curvature-driven 4th-order diffusion from non-invariant...
TRANSCRIPT
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Curvature-Driven 4th-Order Diffusion from Non-Invariant Grain Boundaries.
Philip Broadbridge
Acknowledgements to P. Tritscher, J. M. Goard, M. Lee, D. Gallage,
J. Hill, D. Arrigo, D. Triadis, W. Miller Jr., P. Vassiliou, P. Cesana
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PDEs for evolution of curves and surfaces under isotropic and homogeneous processes should be invariant under Euclidean group.Simple example is evolution by mean curvature.Hypersurface of dimension n-1 embedded in Rn.
� ⇥� ⇤n
n = ‘inward’ unit normal vector.
n · @r(✓, t)@t
= B
This models surface of volatile metals e.g. Mg.Surfaces of stable metals e.g. Au, evolve by 4th ordersurface diffusion. In 2D,
@N
@t= �B
2
@2
@s2
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HRTEM image of grain boundary. Z. Zhang et al, Science 302, 846-49 (2003)
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−3 −2 −1 0 1 2 3−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
�b
�s
�
�s
m = tan(⇥) ; �b(T ) = 2�s(T )sin(⇥)
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Tritscher & Broadbridge, 1995
!
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Metal surface near grain boundary
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� = particle density
� = mean volume per particle
v = mean drift velocity
v =�Ds
kT
���s
Nernst-Einstein Relation
� = chem pot’l per particle;
T = temp; k = Boltzmann const; Ds = surf diff const
J = ��vVolume flux on surface
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� = ⇥ [�s(⇤) + ���s (⇤)]⇥.
Laplace-Herring Equation 1814 - 1950
�s = surface tension
� = arctan yx
Curvature
Sub this flux model in local cons of mass �N
�t+
�J
�s= 0
� =�yxx
(1 + y2x)3/2
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Eq of continuity implies
⇥N
⇥t= B
⇥2�
⇥s2(B constant)
cos(⇥)⇤y
⇤t= B
⇤2�
⇤s2
�y�N
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Now use � =����d2rds2
����
=
�⌥⌥⌃⇤
dx
ds
d
dx
1⇧1 + y2
x
⌅2
+
⇤dx
ds
d
dx
�1⇧
1 + y2x
yx
⇥⌅2
= f (�) �x
�[f �(�)]2 + [�f �(�) + f(�)]2
where � = yx , f(�) =1�
1 + �2= cos ⇥
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Mullins Surface Diffusion Eq + bdry conds for grooving
yt = �B �x
⇤�1 + y2
x
⇥�1/2�x
yxx
(1 + y2x)3/2
⌅
�t = �B ⇥2x
⇤f(�) ⇥x
��xf(�)
⇧[f �(�)]2 + [�f �(�) + f(�)]2
⇥⌅
� � 0 , �x � 0, x�⇥.
⇤x
��xf(�)
⇤[f �(�)]2 + [�f �(�) + f(�)]2
⇥= 0 , x = 0
� = m ,x = 0, t > 0
� = 0 , t = 0 x � 0 ;
yx =
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y� = ��x [D(yx)�x [E(yx)yxx]]
Anisotropic surface diffusion in terms of rescaled dimensionless variables,
� = 0; y = 0
x�⇥; y � 0 , yx � 0
x = 0; J = 0 �⇥ �x [E(yx)yxx] = 0
x = 0 ; yx = m(�).
m = tan(⇥) ; �b(T ) = 2�s(T )sin(⇥)
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Surface tension
D(⇥) =�
� + ⇥, E(⇥) =
��
� + ⇥
⇥3
,
closest to isotropic model
E(�) =1
(1 + �2)3/2D(�) =1
(1 + �2)1/2,
when � = 2.026
Integrable model
�(�), � = tan�1✓,
E / [�00(�) + �]/[1 + y2x
]3/2 (Herring eq.)
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0.5
1
1.5
30
210
60
240
90
270
120
300
150
330
180 0
Polar plot of surface tension vs angle for integrable model
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D = D1(t)D2(yx); E = E1(t)E2(yx)
⌧ =
Zt
0D1(t)E1(t)dt
From explicit time dependence of mobility and surface tension, due to temperature change,
, new time coordinate
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µ =�
� + ⇥z =
� x
0
� + ⇥
�dx
µ� = �µzzzz �1�
R(⇤)µz ,
where R(�) = �y� (0, �).
