Deriving nonequilibrium interface kinetics from variational principles

0.19 MB·English

by  Robert Spatschek

Deriving nonequilibrium interface kinetics from variational principles
∗
Robert Spatschek
Institut fu¨r Festk¨orperforschung, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany
(Dated: February 2, 2008)
Nonequilibrium dynamics at interfaces is generally driven by a chemical potential. Here we
4 demonstrate a generic technique to derive the basic equations of motion, boundary conditions and
0 thechemical potentialin aconsistent way from fundamental variational principles. Asa particular
0 example, we consider a solid surface with elastodynamic effects, together with surface energy and
2 tension. Weapplythegenericresultstoperformalinearstabilityanalysisofaplanarfrontsubjected

n touniaxial stress (Asaro-Tiller-Grinfeld instability [1]), here also with surface tension.
a
J PACSnumbers: 05.70.Np,02.30.Xx,46.25.-y,46.50.+a

1
2
I. INTRODUCTION thatthelackofaconsistentderivationandtheuseofbare

] intuition can easily lead to wrong predictions. Though
i
c The quest for the correct equations of motion describ- the aim of the current paper is mainly to demonstrate
s methodological aspects how to derive consistently these
ing a system is an essential task of each problem in all
-
l branches of physics. The procedure is well-known for expressions from variational principles, we also want to
r
pointoutthattheresultsformabasistotheunderstand-
t mechanical systems, where for example the appropriate
m ing of important topics like solidification of superfluid
Lagrangianhastobefound,fromwhichthe(Newtonian)
. equationsofmotioncanbederivedusingvariationalprin- helium [3] and fracture mechanics [4].
t
a ciples. This method can in particular be applied to con- The Asaro-Tiller-Grinfeld (ATG) instability is a par-
m
tinuum mechanics, generating the usual bulk equations ticularexampleoftheseprocesses[1]. Hereoneconsiders
- and boundary conditions. a solid piece of material which is either in contact with
d
However, for more complicated dissipative systems, its melt such that phase transitions and thus motion of
n
o these straightforward and general methods are usually the interface are possible, or which suffers from surface
c not applicable. Bulk diffusion, friction, and also irre- diffusion. A uniaxial stress at the surface of this solid
[ versiblephasetransitionsareimportantexamples. Inthe givesrisetoamorphologicalinstability;longwavedefor-

frameworkoflinearnonequilibriumthermodynamics,the mations lead to a reduction of the total energy. A relax-
1
v “process velocity” is proportional to the deviation of an ationofelasticstressesinsomeregionsmakesthemmore
9 energy functional from its equilibrium value. favorableincomparisontothevalleysinwhichthestress
7 Ifweconsiderforexamplediffusionprocesses,thenthe is increased; consequently, a corrugated surface is char-
3 flux of material is typically j = −D∇µ with the diffu- acterized by a modulation of the elastic energy density
1 sion coefficient D and a chemical potential µ. Similarly, at the surface. This contribution favors a destabiliza-
0
for phase transformations at an interface, the resulting tion of the interface, counteracted by the surface energy
4
0 interface normal velocity is assumed to be proportional which prefers a flat surface. After the initial develop-
/ to the difference in chemical potentials in both phases, ment, the originallyslowprocessbecomes extremelyfast
at v = κ∆µ (κ is a kinetic coefficient); obviously, in both and leads to an unphysical breakdown of the analytical
n
m casesthechemicalpotentialµistherelevantdrivingforce and numerical predictions: Deep groves with sharp tips
- for this type of dynamics; however,it is not evident how advancing with increasing velocity can form in the sam-
d it can be expressed through the state variables of the ple[5,6]. Herestrongsurfacetensionandalsodynamical
n system. Therefore it is desirable to provide a method effects become relevant, but they are not taken into ac-
o to derive the correct expression for the chemical poten- count in conventionaldescriptions. The main problemis
c
: tialusing firstprinciple techniques. For relativelysimple here that it is not obvious how these effects enter into
v processes, it is easy to guess it correctly, but as soon the expression of the chemical potential.
