Also, the vectors $\vec B$ and $\vec d$ are constant vectors independent of $\hat \zeta$ . If we now take the axis of the helix to be along $(\vec B-\vec d)$ so that $(\vec B-\vec d) = \hat k\left\vert {\vec B-\vec d} \right\vert$ and put the results above into Eq. 11.46, we find that it is satisfied if

Eq. 11.50

$${\frac{\mu b^2}{\sqrt {a^2+p^2}}}=\left\vert {\vec B-\vec d} \right\vert$$

Eq. 11.52

$${\frac{\mu b^2}{\sqrt {a^2+p^2}}}=\left\vert {\vec B} \right\vert = \frac{bkT}{\Omega } \ln \left(\frac{X_V}{X_V^\mathrm{eq}}\right)$$

Page 278 — 2 corrections

Eq. 11.53

$$X_V=X_V^\mathrm{eq} e^{\mu b\Omega/\left(kT\sqrt {a^2+p^2} \right)}$$

Eq. 11.55

$$W\cong W^\circ + \frac{1}{2} \left( \frac{W^\circ } {c^2} \right) v^2$$

Page 280

Eq. 11.58

$$\frac{bkT_a}{\Omega} \ln\left(\frac{X_V}{X_V^\mathrm{eq}} \right) = \frac{\mu b^2}{ R} = \frac{2\mu b^2}{L}$$

Page 282 — 3 corrections

Eq. 11.64

$$c_V(r^\prime) - c_V^\mathrm{eq}(\infty) = a_1 \ln\left(\frac{R}{r^\prime}\right) + a_2$$

Eq. 11.65

$$\begin{split} a_1 \ln\left(\frac{R}{R_\circ}\right) + a_1 \l... ...isl}) \\ a_2 & = \frac{c_V^\mathrm{eq}(\infty)}{2} \end{split}$$

Eq. 11.66

$$c_V(r)- c_V^\mathrm{eq}(\infty) = \frac{ c_V^\mathrm{eq}(\mathrm{... ...thrm{eq}(\infty) } {\ln (R / \sqrt{R_\circ l})} \ln\left(\frac{R}{r}\right)$$

Page 296 — 5 corrections

Eq. 12.17

$$c - c^\infty = a_1 \sinh\left(\frac{2x}{\sqrt {\langle x^2\rangle }}\right) + a_2 \cosh\left(\frac{2x}{\sqrt {\langle x^2\rangle }}\right)$$

Eq. 12.18

$$c(x=0) = c^\mathrm{eq}$$    $$\left. \frac{\partial c}{\partial x} \right\vert _{x = \lambda/2} = 0$$

Eq. 12.19

$$\phi =2 \delta D^S_A\left. \frac{\partial c}{\partial x} \right\vert _{x=0}$$

Eq. 12.20

$$\phi _\mathrm{ideal}=\left( \frac{c^\infty }{\tau} - \frac{c^\ma... ... = \frac{4(c^\infty -c^\mathrm{eq})\lambda \delta D^S_A}{\langle x^2\rangle }$$

Eq. 12.21

$$\eta =\frac{\phi}{\phi _\mathrm{ideal}} = \frac{\langle x^2\rang... ... x^2\rangle }} \tanh\left(\frac{\lambda}{\sqrt {\langle x^2\rangle}}\right)$$

Page 297 — 3 corrections

Eq. 12.23

$$\phi = 2 \delta D_{A}^{S} \left(\FD {c}{x}\right)_{x=0}$$

Eq. 12.24

$$\phi = \frac{4 \delta D_{A}^{S}(c^{\infty}-c^\mathrm{eq})} {\sqrt{\ave x^2}} \tanh \left( \frac{\lambda}{\sqrt{\ave x^2}}\right)$$

Eq. 12.27

$$\begin{split} J_\mathrm{net} = & \frac{\phi}{\lambda} = \fra... ...^\prime (P - P^\mathrm{eq}) = K (P - P^\mathrm{eq}) \end{split}$$

Page 298 — 4 corrections

Eq. 12.29

$$\Delta G=2\pi \! Rg^L-\frac{\pi R^2h}{\Omega} kT\ln \left( \frac{P^\text{eq}}{P}\right)$$

Eq. 12.30

$$\Delta G_c(\mathrm{homo})=\frac{\pi \Omega \left(g^L\right)^2} {hkT\ln\left({P^\text{eq}}/{P}\right)}$$