Change of variables
Transforms governing eq to linear PDE
=⇥ z = 0, µzzz =�R(⇤)
� + m(⇤).
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Z = z +1�
y(0, ⇤) , µ� = �µZZZZ
which has scaling symmetry
Z = e�Z, ⇥ = e4�⇥, µ = µ.
In terms of canonical coords,
Y = Y, S = S + ⇥, µ = µ.
Then separation of variables is possible
µ = F (S)G(Y ).
Y = Z��1/4, S = log(�1/4),
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µ = F (S)G(Y ).
+K2jY 1F3
�⇤14� j
4
⌅,
⇤12,34,54
⌅,
Y 4
256
⇥
+K3jY21F3
�⇤12� j
4
⌅,
⇤34,54,32
⌅,
Y 4
256
⇥
+K4jY31F3
�⇤34� j
4
⌅,
⇤54,32,74
⌅,
Y 4
256
⇥.]
µ0(Y ) +�⇧
j=0
⇥ j/4 K1j 1F3
�⇤�j
4
⌅,
⇤14,12,34
⌅,
Y 4
256
⇥
[x 1
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y(0, ⇥) = �⇥1/4��
i=0
bi⇥i/4
1F3([a], [b1, b2, b3], z) =��
k=0
(a)k
(b1)k(b2)k(b3)k
zk
k!
(a)k = a(a + 1)(a + 2) . . . (a + k � 1)
Generalized hypergeometric function
For m const, we have similarity solution of form
y��1/4 = H(x��1/4)
which has scaling symmetry
For m varying with t, assume
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These expansions for y(0,t) and µ(Z,t) can solve the free boundary problem with boundary conditions specified at unknown location
x = 0 () Z = y(0, ⌧)/�.
In order to implement boundary conditions at infinity,match Laplace transform of power series solution with
µ = exp
��p
14 Z⇥2
⇥⇤C3(p) cos
�p
14 Z⇥2
⇥+ C4(p) sin
�p
14 Z⇥2
⇥⌅+
1p
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µ = exp
��p
14 Z⇥2
⇥⇤C3(p) cos
�p
14 Z⇥2
⇥+ C4(p) sin
�p
14 Z⇥2
⇥⌅+
1p
By matching with , C3(p) and C4(p) can be deduced. With j fixed, K1j, K2j, K3j and K4j are linearly dependent since they are functions of the two independent power series coefficients in
µ(0, ⌧), & µZ(0, ⌧)
L�1C3(p) and L�1p1/4C4(p)
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_________ t=0.0002 ……………t=0.1 ---------------t=1
m(�) = 1/2 + 1/2 �1/4
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Mullins 1957 theory of evaporation-condensation.Lateral mixing of vapour keeps pressure close to that above flat surface. Linear model of non-equilibrium evaporation rate gives
�⌅N
⌅t= �J =
��(peq � p)⇥2⇥mkT
,
Laplace showed that energy per unit volume of surface material is
2�s⇥
Omega being spec vol per particle.
so energy platform per particle is . E = 2��s⇥
Gibbs-Thompson formula
implies evaporation rate proportional to mean curvature.
peq � p = p(eE/kT � 1) ⇡ 2p⌦�/kT
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�y�N
cos(�)⇥y
⇥t=
1(1 + y2
x)1/2B
yxx
1 + y2x
�y
�t= B
yxx
1 + y2x
Around linear groove, z-direction is irrelevant. Evaporation equation reduces to curve-shortening equation.