i
X as more complicated effects are included, the results are Themaingoalofthecurrentpaperisthereforetopro-
sometimes unexpected. A particular example are elasto- vide a method to derive all describing quantities in a
r
a dynamiceffects,andthecorrectexpressionforthechem- consistent way from variational principles. A particular
icalpotentialwillbederivedinthispaper. Theinfluence illustration is the Grinfeld instability with surface ten-
ofsurfacetension,alsocausingcomplicatedcontributions sion. The paper is organizedas follows: In Section II we
tothechemicalpotential,wasalreadyconsideredforpro- presenttheLagrangianwhichisusedtoderiveallfurther
cesses at elastically stressed surfaces in [2]; however, the physical laws. In particular, we consider a linear elastic
author made some systematic mistakes. This illustrates medium with dynamical effects, surface tension and sur-
face energy, under the influence of external forces. In
Section III we derive the basic equations of motion and
boundaryconditionsforthe“mechanical”degreesoffree-
∗Electronicaddress: [e-mail] dom, the elastic displacements, from the Lagrangian. In
2
Section IV the expression for the chemical potential at length element dτ. Surface energy gives only an addi-
the surface of a solid body is derived using variational tional contribution to the total energy, surface tension
principles. The surface tension enters into the final ex- also couples to the elastic fields and changes the elastic
pression in a complicated way, and also inertial effects boundary conditions, as will be seen in the next section.
appearunexpectedly. Therefore,inSectionVthe rateof Also, in Section VII, it will be shown how the surface
changeofthetotalenergyduringareshufflingofthesur- tension can be interpreted in a more intuitive way, by
face is calculated; this confirms the preceding results for relatingittosurfaceforceswhichareproportionaltothe
the chemical potential. A specific application of these local curvature of the boundary. In general, both coeffi-
general techniques is the Grinfeld instability with sur- cients α and β are of the same order of magnitude, and
face tension (Section VI). We derive the spectrum of for aliquid they canbe shownto be identical [8]; in gen-
the initial stage of this instability. Additionally, in Sec- eral,β canevenbe negative. For three-dimensionalsitu-
tion VII the energetic change during the morphological ations, the surface tension contribution in Eq. (4) reads
transformations is calculated, correcting inconsistencies in general U = β u with the summation of i,j being
β ij ij
in [2]. Appendix A explains the fundamental conser- restrictedto the two tangentialdirections of the surface.
vation laws for momentum and angular momentum for Inthe case ofsurfaceswith a symmetry axishigher than
elastic systems with surface tension; Appendix B con- twofold,thesurfacetensiontensorisdiagonal,β =βδ
ij ij
tains the mathematical details to express the dissipation [7].
rateandthefunctionalderivativeforagenericfunctional Stress and strain are connected by Hooke’s law. For
whichisusedtoderivethechemicalpotentialandtheen- simplicity, we restrict it to the isotropic case,
ergy dissipation.
E ν
σ = u + δ u , (5)
ij 1+ν ij 1−2ν ij kk
(cid:18) (cid:19)
II. ACTION AND LAGRANGIAN
withE beingtheelasticmodulusandν thePoissonratio.
The inverse transformationis
Weconsiderasolidbodyinvacuumundertheinfluence
of external forces, linear elastodynamics, surface energy 1
u = [(1+ν)σ −νδ σ ]. (6)
and tension. The Lagrangianis given by ij E ij ij kk
The action is given by the time integral
L= (T −U )dV − U dτ =L +L , (1)
b s b s
VZ(t) ∂VZ(t) t1
S = Ldt=S +S (7)
b s
where V(t) is the time-dependent volume of the elastic
Z
body and ∂V(t) its boundary with elements dτ. The t0
expressionsfor kinetic and potentialenergy densities are
with arbitrary initial and end times t ,t .
0 1
SincetheLagrangianistranslationalandrotationalin-
1
T = ρu˙iu˙i, (2) variantintheabsenceofexternalforces,totalmomentum
2
and angular momentum are conserved. This property is
1
U = σ u , (3) notobviousifnonequilibriumprocessesatthesurfacelo-
b ik ik
2
callychangetheamountofmaterial;afurtherdiscussion
Us = α+βuττ −σn(0i)ui. (4) is given in Appendix A.