Eq. 12.31

$$\Delta G=\pi \! Rg^L-2Rh\gamma ^S-{\frac{\pi R^2h}{2\Omega }} kT\ln \left(\frac{P^\text{eq}}{P}\right)$$

Eq. 12.33

$$\frac{\Delta G_c(\mathrm{edge})}{\Delta G_c(\mathrm{homo})}= \frac{1}{2}\left( {1- {\frac{2h\gamma ^S}{\pi g^L}}} \right)$$

Page 300 — 6 corrections

Eq. 12.36

$$P=\frac{N!}{\left( N_k^+ \right)! \left( N_k^- \right)! \left( N - N_k^+ - N_k^- \right)!}$$

Eq. 12.37

$$\begin{split} d \Delta \mathcal{G} & = G_k^f dN_k^+ + ... ...t(\frac{N_k^-}{N - N_k^+ - N_k^-} \right) dN_k^- \end{split}$$

Eq. 12.38

$$\PFD {\Delta \mathcal{G}}{N_k^+} = 0 = 2 G_k^f + 2kT \ln\left(\frac{N_k^+}{N - N_k^+ - N_k^-} \right)$$

Eq. 12.39

$$\left[ \frac{N_k^+}{N} \right]^\mathrm{eq} = e^{-G_k^f/(kT)}$$

Eq. 12.40

$$\frac{N_k^\mathrm{eq}}{N} = \left[ \frac{N_k^+}{N} \right]^\mathrm{eq} + \left[ \frac{N_k^-}{N} \right]^\mathrm{eq} = 2 e^{-G_k^f/(kT)}$$

Last equation on page

$$\begin{split} \nonumber d \Delta \mathcal{G} & = G_k^f dN... ...\theta a N/d) - N_k^+ - N_k^- } \right] d N_k^- \end{split}$$

Page 301 — 2 corrections

Eq. 12.41

$$\frac{d \Delta \mathcal{G}}{dN_k^+} = 0 = 2G_k^f + kT \ln \lef... ...c{[(\theta a N/d) +N_k^+]N_k^-}{[N - (\theta a N/d) -N_k^+ - N_k^-]^2}\right\}$$

Eq. 12.42

$$\left[ \frac{N_k^+}{N}+\frac{\theta a}{d} \right]^\mathrm{eq} \left[ \frac{N_k^-}{N} \right]^\mathrm{eq} = e^{-2G_k^f/(kT)}$$

Page 317

Caption to Fig. 13.9

Experimentally determined plot of $\ln M^B$ vs.tex2html_image_mark>#tex2html_wrap_inline8542#
for a (111) tilt boundary with a $46.5^\circ$ tilt angle in Al containing Fe solute atoms. $M^B =$ boundary mobility. From Molodov et al. [21].

Page 329

Eq. 13.38

$$\ln M_B(T_1) - \ln M_B(T_2) = -\frac{E^B}{k} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)$$

Page 331 — 5 corrections

Eq. 13.46

$${{\partial c} \over {\partial t}}=\nabla ^2c = D_B \left( \PSD {c}{r}+{2 \over r}{{\partial c} \over {\partial r}} \right)$$

Eq. 13.49

$$u(r,t)=a_1\left[ {1-\erf \left( {{{r-R} \over {2\sqrt {D_B t}}}} \right)} \right]+a_2$$

Eq. 13.50

$$c-c_\circ ={{a_1} \over r}\left[ {1-\erf \left( {{{r-R} \over {2\sqrt {D_B t}}}} \right)} \right]+{{a_2} \over r}$$

Eq. 13.51

$${{c-c_\circ } \over {c_{\mathrm{eq}}^{\alpha \beta}-c_\circ }}={R... ...er r}\left[ {1-\erf \left( {{{r-R} \over {2\sqrt {D_B t}}}} \right)} \right]$$

Eq. 13.52

$$I=4\pi R^2D_B\left({{{\partial c}\over{\partial r}}} \right)_{r=R... ...athrm{eq}}^{\alpha \beta}\right)\left[ {1+{R \over {2\sqrt {D_B t}}}} \right]$$

Page 333

Eq. 13.63

$$\ln \left( {{{\sqrt {R^2+R_\infty R+R_\infty ^2}} \over {R_\infty... ...} \right)+{\pi \over {2\sqrt 3}}={{3R_\infty \widetilde D } \over {R_c^3}} t$$

Page 335

Eq. 14.50

$$\PFD {h}{t} = \vec J_A \cdot \hat \jmath \Omega = -\frac{^\star \! D^{XL}}{kT\corf } \left( \PFD {\Phi_A}{y} \right)_{y=0}$$