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CSE in differentiated form
Fujita 1952-54 presented exact similarity solutions for u(x,t) with
Bluman and Reid 1988: these come from “potential” symmetries, or from point symmetries of the system
yt
= D(yx
)yxx
✓t
= @x
[D(✓)✓x
] = @2x
[arctan(✓)]; ✓ = yx
; D(✓) = B/(1 + ✓2)
yx
= ✓ ; yt
= D(✓)✓x
D(✓) =1
a✓ + b, D(✓) =
1
(a✓ + b)2, D(✓) =
1
(a✓ + b)2 + c
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P. Tritscher & PB,I.J. Heat Mass Transf 1994
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Broadbridge 1989: Exact CSE parametric similarity solution for grain boundary groove � ⇥� (�, ⇥) = (x/2
⇤t, y/2
⇤t)
⇥ = ��1/2⇤sin[F (⇤; �)] + (1� ⇤2 � �ln⇤)1/2cos[F ] ; 0 ⇥ � ⇥ �⇥
⇥ = ��1/2⇤sin[G(⇤; �)] + (1� ⇤2 � �ln⇤)1/2cos[G] ; �⇥ ⇥ � ⇥ 1
⌅ = m[⇥��H(⇤; �)]
F (⇥; �) =� �
0(1� q2 � �lnq)�1/2dq
G(⇥; �) = tan�1m� F (⇥; �) + F (⇥m; �)
H(⇥; �) = ��1/2m�1⇥ sec[F (⇥; �)] ; 0 � � � �⇥
H(⇥; �) = ��1/2m�1⇥ sec[G(⇥; �)] ; �⇥ � � � 1
�⇥ = m�1tan[F (1; �)] ; � =⇥2
m � 1�m2
⇥2mln ⇥m
2F (1; �)� F (⇥m; �) = tan�1m
� = m�1tan[F (⇥; �)] ; 0 � � � �⇥
� = m�1tan[G(⇥; �)] ; �⇥ � � � 1
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Arrigo et al 1997: for classical diffusion, groove depth F(0) proportional to m but for CSE with m sufficiently large,
�ln(
m
2⌅
�) +
14
⇥1/2
� 12⇥ |y(0, t)|⇤
(Bt)⇥
�2ln(
m⌅�
)⇥1/2
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Groove depth due to surface diffusion.
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zt = B[zrr
(1 + z2r )1/2
+1
rzr]
Evaporation at axi-symmetric surface by mean curvature:
✓ = zr; ⇢ = rt�1/2; ✓ = f(⇢)
zr = m, r = at1/2 ; z = 0, t = 0 ; z ! 0, r ! 1
r=0 z=0 r=at1/2
phase boundary
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Gallage, PB, Triadis and Cesana use inverse methodpreviously applied to 1D nonlinear diffusion J. Philip (1960).
D(✓) =�0.5B�1 d⇢
d✓
R ✓0 ⇢d✓
1 + ✓(1 + ✓2)d ln(⇢)/d✓
Isotropic model
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Well-known solutions (mostly trivial) for curve-shortening eq.
Straight lines are steady states.
Shrinking circles radius R(t)=[2B(tc-t)]1/2 .Tubular pipe of outer,inner radii R,r becomes simply connectedat time r2/2B and disappears at time R2/2B
Calabi “grim reaper” travelling wave
y + ct =1clog(cos(cx))
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−9
−8
−7
−6
−5
−4
−3
−2
−1
0
.
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−5 0 5−6
−5
−4
−3
−2
−1
0
Obtuse open-angle solution = grain boundary solution.
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Other similarity reductions give rotating spirals, or Abresch-Langer 1986 “flowers”with n intersecting petals.
All of these solutions can be recovered by symmetry reductions from potential symmetries of CSL, i.e. symmetries of system
yx = u ; yt = D(u)ux
=� yt = D(yx)yxx
and ut = �x[D(u)ux]
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Doyle & Vassiliou 1998 classified nonlinear diffusion equations
ut = �x[D(u)ux]
according to compatibility with functional separation of variables
�(u) = f(x)g(t)
For D(u)=1/(1+u2), this leads to
(1) u=yx=tan(x) -> y=log(cos x) +t +const (vertical grim reaper),
(2) u=x/[2(t0-t) -x2]1/2 -> shrinking circle
(3) u=[e2(x-t) -1]-1/2 -> rotated (horiz) grim reaper
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u2 + 1u2
= (1 + e2t) cosec2(x),(4)
->
(K,x0,t0 arbitrary)
y =1K
log
�⇤exp(2BK2[t� t0]) + cos2(K[x� x0]) + cos(K[x� x0])
exp(BK2[t� t0])
⇥.