Here ρ denotes the mass density, which we assume to be
constant. Thereforewe neglectcompressiveeffects, since
III. ELASTODYNAMIC EQUATIONS AND
∆ρ = ρuii ≪ ρ. The elastic displacement field is de- BOUNDARY CONDITIONS
noted by u , σ is the stress and u the strain tensor,
i ik ik
α is the surface energydensity andβ the surface tension
The Euler-Lagrange conditions demand that the vari-
coefficient. Therepresentationofthesurfacetensioncon-
ationofthe actionwithrespectto the displacements has
tribution is unusual, and we will further comment on it
to vanish [9]. Here we also take into account that the
below [7]. The stresses σi(j0) are the given external forces volume can change in time due to other processes. We
which may act on some parts of the boundary; every- obtain for the bulk contribution
where else they are zero. We assume the volume to be
two-dimensional; τ is the tangential and n the normal t1
direction at the interface. δS = dt (ρu˙ δu˙ −σ δu )dV
b i i ij ij
The surface energy in Eq. (4) expresses the energy of Z Z
t0 V(t)
theoriginalundeformedsurface,whereasthesurfaceten-
sioncontributiongivestheadditionalenergywhicharises t1 ∂σ
ij
from elongation of the surface due to elastic effects. No- = dt (ρu˙iδu˙i+ δui)dV
∂x
tice that uττdτ is the elongation or compression of the tZ0 nVZ(t) j
3
− σ δu dτ . Thus we get for the variation of the boundary action
ni i
∂VZ(t) o t1
δS = dt (βκ+σ(0))δu +σ(0)δu dτ,
Now we examine the first term s nn n nτ τ
t1 tZ0 ∂VZ(t) (cid:16) (cid:17)
δA= dt ρu˙iδu˙idV since the term containing dδuτ/dτ can be integrated
tZ0 VZ(t) along the boundary and does not contribute. Vanishing
total variation, δS +δS =0, requires therefore
b s
in more detail. We start from the expression
∂σ
d ij = ρu¨ , (10)
dt ρu˙iδuidV = (ρu¨iδui+ρu˙iδu˙i)dV ∂xj i
VZ(t) VZ(t) σnn = −vnρu˙n+βκ+σn(0n), (11)
+ vnρu˙iδuidτ, (8) σnτ = −vnρu˙τ +σn(0τ). (12)
Z
∂V(t) These are the known elastodynamic equations, together
with the boundary conditions, now also considering sur-
wherev isthe normalvelocityoftheinterface;itispos-
n
face tension. The velocitydependent termin the bound-
itive if the solid extends. We point out that the elastic
ary conditions ensures momentum conservation. Notice
velocity u˙ at the surface and v are different quanti-
n n
that we had chosen for simplicity the particular case of
ties: The latter one describes the morphologicalchanges
a solid body inside the vacuum, and not for example in
of the shape due to nonelastic effects, for example by
contact with a liquid or different solid phase. However,
surface diffusion or the growth of a solid body from its
the calculations can easily be extended to these slightly
undercooled melt. On the other hand, the normal dis-
more complicated case; then in Eqs. (11) and (12) the
placementrateu˙ doesnotcontributetov ,sinceinthe
n n
difference [σ + v ρu˙ ] between the values in the two
framework of the linear theory of elasticity one does not ni n i
phases appears instead [10].
distinguishbetweentheundeformedanddeformedframe
of reference. Therefore we obtain
t1 IV. THE CHEMICAL POTENTIAL
d
δA = dt ρu˙ δu dV − ρu¨ δu dV
i i i i
dt
(
Z Z Z Intheprevioussectionthedynamicsofthemechanical
t0 V(t) V(t)
variables, the elastic displacements, was studied, requir-
− v ρu˙ δu dτ . ing that the variation of S with respect to ui vanishes.
n i i
) The elastic body was already considered as moving or
Z
∂V(t) transforming,whichleadstotheappearanceoftheinter-
The first bulk integraldoes not contribute since we have face normal velocity in the elastic boundary conditions.
todemandthatallvariationsdisappearatt andt . Con- The reason for this surface rearrangement is not yet
0 1
sequently, we obtain the variation of the bulk action specified and is not of mechanical origin. A specific ex-
ample would be surface diffusion. The elastic motion as
t1 “mechanical” process describes the motion of individual
∂σ
δS = dt [−ρu¨ + ij]δu dV atoms due to Newton’s law; in contrast, the “chemical”
b i i
∂x
Z ( Z j processes are related to a (local) change of the number
t0 V(t)
of atoms and consequently of the shape of the body.