Page 357

Eq. 14.55

$$\begin{split} \kappa_1 = & -\frac{\SD {r}{z}}{\left[ 1 + \le... ...^2 (z/R_w)]^{3/2}} = \frac{-1}{R_w \cosh^2 (z/R_w)} \end{split}$$

Page 358 — 3 corrections

Eq. 14.56

$$\cos \phi = \frac{1}{\sqrt{1+\left( \FD {r}{z} \right)^2}}$$

Eq. 14.57

$$\kappa_2 = \kappa_S \cos \phi = \frac{1}{r(z)} \frac{1}{\sqrt{1+\left( \FD {r}{z} \right)^2}}$$

Eq. 14.60

$$\Total {G} = 2 \pi \gamma^B \int_0^{h/2} r \sqrt{1 + (dr/dz)^2}\; dz = \frac{\pi \gamma^B}{2} \left( h R_w + R_w^2 \sinh \frac{h}{R_w} \right)$$

Page 361 — 6 corrections

Eq. 14.70

$$\left( {{{\partial k} \over {\partial t}}} \right)_x = -\nabla \cdot q = -\left( {{{\partial q} \over {\partial x}}} \right)_t$$

Eq. 14.71

$${{dk} \over {dt}} =\left( {{{\partial k} \over {\partial x}}} \right)_t{{dx} \over {dt}}+\left( {{{\partial k} \over {\partial t}}} \right)_x$$

Eq. 14.72

$$\left( {{{\partial q} \over {\partial x}}} \right)_t ={{dq} \over... ...partial x}}} \right)_t = -\left( {{{\partial k} \over {\partial t}}} \right)_x$$

Eq. 14.73

$${{dk} \over {dt}} =\left( {{{\partial k} \over {\partial x}}} \right)_t \left( {{{dx} \over {dt}}-{{dq} \over {dk}}} \right)$$

Eq. 14.74

$$dy = \left(\PFD {y}{x}\right)_t dx + \left(\PFD {y}{t}\right)_x dt$$

Eq. 14.75

$$\frac{dy}{dx} = \left(\PFD {y}{x}\right)_t + \left(\PFD {y}{t}\right)_x \frac{dt}{dx} = -h k + q h \frac{dt}{dx}$$

Page 366

Fig. 15.2

Page 384 — 3 corrections

Third equation on page

$$\frac{dR}{dt} = M \left( \frac{\ave {R}}{\ave {R^2}} - \frac{1}{R}\right) = M \left( \frac{5}{3 R_\text{max}} - \frac{1}{R}\right)$$

Fourth equation on page

$$\frac{\partial f}{\partial t} = - \frac{\partial}{\partial R} \le... ...}{t} \right] = \frac{2 A M}{3 R_\text{max}} \left( 5R - 4 R_\text{max} \right)$$

Eq. 15.52

$$\FD {R_\mathrm{cut-off}}{t} = \frac{2 \widetilde D\gamma \Omega c... ...hrm{cut-off}} \left( \frac{1}{\ave {R}} - \frac{1}{R_\mathrm{cut-off}} \right)$$

Page 385

Eq. 15.56

$$\frac{dA}{dt} \propto \left[ (N-1)\frac{\pi}{3} - \frac{\pi}{2}\right] = \frac{\pi}{3} \left( N-\frac{5}{2} \right)$$

Page 411 — 6 corrections

Eq. 16.71

$$\begin{split} \sigma_s^A & = \left[ -\sqrt{3} \cos^2 \the... ... \sigma_s^C & = 2 \sin \theta \cos \theta \sigma \end{split}$$

Eq. 16.72

$$\dot \varepsilon = \frac{1}{d} \left[ \frac{1}{\sqrt{3}}\FD {(S_s... ... + \frac{1}{3} \FD {(2S_s^C-S_s^A-S_s^B)}{t} 2\sin \theta \cos \theta \right]$$

Page 412 — 2 corrections

Eq. 16.76

$$\Phi_A\left(\frac{L}{2}\right) = \mu_A^\circ$$

Eq. 16.77

$$\Phi_A - \mu_A^\circ = \pm \frac{\sigma \Omega}{\ln [L/(2R_\circ)]} \ln\left( \frac{L}{2r} \right)$$

Page 414

Eq. 16.86

$$\begin{split} \frac{\mathrm{rate}_B}{\mathrm{rate}_S} = & \f... ...S w_S - \sin \frac{\psi}{2})\right]} = \lambda^{-4} \end{split}$$

