Asymptotically periodic L^2 minimizers in strongly segregating diblock copolymers

0.12 MB·

by  Adam Chmaj

ASYMPTOTICALLY PERIODIC L2 MINIMIZERS
IN STRONGLY SEGREGATING DIBLOCK COPOLYMERS
9
0 ADAMCHMAJ
0
2
Abstract. Using the delta correction to the standard free energy [4] in the

n elasticsettingwithaquadraticfoundationtermandsomeparameters,wein-
a troduce aonedimensiononlymodelforstrongsegregationindiblockcopoly-
J mers,whosesharpinterfaceperiodicmicrostructureisconsistentwithexperi-

7 ment inlow temperatures. The Green’s function pattern formingnonlocality
isthesameasintheOhta-Kawasakimodel. Thuswecompletethestatement
] in[31,p. 349]∗: “Thedetailedanalysisofthismodelwillbegivenelsewhere.
M
Our preliminary results indicate that the new model exhibits periodic mini-
mizerswithsharpinterfaces.” Westressthattheresultisunexpected, as the
G functional is not well posed, moreover the instabilities in L2 typically occur
. onlyalongcontinuous nondifferentiable“hairs”.
h
We alsoimprovethederivation donebyvan derWaals and useitand the
t
a above to show the existence of a phase transition with Maxwell’s equal area
m rule. However,thismodeldoesnotpredicttheuniversalcriticalsurfacetension
exponent, conjectured to be 11/9. Actually, the range (1.2,1.36) has been
[
reportedinexperiments[21,p. 360]. Bysimplytakingaconstantkernel,this

1 exponent is2. This isthe experimentally (±0.1) verifiedtricritical exponent,
v found e.g., at the consolute 0.9 K point in mixtures of 3He and 4He. Thus
8 thereisathirdunseenphaseatthephasetransitionpoint.
9
8
0
Keytosomerecentdevelopmentsinmodernmicroelectronicshasbeentheability
.
1 to create high-quality atomically abrupt interfaces between different semiconduc-
0
tors, which can produce new quantum states and uncover unexpected phenomena
9
(see http://en.wikipedia.org/wiki/Nanotechnology). Several experiments contra-
0
: dictingcommonknowledgeaboutbulkmaterialshavingonlydiffuseinterfaceshave
v
been reported, see the References, especially [7], where a new experimental tech-
i
X nique is explained.
r It is remarkablethat chemically diverse diblock copolymers and microemulsions
a admit very similar characteristic structures, e.g., the bicontinuous ordered double
diamond[27]. Atheoryformallypredicting the phase transitionbetweenweakand
strongsegregationwasderivedin[4]. Hereweshowhowtouseelasticityandchoose
parametersintheVasersteinpseudoassociationpotentialtogetlocalL2 minimizers
resembling the square wave [38]. As a by-product we get a rigorous proof of the
gas to liquid phase transition.
To make this note self-consistent, we review the history of this modeling.
The ideal gas law
RT
(0.1) P = ,
V
∗ A preliminary version of this result was announced at the “Defects and their dynamics”
workshop(09.08-16.08.2003), BanffInternational ResearchStation, Alberta,Canada.
1
2 ADAMCHMAJ
where P is the internal pressure, V the volume per mole, T the absolute tempera-
ture,Rthegasconstant,doesnotpredictaphase transitionbetween gas and liquid,
definedaccordingtoexperimentas twodensities coexistingatthesamepressure(see
e.g., the isotherms for carbon dioxide in [36]).
Van der Waals formulated the modified equation of state as
RT a
(0.2) P = − ,
V −b V2
whereaistheattractionparameterwhicharisesfrompolarizationofmoleculesinto
dipoles and b is the volume enclosed within a particle (the repulsion parameter).
Below the critical temperature isotherms for (0.2) have a wiggle [36], which is
unphysical. Maxwell’s construction in which the fluid is taken around a reversible
cycle of states from a logicalpoint of view is worthless [36], as states on the wiggle
have no meaning. However, such soft reasoning can give other interesting results
[22, 36].