Demanding that the solid body is in equilibrium not
− [σ +v ρu˙ ]δu dτ .
ni n i i
onlywithrespecttovariationsofthedisplacementvector
)
Z
∂V(t) but also with respect to this new degree of freedom, the
interface position, leads to the requirement that δS = 0
We define the curvature of the surface to be negative for
holds also for variations of the surface position in nor-
a convex body. The tangential strain is
mal direction by an interface shift δn. Otherwise, small
u = ∂uτ = duτ −u κ. (9) deviations from the condition δS =0 initiate thermody-
ττ ∂τ dτ n namical processes, driven by the chemical potential
Here, the definitions of the partial and the total tangen-
δL
tial derivative are µ=−Ω . (13)
δn
∂u u (τ +∆τ)−u (τ)
τ τ τ
= lim , From the bulk Lagrangianwe gain
∂τ ∆τ→0 ∆τ
du u (τ +∆τ)−u (τ) δL
τ = lim τ+∆τ τ . µ =−Ω b =Ω(U −T).
dτ ∆τ→0 ∆τ b δn b
4
Noticethatthisexpressioncontainsinparticularthedif- altered. It consists basically of three contributions: A
ference between potential and kinetic bulk energy. A change of the stressed volume reduces or increases the
naive guess would be that the chemical potential is the energy by the density Ω(U +T); also, a change of the
b
totalenergydensity,containingthusthesumofbothcon- shape modifies the boundary conditions and leads to an
tributions. However, the above explanation shows that additional energy contribution −2ΩT. Finally, compli-
this is not true. A further discussion on this point is cated“boundary”termsarisingfromsurfacetensionand
given in the next section. energy appear.
For the boundary contribution we assume here that Tocalculatethedissipationrate,westartwiththebulk
the surface of consideration is free of external tractions. contribution,
Then we get from Eq. (B4)
µ = −ΩδLs Eb = (T +Ub)dV. (17)
s δn VZ(t)
∂u du
= Ω −ακ+β ττ −uττκ−2 nτ Then we obtain
∂n dτ
(cid:20) (cid:18) (cid:19)(cid:21)
dE ∂
where we used the rotation rule b
= v (T +U )dτ + (T +U )dV
n b b
dt ∂t
∂uττ =2u (14) ∂VZ(t) VZ(t)
nτ
∂θ
= v (T +U )dτ + (ρu˙ u¨ +σ u˙ )dV
and the expression for the total derivative n b i i ij ij
Z Z
∂V(t) V(t)
d ∂ ∂
= +κ . (15) ∂σij
dτ ∂τ ∂θ = vn(T +Ub)dτ + u˙i
∂x
Z Z j
Thereforethetotalchemicalpotential,µ=µb+µs,reads ∂V(t) V(t) (cid:0)
∂ ∂σ
µ = Ω"12σikuik− 21ρu˙2i −ακ+β ∂∂unττ +∂xj(σiju˙i)−u˙i ∂xijj(cid:1)dV
= v (T +U )dτ + σ u˙ dτ
n b in i
du
−uττκ−2 nτ . (16) ∂VZ(t) ∂VZ(t)
dτ
!#
= v (T +U )dτ + (−v ρu˙ +βκ)u˙
n b n n n
Obviously,oneimmediatelyreproducestheknownelasto-
Z Z
static case with surface energy, without inertial and sur- ∂V(t) ∂V(t) (cid:0)
facetensioneffects[1];noticethatthechemicalpotential −v ρu˙2 dτ
n τ
is then nothing else than the energy density at the sur-
face. Inertialeffectsenterinanunexpectedway,because = vn((cid:1)Ub−T)dτ +β κu˙ndτ.
instead of the sum of elastic and kinetic energy density Z Z
∂V(t) ∂V(t)
their difference appears here. Surface tension gives rise
to three contributions: At first, the term ∂u /∂n re- The boundary energy is
ττ
flectsthechangeofthestrainu whenthesolidextends
ττ
in normal direction, and uττκ expresses the elongation E = α+βu −σ(0)u dτ, (18)
of a curved surface element. Without dynamical effects s ττ ni i
and external forces, the last term in the surface tension ∂VZ(t) (cid:16) (cid:17)
contribution vanishes, because, by virtue of the bound-
and its dissipation rate according to Eq. (B3)
ary condition (12) and Hooke’s law, also u = 0. This
nτ
full derivative appears only if both inertial contributions
and surface tension are taken into account, leading to a dEs = v −ακ+β ∂uττ −u κ− ∂uττ
n ττ
coupling of both effects. dt ∂n ∂τ∂θ
" (
Z
∂V(t)
∂2u
V. THE DISSIPATION RATE −κ ττ dτ + βu˙ +σ(0)u˙ dτ.
∂θ2 ττ ni i
)#
∂VZ(t) h i
An alternative approach to derive the expression for
the chemical potential is to express the dissipation of Here the first integral is the contribution due to the
energy due to morphological changes of the solid phase change of the shape and constant fields, whereas the
in terms of boundary integrals. Then the chemical po- second one does the opposite, fixed shape and changing
tential gives the amount of energy change at the trac- fields. We assume explicitly that the external forces are
tion free surfaces (per atom) if the shape of the solid is constant in time and that v = 0 on the interface parts
n
5
y
where these forces act. From (9) follows for fixed shape
(thus κ˙ =0) in the second integral
y(x,t)
x
du˙
u˙ = τ −u˙ κ.