(5) u=cosh(x)/[e-2t-sinh2 x]1/2 -> rotation of (4) above
- A periodic solution that can satisfy boundary conds y=0 at x=0,L -J. R. King,
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5
−1
−0.5
0
0.5
1
1.5
x/l
y/l
Each peak or valley is asymptotic to a grim reaper as Initial conds may resemble diffraction grating.
t⇥ �⇤
---- approx bounds y = ±K [t0 � t] + log(2)/K
B(t-t0)=-0.07
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(6) u= ± sin x [cos2 x-e2t ]-1/2 ->
cosh(y)� et0�tcos(x) = 0
x2 + y2 = 2(t0 � t) + O([t0 � t]2)
x0 1
y 0
1
2
Figure 1: Evolution by heat shrinking flow of cosh y-5cos x=0
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−10 −5 0 5 10
−10
−8
−6
−4
−2
0
2
4
6
8
10
asymptotically approaches two coalescing grim reapers as .t⇥ �⇤
t-t0=-10
grimgrim
Long before extinction, ymax ⇥ t0 � t + log(2),and near extinction, ymax ⇥
�2(t0 � t).
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Anisotropy
Physical requirement:
After rotation of axes by !/2, 0 < B0 = D(0) <�
0 < B0 = D(0) <�
D(yx) =1
yx2D
��1yx
⇥⇥ B0y
�2x
After rotating back,
yt = B(yx)yxx
1 + y2x
= D(yx)yxx
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Doyle-Vassiliou classification of diffusion eqs compatible with functional separation
log �(u) = v(x) + w(t)
gives one more physically realistic model:
D(u) = D0cos(z(Au))
Au =� z
0(cos s)�3/2ds ; � �/2 < z < �/2
D(u) =1
1 + u2+ O(u4) u small
= (u/�
2)�2 + O(u�4) u large
With A= 20.5,
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−6 −4 −2 0 2 4 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u
D
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−3 −2 −1 0 1 2−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
x
y
t=−2.0
−15 −10 −5 0 5 10 15
−10
−5
0
5
10
x
y
t=−8.0
Functionally separated solution approaches grim reaper at large negative times, like others that we saw above !!
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Grooving by evaporation-condensation solved exactly: PB 1989
Grooving by surface diffusion, with slope yx(0,t) constant, on nearly isotropic surface, solved by Tritscher & PB 1995.
Solved exactly when m depends on t, due to temperature dependence of surface tension-PB & Goard 2010.
For what anisotropic materials do Angenent ovals, grim reapersand diffraction grating solutions exist ?—PB & P. Vassiliou 2011
Set up reliable semi-discrete approximationfor boundary conditions that can’t be treated analytically- Zhang & Schnabel 1993, Lee 1997, PB & Goard 2012.
Integrable nonlinear discretised model with self-adapting non-uniform grid- PB, Kajiwara, Maruno & Triadis, in progress.
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P. B. & J. M. Goard, “Temperature-dependent surface diffusion near a grain boundary”, J. Eng. Math., 66(1-3), 87-102 (2010).
D.J. Arrigo, P. B., P. Tritscher & Y. Karciga, "The Depth of a Steep Evaporating Grain Boundary Groove: Application of Comparison Theorems", Math. Comp. Modelling, 25 (10), 1-8 (1997).
P. Tritscher & P. B., "Grain Boundary Grooving by Surface Diffusion : an Analytic Nonlinear Model for a Symmetric Groove", Proc. Roy. Soc. A, 450, 569-587 (1995).
P. B. & P. Tritscher, " An integrable fourth order nonlinear evolution equation applied to thermal grooving of metal surfaces", I.M.A. J. App Math., 53,249-265(1994).
P. B., "Exact Solvability of the Mullins Nonlinear Diffusion Model of Groove Development", J. Math. Phys. 30, 1648-51 (1989).
P. B. & J. M. Goard, “When central finite differencing gives complex values for a real solution !”, Complex Variables and Elliptic Equations, 57 (2-4), 455-467 (2012).
J. M. Goard & P. B., "Exact solution of a degenerate fully nonlinear diffusion equation", Journal of Appl. Math. and Physics (Z.A.M.P.) 55(3), 534-538 (2004) .
P. B. & P. J. Vassiliou, “The Role of Symmetry and Separation in Surface Evolution and Curve Shortening”,Symmetry, Integrability & Geom: Methods & Applics (SIGMA), 7, 052, 19 pages (2011).