Page 415 — 4 corrections

Eq. 16.88

$$\nabla^{2} \sigma_{nn} = \constant = A = \frac{1}{r} \FD {}{r} \left( r \FD {\sigma_{nn}}{r}\right)$$

Eq. 16.89

$$\left[ \FD {\sigma_{nn}}{r}\right]_{r = R_{c}}$$

Eq. 16.90

$$\sigma_{nn} =\frac{A}{4}\left[ (r^{2} - R^{2} \cos^{2} \theta) + 2 R_{c}^{2} \ln \left( \frac{R \cos \theta}{r}\right)\right] - \gamma^{S}\kappa$$

Eq. 16.91

$$A = - \frac{8(F_\text{app} + \Upsilon)}{\pi \left\{ R_{c}^{4} \le... ...ight] + R^{2} \cos^{2} \theta (4 R_{c}^{2} -R^{2} \cos^{2} \theta ) \right\} }$$

Page 416

Eq. 16.98

$$dV = d\left[\pi R_{c}^{2} L -\frac{2 \pi R^{3}}{3}(2 - 3 \cos \theta + \sin^{3} \theta)\right] = 0$$

Page 446

Footnote 9

The last sentence of this footnote should cite reference [4].

Page 452

Reference 16

This publication was listed incorrectly. The correct citation may be found at reference [4].

Page 453 — 3 corrections

Eq. 18.49

$$\begin{split} \vec J_A^C = & -c_A \left[ \frac{L_{AA}}{c_A} ... ...A} \right] \nabla \mu_B = - c_B L_B \nabla \mu_B\\ \end{split}$$

Eq. 18.57

$$\Omega_A \nabla \mu_B - \Omega_B \nabla \mu_A = \Omega_A \nabla \! \left( \PFD {f}{c_B} \right)$$

Eq. 18.58

$$\vec J_B^V = -{\Omega_A}^2 c_A c_B \left( L_B c_A + L_A c_B \righ... ...\! \left( \PFD {f}{c_B} \right) = -L \nabla \! \left( \PFD {f}{c_B} \right)$$

Page 454

Eq. 18.60

$$\vec J_B^V = - \left( L \PSD {f}{c_B} \right) \nabla c_B$$

Page 474

Fig. 19.10

Page 490 — 3 corrections

Eq. 19.82

$$\Delta \mathcal{G}'' = \frac{2\mu (1+\nu )N^2\Omega ^2}{9(1-\nu )V} -NkT\ln \left(\frac{X_V}{X_V^\mathrm{eq}}\right)$$

Eq. 19.83

$$\Delta \mathcal{G}''_{\min } = -\frac{9(1-\nu )V}{4E\Omega^2} \left[kT\ln \left(\frac{X_V}{X_V^{\mathrm{eq}}} \right)\right]^2$$

Eq. 19.84

$$\Delta g = - \frac{3\varepsilon _{xx}^T}{\Omega} kT \ln \left(\fr... ...{4\Omega ^2E} \left[kT\ln \left(\frac{X_V}{X_V^{\mathrm{eq}}} \right)\right]^2$$

Page 491 — 4 corrections

Text before Eq. 19.86

Standard relationships for volumes and areas show that the volume of the nucleus is given by $V=2\pi R^3 \left[ \cos \alpha - (\cos^3 \alpha)/3 \right]$ while the area of the two facets is $2\pi R^2 \sin^2 \alpha$ and the area of the spherical portion of the interface is $4\pi R^2 \cos \alpha$ . The free energy to form the nucleus is therefore

Eq. 19.86

$$\Delta \mathcal{G} = 2\pi R^3 \left( \cos \alpha - \frac{\cos^3 ... ...elta g_B + 2\pi R^2 \sin^2 \alpha \gamma^f + 4\pi R^2 \cos \alpha \gamma$$

Eq. 19.87

$$\Delta \mathcal{G} = \left( 2\pi R^3 \Delta g_B + 6 \pi R^2 \gamma \right) \left(\cos \alpha - \frac{\cos^3 \alpha}{3} \right)$$

Eq. 19.88

$$\frac{\Delta \mathcal{G}_c}{\Delta \mathcal{G}_c(\textrm{sphere})} = \frac{1}{2} \left( 3 \cos \alpha - \cos^3 \alpha \right)$$