ττ n
dτ
σ σ
Using again Eqs. (14) and (15), we do not get a contri- 0 0
butionfromthe fullderivative,since itcanbeintegrated
along the whole (closed) surface. Thus we have solid
dE ∂u
s = v −ακ+β ττ −u κ
n ττ
dt ∂n
( "
Z
∂V(t)
du FIG. 1: The Grinfeld instability.
−2 nτ + −βκu˙ +σ(0)u˙ dτ.
dτ n ni i
!# )
(cid:16) (cid:17)
Hence the total change of energy, E =Eb+Es, is Here the boundary conditions are accordingto Eqs. (11,
12) σ = βκ and σ = 0. The latter one implies
nn nτ
dE ∂u
= v U −T −ακ+β ττ −u κ unτ = 0 along the interface. This simplifies the general
n b ττ
dt " ∂n expression for the chemical potential, and we obtain
Z
∂V(t)
1 ∂u
−2dduτnτ!#+ σn(0i)u˙idτ. µ=Ω(cid:18)2σikuik−ακ+β(cid:20) ∂nττ −uττκ(cid:21)(cid:19).
Z
∂V(t) The strategy is to solve the elastostatic problem for a
slightly perturbed surface first, then to calculate the
The last integral gives the trivial contribution resulting
chemical potential, and finally to implement a dynam-
from the external forces. We are mainly interested in
local deformations of traction free surfaces (σ(0) = 0), ical process at this surface, driven by spatial variations
ni of the chemical potential.
leading to an alternative definition of the chemical po-
Curvature in Cartesian coordinates is given by
tential,
′′
y
dE 1 κ= .
= µvndτ (19) (1+y′2)3/2
dt Ω
Z
∂V(t)
Fora two-dimensionalplane-strainsituationwith isotro-
with the same expression (16). In particular, we clearly pic elasticity we have
seethatthedensityofthebulkLagrangianinsteadofthe
σ =σ =0, σ =ν(σ +σ ).
total energy is relevant for the dynamics. xz yz zz xx yy
Strains are
1
VI. GRINFELD INSTABILITY WITH SURFACE u = (1−ν2)σ −ν(1+ν)σ ,
xx xx yy
TENSION E
1 (cid:0) (cid:1)
u = (1−ν2)σ −ν(1+ν)σ ,
yy yy xx
A straightforward application of the preceding results E
istheGrinfeldinstability[1];similarto[2]wealsoinclude 1+(cid:0)ν (cid:1)
u = σ ,
xy xy
surface tension here and neglect inertial effects. Usually, E
thedynamicsofthisinstability,eitherdrivenbyevapora- uxz = uyz =uzz =0.
tionandcondensationorbysurfacediffusion,isslow,and
We use an Airy function U(x,y) to express stresses,
thereforekinetictermsneednotbeconsideredhere. This
is at least valid in the initial stage of the instability, but ∂2U ∂2U ∂2U
hastobecorrectedinthelateregimewhenthisomission σ = , σ =− , σ = ,
xx ∂y2 xy ∂x∂y yy ∂x2
leads to unphysical finite time singularities [4, 5].
The geometrical situation is sketched in Fig. 1. The fulfilling the biharmonic compatibility condition ∆2U =
horizontally applied uniaxial loading σx(0x) = σ0 can be 0andtheelasticboundaryconditionsσnτ =0,σnn =βκ.
compressive or tensile. Our goal is to calculate the sta- We make the ansatz [11]
bility of an originally flat surface. Therefore we study
the evolution of a slightly perturbed interface, U = σ0y2+(A y+B )ekycoskx+(A y+B )e2kycos2kx,
1 1 2 2
2
y(x,t)=∆coskx. (20)
6
with A ,B and A ,B being of first and second order We compute the energy which is contained in a strip
1 1 2 2
in the surface profile amplitude ∆ respectively. Then we of the width 2π/k. By symmetry, only even order con-
obtain from the boundary conditions up to second order tributions in ∆ can appear. The total elastic energy,
A =−∆(βk+σ ), B =β∆, x0+2π/k y(x)
1 0 1
1 1 1 E = U dydx
A = k∆2(σ + kβ), B = ∆2σ . b b
2 2 0 2 2 4 0 x=Zx0 y=Z−∞
The total chemical potential at the interface is therefore is infinite due to the zeroth order contribution (homoge-
to first order in ∆ neous strain). We introduced a shift x which is irrele-
0
vantheresincetheenergycontentisthesameinallstrips
µ = Ω (1−ν2)σ02 +k∆coskx −2σ21−ν2 +αk ofwidthofoneperiod. The secondordercontributionin
2E 0 E ∆ is

h
−kβ(1E−ν2)(3σ0+2βk) +O(∆2). Eb(2) =−πE∆2(1−ν2)(σ02−k2β2).