Foruniformsystems,theHelmholtzfreeenergyisdefinedasΨ=U−TS,where
U isthe internalenergyandS the entropy. The firstlawofthermodynamicsstates
dU = δQ−PdV, where δQ is heat change, the second law for reversible systems
states δQ = TdS. Thus dΨ = dU − TdS − SdT = δQ − PdV − TdS −SdT,
so P = −∂Ψ and Ψ = − PdV. Since the isothermal compressibility β satisfies
∂V
β ≡−1 ∂V ≥0, we get ∂2RΨ ≥0. From (0.2) we get
V ∂P ∂V2
a
(0.3) Ψunif =− PdV =−RTln(V −b)− ,
Z V
and the convex envelope of (0.3) gives Maxwell’s equal area rule for the isotherms
of (0.2). However, this and related statistical mechanics arguments lack a proof
of phase transition as defined above. To be more precise, suppose that Maxwell’s
envelope is on the line segment [V ,V ]. Why does the system take on only V and
1 2 1
V , but not other values inside the segment? In other words, the implicit meaning
2
ofarguingthis wayisthatassuming there is a phase transition, it is determined by
the equal area rule.
As is often the case in mathematics, we can get a proofof phase transition with
Maxwell’s equal area rule by studying a related larger structure. This was done in
[20,p. 40]. Inarelatedwork,the authoralsoobtained4/3as the criticalexponent
[21, p. 360]. Other methods have reported 3/2 (van der Waals), 1.38 [28] and
(1.21,1.32)[42].
We now improve the derivation of van der Waals.
Let ρ = 1 denote the density. The free energy of a nonuniform state ρ(x) of a
V
nonreversible process is a functional of the Helmholtz form I(ρ) = U(ρ)−TS(ρ).
The cumulative free energy 1ρΨunif(ρ)dx is:
0
R
1 1
(0.4) Ψnonunif(ρ)= ρΨunif(ρ)= [−RTρln(ρ−1−b)−aρ2].
Z Z
0 0
1 1 1
U(ρ) cannot be only (C − aρ)ρ, but rather − J(x − y)ρ(x)ρ(y)dxdy ≃
0 0 0
1ρ(C −aρ− c2ρ′′)dxR. Since the term 1Cρ canRbeRneglected due to the mass
0 2 0
cRonstraint, one can conclude that R
1 c 1
(0.5) I(ρ)=Ψnonunif(ρ)= − 2ρ′′ρ+ [−RTρln(ρ−1−b)−aρ2]
Z 2 Z
0 0
PERIODIC MICROSTRUCTURE 3
Note that −aρ2−RTρ(ln(ρ−1−b)) is concave(double-well) for ρ and T satisfying
RT ≥(<)2aρ(1−ρb)2. However,notethatthenonlocalandthe− 1aρ2termsap-
0
pearedquiteseparately,thereforeitmaynotbeentirelyconvincingtRhatoneisanex-
tensionofthe other. Also,if− 1aρ2 isreplacedby− 1 1J(x−y)ρ(x)ρ(y)dxdy,
0 0 0
the solutions of the EL equatioRn are always continuousR, sRo we have no phase tran-
sition as defined above.
InhisNobellecture,vanderWaalsexpressedhisabsoluteconvictionthatmolecules
associate in complexes not of chemical origin. He called them pseudoassociations.
Thus the derivation can be improved by adding the nonlocal term to (0.4):
1 1 1
(0.6) Imod(ρ)=− J(x−y)ρ(x)ρ(y)dxdy+ [−RTρln(ρ−1−b)−aρ2].
Z Z Z
0 0 0
The mass constraint makes possible an addition of a linear term, which we choose
so that −RTρln(ρ−1 −b)−aρ2 has equal depth wells. Also, on a bounded in-
terval boundary effects may come into play, so J is not necessarily translationally
invariant: J =J(x,y). Now (0.6) is qualitatively the same as derived in [4]
1 1 1 1
(0.7) I(u)= J(x,y)(u(x)−u(y))2dxdy+ W(u(x))dx,
4Z Z Z
0 0 0
where W is the double-well function
1 1
(0.8) W(u)=− ju2− u2+kT[(1+u)ln(1+u)+(1−u)ln(1−u)],
2 2
1
with j(x)= J(x,y)dy and k the Boltzmann’s constant. For some works on this
0
and related Rmodels see the References, especially the collective diffusion kinetics
of the phase transition in polymer gels, what the authors in [24] consider one of
the most exciting problems in currentcondensedmatter physics, and a nonlocalin
time evolution discussed in [37].