Page 496 — 5 corrections

Eq. 19.101

$$\Delta \mathcal{G}_\mathrm{tri} = \frac{l^2}{2} \Delta g_a + 2 \g... ...+\left( \gamma^{11/\mathrm{sub}} - \gamma^{v/\mathrm{sub}}\right) \sqrt{2} l$$

Eq. 19.102

$$\PFD {\Delta \mathcal{G}_\mathrm{tri}}{l} = 0 = 2 \Delta g_a \fra... ...} + \left( \gamma^{11/\mathrm{sub}} - \gamma^{v/\mathrm{sub}}\right) \sqrt{2}$$

Eq. 19.103

$$l_c = -\frac{2 \gamma^{10/v} + \sqrt{2}\left( \gamma^{11/\mathrm{sub}} - \gamma^{v/\mathrm{sub}}\right)}{\Delta g_a}$$

Eq. 19.104

$$\Delta \mathcal{G}_\mathrm{rect} = bc \Delta g_a + (2b + c) ... ...ma^{10/v} + c \left( \gamma^{10/\mathrm{sub}} - \gamma^{v/\mathrm{sub}}\right)$$

Eq. 19.106

$$\PFD {\Delta \mathcal{G}_\mathrm{rect}}{c} = 0 = b_c \Delta g_a + \left( \gamma^{10/v}+\gamma^{10/\mathrm{sub}} - \gamma^{v/\mathrm{sub}}\right)$$

Page 497

Eq. 19.108

$$A_$$tri$$= \frac{1}{2} l_c^2 = \frac{1}{2} \left[ -\frac{2 \gamma^{10/v... ...gamma^{11/\mathrm{sub}} - \gamma^{v/\mathrm{sub}}\right)}{\Delta g_a}\right]^2$$

Page 499 — 3 corrections

text before Eq. 19.115

Let $\gamma^{LP}$ , $\gamma^{LS}$ , and $\gamma^{SP}$ be the energies (per unit area) of the liquid/particle, liquid/solid, and solid/particle interfaces, respectively. From Section 19.2.1 the volume of the solid nucleus is $V^S = \left( \pi R^3 /3 \right)\left( 2 - 3\cos \theta + \cos^3 \theta \right)$ , the spherical liquid/solid cap area is $A^{LS} = 2 \pi R^2 (1 - \cos \theta)$ , and the solid/particle area is $A^{SP} = \pi R^2 \sin^2 \theta$ . The free energy of nucleus formation on the particles is then

Eq. 19.117

$$\Delta \mathcal{G}_c^P = \frac{4\pi\left( \gamma^{LS}\right)^3}{3( \Delta g_B)^2}\left( 2 - 3\cos \theta + \cos^3 \theta \right)$$

text after 19.117

On the other hand, for the homogeneous nucleation of a spherical solid nucleus in the bulk liquid, $\Delta \mathcal{G}_c^H = 16 \pi \left( \gamma^{LS} \right)^3 / \left[ 3( \Delta g_B)^2 \right]$ .

Page 526 — 4 corrections

Eq. 20.81

$$c_B^{\alpha \infty} = c_B^{\alpha \beta}(R_{c}) = c_B^{\alpha \beta} (\infty) \left( 1 + \frac{\gamma \Omega}{kTR_c} \right)$$

Eq. 20.82

$$c_B^{\alpha \infty} - c_B^{\alpha \beta}(R) = \frac{c_B^{\alpha \... ...fty} - c_B^{\alpha \beta}(\infty) \right] \left( 1 - \frac{R_c}{R} \right)$$

Eq. 20.83

$$\FD {\IP _1}{t}\left( c_\mathrm{eq}^{\beta \alpha} - c_\mathrm{eq... ...a_2 \left( c_\mathrm{eq}^{\beta \alpha} - c_\mathrm{eq}^{\beta \gamma} \right)$$

Eq. 20.84

$$\FD {\IP _2}{t}\left( c_\mathrm{eq}^{\beta \alpha} - c_\mathrm{eq... ...a_2 \left( c_\mathrm{eq}^{\beta \alpha} - c_\mathrm{eq}^{\beta \gamma} \right)$$

Page 527 — 4 corrections

Eq. 20.85

$$\frac{c_B^\beta - c_B^{\beta 0}}{c_B^{\beta \alpha} - c_B^{\beta ... ...rf ({A_1}/{\sqrt{4 \tilde D^\beta}}) - \erf ({A_2}/{\sqrt{4 \tilde D^\beta}})}$$