!
i
The surface energy,
In the particular case of surface diffusion,
x0+2π/k
D ∂2µ E = α 1+y′2dx,
v = , (21) α
n αΩ∂τ2
x=Zx0 q
withakineticcoefficientD,theamplitudeevolution,∆=
is to second order
∆ exp(λt), leads up to linear order to the spectrum
0
αkπ∆2
E(2) = .
λ=Dk3 2σ21−ν2 −k+kβ(1−ν2)(3σ +2βk) , α 2
0 Eα Eα 0
(cid:18) (cid:19) Finally, the surface tension contribution
(22)
assuming k ≥ 0. Obviously, the surface tension favors
x0+2π/k
the Grinfeld instability. Apart from the Grinfeld length,
E = βu 1+y′2dx (25)
β ττ
L = Eα , (23) x=Zx0 q
G 2(1−ν2)σ2
0 gives
we have now an additional lengthscale in the system,
βkπ∆2
E(2) =− (1−ν2)(4kβ+3σ ).
β β 2E 0
L := . (24)
β
σ0 Theexpression(25)canbepresentedinamoresuggestive
form by the means of Eq. (9),
Consequently, the spectrum can be expressed as
U = β[u (x +2π/k,y(x +2π/k))−u (x ,y(x ))]
β τ 0 0 τ 0 0
3 L
λL4G/D = (kLG)3 1−(kLG)+ 2(kLG)Lβ x0+2π/k
G
(cid:16) − βκu dτ. (26)
L2 n
+(kL )2 β . Z
G L2 x=x0
G
(cid:17)
The firstcontributionvanishes for properlychosenoffset
Typically, L ≪ L , since σ ≪ E, and thus the sur-
β G 0 x =0. Also,itdisappearsforaclosedboundarywithout
0
face tension correction can be neglected. However, the
corners. Thesecondcontributionistheworkdonebythe
self-stress due to surface tension in Eq. (22) leads to an
total displacement against the surface tension βκ.
instability even without external loading σ0. Altogether, E(2) =E(2)+E(2)+E(2) results in
b α β
π∆2 1−ν2
VII. ENERGY CALCULATION E(2) = −2σ2 +αk
2 0 E
β((cid:16)1−ν2)
In addition to the local chemical potential, also the −k (3σ0+2kβ) . (27)
E
total change of energy due to the morphological defor- (cid:17)
mation of the surface can be calculated, leading to the Theenergyhasthe samestructureasthe spectrum(22),
same result. therefore the prediction of the threshold of instability is
7
thesame. Wepointoutthattheelasticconstantsappear Acknowledgments
only in the combination E/(1−ν2), since the scenario
described here is plane strain. IappreciatemanyusefuldiscussionswithEfimBrener.
At a first glance, the description of the Grinfeld insta- This work has been supported by the Deutsche For-
bility using the chemical potential and the energy cal- schungsgemeinschaft(Grant No. SPP 1120).
culation seem to differ in a very important sense: The
chemical potential is first order in the perturbation am-
plitude ∆; in particular, it is sufficient to calculate all APPENDIX A: CONSERVATION OF
elastic fields only up to first order. The use of energetic MOMENTUM AND ANGULAR MOMENTUM
arguments always requires the calculation of second or-
der terms. In principle, it needs the elastic fields also up
Herewederivethebasicconservationlawswhichresult
to second order,because for example in the bulk energy,
from continuous symmetries of the Lagrangian (1). We
also terms like σ(0)u(2) appear. This seems to contradict performavariationofthefundamentalfieldu ,assuming
ij ij i
the former results using the chemical potential, which prescribedevolutionofthe volumeV(t)and∂V(t). This
should containthe same physicalinformation. In fact, it means that a reshuffling of material by other processes
turnsoutthatthesecondordertermsintheelasticfields thanelasticdisplacementispossible;thetotalnumberof
do not enter into the final energy terms; in other words, atoms does not need to be conserved. For simplicity, we
thesameresultscanbeobtained,settingA2 =B2 =0in assumethemassdensityρtobeconstantandnoexternal
the Airy function above. The explanation is now rather forces σ(0) to be present.