Let G(u) = −1u2+kT[(1+u)ln(1+u)+(1−u)ln(1−u)], G∗ be the convex
2
envelope of G and
1 1 1
(0.9) I∗(u)=− J[u](x)u(x)dx+ G∗(u(x))dx.
2Z Z
0 0
Let g∗ = G∗′, [u,u] be the interval on which g∗ is constant and v∗ = g∗(u) =
g∗(u). We define
g∗−1(v), v 6=v∗, g∗−1(v), v 6=v∗,
s(v)= s(v)=
(cid:26) u, v =v∗, (cid:26) u, v =v∗.
Let u be a local minimizer of (0.7). Let v = g(u) and x be such that v(x )= v∗.
0 0
We now show that under some conditions v′(x )6=0, which implies that the set of
0
discontinuities of u is finite.
Let δ >0 be such that J(x,y)> 1J(x,x) for all (x,y) such that |x−y|<2δ .
0 2 0
Let δ ∈(0,δ ) and I =[x −δ,x +δ]. We define
0 δ 0 0
v+ =δ2+max{v(x)|x∈I }, v− =min{v(x)|x∈I }−δ2.
δ δ
Let
1
(0.10) φ=(u−s(v−))χ − (u−s(v−) χ .
Iδ 1−2δ Z [0,1]\Iδ
(cid:0) Iδ (cid:1)
4 ADAMCHMAJ
1
Since φ=0, we have I(u)≤I(u−φ), or
0
R
1 1
0≥ G∗(u)−G∗(u−φ)−φJ[u]+ φJ[φ] .
Z 2
0 (cid:8) (cid:9)
On I we have
δ
G∗(u)−G∗(u−φ)−vφ≥(g∗(u−φ)−v)φ≥(v−−v+)φ
and
1 φJ[φ]≥ J(x0,x0) φ 2− maxx∈Iδj(x) φ 2
2Z 4 Z 2(1−2δ) Z
Iδ (cid:0) Iδ (cid:1) (cid:0) Iδ (cid:1)
On [0,1]\I we have
δ
−1
G∗(u)−G∗(u−φ)−vφ≥(g∗(u−φ)−g∗(u)) φ
1−2δ Z
Iδ
−g′(u) 2 3
≥ φ +O φ
(1−2δ)2(cid:0)ZIδ (cid:1) (cid:16)(cid:0)ZIδ (cid:1) (cid:17)
and
1 φJ[φ]≥−maxx∈Iδj(x) φ 2+ [0,1]\Iδ [0,1]\IδJ φ 2
2Z 2(1−2δ) Z R 2(1−R2δ)2 Z
[0,1]\Iδ (cid:0) Iδ (cid:1) (cid:0) Iδ (cid:1)
Taking into account these inequalities we obtain
0≥ J(x0,x0) + [0,1]\Iδ [0,1]\IδJ − maxx∈Iδj(x) − [0,1]\Iδg′(u) φ 2
n 4 R 2(1−R2δ)2 1−2δ R(1−2δ)2 o(cid:0)ZIδ (cid:1)
+(v−−v+) φ+O φ 3 .
ZIδ (cid:16)(cid:0)ZIδ (cid:1) (cid:17)
Denote the expression in the curly brackets by C(J,g,δ). After dividing by φ
Iδ
we get R
v+−v− ≥C(J,g,δ) (u−s(v−))+O(δ2).
Z
Iδ
In a similar manner, we take
1
φ=(u−s(v+))χ − (u−s(v+) χ
Iδ 1−2δ Z [0,1]\Iδ
(cid:0) Iδ (cid:1)
and obtain
v+−v− ≥C(J,g,δ) (s(v+)−u)+O(δ2).
Z
Iδ
Adding these two inequalities and dividing by 2δ we get
v+−v− C(J,g,δ)
≥ (s(v+)−s(v−))+O(δ).
2δ 2
Letting δ →0 we get
C(J,g)
|v′(x )|≥ (u−u),
0
2
where
J(x ,x ) 1 1J 1
C(J,g)= 0 0 + 0 0 −j(x )− g′(u).