Eq. 20.88

$$\begin{split} A_1 & = \frac{Q^\beta \sqrt{4 \tilde D^\beta}}... ...- \erf \left( A_1/\sqrt{4 \tilde D^\alpha} \right)} \end{split}$$

Eq. 20.89

$$\begin{split} A_2 & = \frac{\sqrt{4 \tilde D^\beta}}{\sqrt{\... ... - \erf \left( A_2/\sqrt{4 \tilde D^\beta} \right)} \end{split}$$

Eq. 20.91

$$\frac{d^2 c_B^\alpha (\eta)}{d\eta^2} + \left( \frac{1}{\eta} + 2\eta \right) \FD {c_B^\alpha (\eta)}{\eta} = 0$$

Page 528 — 3 corrections

Eq. 20.96

$$v \left( c_B^{\beta \alpha} - c_B^{\alpha \beta} \right) = \widetilde D^\alpha \left( \FD {c_B^\alpha}{r} \right)_{r = R}$$

Eq. 20.97

$$2 \eta_R \left( c_B^{\beta \alpha} - c_B^{\alpha \beta} \right) = \left( \FD {c_B^\alpha}{\eta} \right)_{\eta = \eta_R}$$

Eq. 20.98

$$\eta_R^2 e^{\eta_R^2} E_1\left( \eta_R^2 \right) = \frac{c_B^{\alpha \infty} - c_B^{\alpha \beta}} {c_B^{\beta \alpha} - c_B^{\alpha \beta}}$$

Page 529 — 5 corrections

Eq. 20.101

$$\FD {\IP }{t} \left( c_B^{\beta \alpha} - c_B^{\alpha \beta}\right) = \widetilde D^\alpha \left(\PFD {c_B^\alpha}{x} \right)_{x=\IP }$$

Eq. 20.102

$$\eta_{\IP } = \frac{\left( c_B^{\alpha \infty} - c_B^{\alpha\beta... ...a} - c_B^{\alpha\beta}\right) \sqrt{\pi} \erfc \left( \eta_{\IP } \right)}$$

Eq. 20.103

$$\widetilde D^\alpha \left( \PFD {c}{x}\right)_{x=\IP } = \FD {\IP }{t} \left( c_B^{\beta \alpha} - c_B^{\alpha \beta}\right)$$

Text after 20.103

Using the linear approximation in Fig. 20.14, $\partial c/ \partial x = \left( c_B^{\alpha \infty} - c_B^{\alpha \beta}\right)/(Z-\IP )$ . Also, the conservation of $B$ atoms requires that the two shaded areas in the figure be equal. Therefore, $\left( c_B^{\beta \alpha} - c_B^{\alpha \infty}\right) \IP = \left( c_B^{\alpha \infty} - c_B^{\alpha \beta}\right)(Z-\IP )/2$ . Putting these relationships into Eq. 20.103 gives

Eq. 20.104

$$\IP \; d\IP = \frac{\widetilde D^\alpha \left( c_B^{\alpha \infty... ...pha \beta}\right) \left( c_B^{\beta \alpha} - c_B^{\alpha \infty} \right)}\;dt$$

Page 530

Eq. 20.106

$$\begin{split} \Delta c^\alpha = & c^{\alpha\beta}_\mathrm{... ...\frac{4 \gamma \Omega c^\alpha (\infty)}{kT \ave R} \end{split}$$

Page 542

Eq. 21.24

$$\begin{split} \ave {N}_c = & \int_{{T_{c}}/{2\beta}}^{t} J(t... ...2 \beta t) (T_c^2 + 4 \beta^2 t^2)}{128 \beta^{4}} \end{split}$$

Page 582

Eq. 24.13

$$\left( 1 - \frac{1}{\eta_1^2} \right) {x_1^\prime}^2 + \left( 1 ... ...ght) {x_2^\prime}^2 + \left( 1 - \frac{1}{\eta_3^2} \right) {x_3^\prime}^2 = 0$$

Page 584 — 2 corrections

Eq. 24.21

$$0 = - \Delta S dT + \Delta V dP - \frac{V_{\circ}}{2} \left... ...ext{mart}}} - \frac{1}{E_{\text{par}}}\right) d (\sigma^{\text{app,uni}})^2$$

Eq. 24.22

$$\frac{dT}{d (\sigma^{\text{app,uni}})^2} = -\frac{V_{\circ}}{2 ... ...ext{{par}} -E_{\text{mart}}) } { 2 E_\text{{par}} E_{\text{mart}} \Delta H}$$

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