in
simple for the surface tension integral: In the first rep-
Using the expressions (1)-(4) and (8), one gets
resentation (25), it seems that u has to be evaluated
ττ
up to second order. In the second representation (26),
δL = (ρu˙ δu˙ −σ δu )dV + βκδu dτ
assuming that the first term vanishes, the curvature is i i ik ik n
already of first order in ∆, and hence only the first or- VZ(t) δVZ(t)
der of u is required. For the elastic bulk energy it is
n ∂σ
ik
known that in the present case of linear elasticity there = ρu˙iδu˙i+ δui dV
∂x
is no interactionbetween the stress caused by externally Z (cid:18) k (cid:19)
V(t)
appliedforcesandinternallocked-instressbysurfaceten-
sion [12], and therefore only first order terms are needed + (βκδu −σ δu ) dτ
n in i
here. Z
∂V(t)
Similar calculations of the total energy of a wavy sur-
d
face were done in [2]. However, the displacement fields = ρu˙ δu dV, (A1)
i i
dt
withoutsurfacetension,asexpressedbyEq.(5)there,are
Z
V(t)
incorrect; the correct expressions can in principle easily
be obtained from the Airy function (20). In particular,
where the last step follows from the bulk equation (10)
it would be necessary to include also third order har-
and boundary conditions (11, 12).
monics cos3kx to solve the elastic problem to the third
Obviously,theLagrangian(1)isinvariantwithrespect
order. Furthermore, the vertical displacement becomes
to a rigid body translation, δu = ε = const., since
i i
correctly to first order (without surface tension)
δu˙ = 0 and δu = 0. On the other hand, by Eq. (A1),
i ik
1−ν2 δL=εidPi/dt, with the total momentum
u =− σ ∆coskx+O(∆2),
y 0
E
P = ρu˙ dV. (A2)
in contrast to Eq. (5) in [2]. Additionally, in the expres- i i
sionforthetotalenergyofasystemwithsurfacetension, VZ(t)
the work done by the surface tension againstthe (zeroth
order)homogeneousdisplacementfieldwasoverlookedin Therefore,thetotalmomentumisconservedandthetotal
[2]. Foraclosedcontour,thisenergycontributioncanbe surface tension force is zero.
written as Similarly, we can consider rigid body rotations, δui =
ǫ ε x , with a constant axial vector ε ; ǫ is the to-
ijk j k i ijk
Uβ = βκundτ (28) tallyantisymmetrictensor. Theserotationsdonotaffect
∂ZV strain, δuik = 0, and therefore again the Lagrangian (1)
is invariant, δL = 0, since also δu˙ = 0. By (A1), we
with the total normaldisplacement u . For a non-closed i
n have δL=ε dℓ /dt with the angular momentum
surface also the corner contributions in Eq. (26) have to j j
be taken into account.
Thisunderlines thesuperiorityofourformalapproach ℓi = ρǫijkxju˙kdV. (A3)
to derive the total energy and the chemical potential, Z
V(t)
because intuitive guesses for the work done for example
by surface tension are not required. Thus the total angular momentum is conserved, too.
8
y
Noticethatbothconservationlawsarenottrivial,since
the body can change its size and therefore its number of y+δy
atoms: the normal velocity describes the local extension
or shrinkage of the body, and is not specified here.
An exampleis a soliddropletinits melt phase,for ex-
dn
ample in a temperature gradient. By crystallization the y(x,t)
dy
body can growat one part of the interface and recede at
dτ
another;thismeansthatthedropletcanwander,though
allits“solid”atomsareatrest. Inotherwords,thebody τ V(t)
moves, but its total momentum (A2) vanishes, since the n θ
displacement is constant.
APPENDIX B: AN INTEGRAL THEOREM
x
We look at an energy functional of the form FIG.2: Theinterfaceattimetisdepictedbythesolidcurve,
theshifted interface at t+δt by thedashed curve.
x1
W[y(x,t),t]= fdτ = f(x,y,y′,t) 1+y′2dx
VZ(t) xZ0 q
in Cartesian coordinates. Here the prime denotes the
∂y ∂2y
derivative with respect to x. We consider now the rate δy = δt, δy′ = δt,
of change of the energy W: During a timestep δt both ∂t ∂x∂t
the function value f andthe interface position y change.
Using the notations the rate of change is to first order in δt
δW = W[y(x,t+δt),t+δt]−W[y(x,t),t]
x1 ′
y ∂f ∂f ∂f
= f δy′+(1+y′2)1/2 δy+ δy′+ δt dx
Z ( (1+y′2)1/2 (cid:20)∂y ∂y′ ∂t (cid:21))
x0
x1 ′ ′ ′
y ∂f ∂f ∂f
= − f δy+ δy+ δt (1+y′2)1/2− (1+y′2)1/2 δy ,
Z ( (cid:20) (1+y′2)1/2(cid:21) (cid:20)∂y ∂t (cid:21) (cid:20)∂y′ (cid:21) )
x0
where we performed an integration by parts, assuming that the boundary contributions vanish; this is automatically
fulfilled for a closed contour. The prime denotes the total spatial derivative (with fixed time t), i.e.
∂[...] ∂[...] ∂[...]
′ ′ ′′
[...] = + y + y .
∂x ∂y ∂y′
Therefore
x1 ∂f ∂f ∂f y′ y′′ y′2y′′ ∂f
δW = − + y′+ y′′ δy−f − δy+ (1+y′2)1/2δy
( "∂x ∂y ∂y′ #(1+y′2)1/2 "(1+y′2)1/2 (1+y′2)3/2# ∂y
Z
x0
∂2f ∂2f ∂2f ∂f y′y′′ ∂f
− + y′+ y′′ (1+y′2)1/2δy− δy+ (1+y′2)1/2δt dx.
(cid:20)∂x∂y′ ∂y∂y′ ∂y′2 (cid:21) ∂y′(1+y′2)1/2 ∂t )
Curvature is given by and we have normal and tangential vectors
y′′ 1 −y′ 1 1
κ= (1+y′2)3/2, n= (1+y′2)1/2 1 , τ = (1+y′2)1/2 y′ .
(cid:18) (cid:19) (cid:18) (cid:19)
9
′ ′
Instead of Cartesian coordinates (x,y,y ) (where y is Orthogonalityofnormalandtangentialvariationrequires
usedtodescribethe slope),we usenowlocalcoordinates
(n,τ,θ). nisthenormal,τ thetangentialcoordinateand
θtheinclinationangleofthesurface. Withθ =arctany′, δn2 =δn2x+δn2y =δyδny.
we get the following transformation rules:
Furthermore, the normal vector is perpendicular to the
∂ ∂
= (1+y′2) , interface (dx,dy), and therefore
∂θ ∂y′
∂ −y′ ∂ 1 ∂
= + , dx
∂n (1+y′2)1/2∂x (1+y′2)1/2∂y δny = δn.
dτ
′
∂ 1 ∂ y ∂
= + .
∂τ (1+y′2)1/2∂x (1+y′2)1/2∂y Combining the preceding expressions gives
Comparing this with the above expression gives
dx
δn= δy,
x1 ∂f ∂2f ∂2f dτ
δW = −f ·κ− −κ δy
∂n ∂τ∂θ ∂θ2
(" #
xZ0 which, together with (B2), proves the statement (B1)
above.
∂f
+ (1+y′2)1/2δt dx. Using this result, we finally obtain
∂t
)
Thenextstepistoshowthataverticalshiftδy isrelated dW ∂f ∂2f ∂2f ∂f
= v −f ·κ− −κ + dτ.
to a normal shift δn according to dt n ∂n ∂τ∂θ ∂θ2 ∂t
( " # )
Z
V(t)
δydx=δndτ =vnδtdτ, (B1) (B3)
The expression in brackets reflects the change of energy
where the last relation defines the interface normal ve- due to motion of the interface; in particular, it consists
locity vn. By substitution ofacontributioncausedbythemotionoftheinterfacein
normal direction, the elongation of a curved length ele-
dx
δydx=δy dτ (B2) mentandatangentialtranslationalandrotationalcontri-
dτ
bution;the lasttermisduetothe changeofthe function
with the arclengthelement dτ2 =dx2+dy2 (see Fig. 2). f in time.
Normal, tangential and vertical displacements are Similarly, we can derive the functional derivative
δW/δn, now at a fixed time t. It corresponds to v =
n
δτ = δτx , δn= δnx , δy = 0 δ(xn), and thus
δτ δn δy
y y
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
δW ∂f ∂2f ∂2f
with δn+δτ =δy, thus = −f ·κ− −κ . (B4)
δn ∂n ∂τ∂θ ∂θ2
−δn
δτ = x .
δy−δn
y
(cid:18) (cid:19)
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