4 R R2 0 Z
0
Thus v′(x ) 6= 0 if C(J,g) > 0. For now this shows that any local minimizer does
0
not take values in (u,u) and proves the existence of phase transition.
PERIODIC MICROSTRUCTURE 5
The well-balanced scaling
1 x−y
(0.11) J (x,y)= Js −ǫJl(x,y),
ǫ
ǫ ǫ
(cid:16) (cid:17)
proposed in [31], where Js ≥ 0 and |x|Js(x)dx < ∞, can be justified as repre-
R
senting an interplay of attractive (JsR) and repulsive (Jl) chemical forces. This is
consistent with an established view in physical chemistry, e.g., “But the establish-
mentofawell-definedperiodicitybetweenthelamellae,ortherods,dependsonthe
existenceoflong-rangeforcesbetweenthem: attractive(vanderWaals)orrepulsive
(electrostatic, steric, plus the short-range Marcelja repulsion)” [12, p.2296].
Its qualitative shape of that of the “mexican hat”, a notion that comes from
mathematical biology, where such kernels are also often used. However, it is in-
teresting to observe that since we assume only |x|Js(x)dx < ∞, J can be
R ǫ
nonnegative. It is then its shape, not sign, that givRes pattern formation.
The construction of periodic minimizers with discontinuous interfaces now fol-
lows from a sequence of lemmas. Let I denote I with J given by (0.11). Then
ǫ ǫ
ǫ−1I Γ-converges as ǫ→0 to a singular limit defined on L2(0,1) by
ǫ
c ||Du||(0,1) − 1 1 1Jl(x,y)(u(x)−u(y))2dxdy u∈BV((0,1),{−1,1}),
I (u)≡ 0 2 4 0 0
0
(cid:26) ∞ R R otherwise,
wherec ≡inf Js(x−y)(u(x)−u(y))2dxdy+ W(u(x))dx and
0 {u:u(±∞)=±1} R R R
||Du|| is the variation measuRreRof u [1]. Here Γ-convergence is deRfined as
1. For every {u }⊂L2(0,1) with lim u =u, liminf ǫ−1I (u )≥I (u);
ǫ ǫ→0 ǫ ǫ→0 ǫ ǫ 0
2. For every u∈L2(0,1)∩BV((0,1),{−1,1}), there exists a family
{u }⊂L2(0,1) such that lim u =u, and limsup ǫ−1I (u )≤I (u).
ǫ ǫ→0 ǫ ǫ→0 ǫ ǫ 0
These two inequalities often go together the compactness property
3. Let ǫ be a sequence of positive numbers converging to 0, and {u } a
n n
sequence in L2(0,1). If ǫ−1I (u ) is bounded above in n, then {u }
n en n n
is relatively compact in L2(0,1) and its cluster points belong to
BV((0,1),{−1,1}).
Afurtherrelatedpropertyisthatastrictlocalminimumu ofI perturbstoalocal
0 0
minimum u of ǫ−1I . This was shown and used in [23] to obtain local minimizers
ǫ ǫ
on dumbbell domains. In the proof, by a standard argument, a minimizer u of
ǫ
ǫ−1I is first constructed in a small closed ball around u . Then using properties
ǫ 0
1−3itis shownthat forsmallenoughǫ>0,u lies in the interiorofthe ball,thus
ǫ
is a local minimizer of ǫ−1I in L2.
ǫ
When G is nonconvex this argument goes through a convexification and the
calculation above. It is shown that a local minimum of I∗ in a subspace of
0
BV((0,1),{−1,1}) having a fixed number of jumps is also a local minimum of
I∗ in L2(0,1). This observation reduces the problem of determining local minima
0
of I∗ to a finite dimensional one. Since 1(||Du||(0,1)) is equal to the number of
0 2
jumps of u, it is enough to investigate only the second term of I∗. In general, its
0
critical points are determined from a system of nonlinear algebraic equations and
this is not an easy, or indeed, pleasant task.
The Ohta-Kawasaki functional follows from [2], though it has been shown in
[27]thatitis notsophisticatedenoughtopredictthecomplicatedphaseseparation
morphologies. However, to confirm that pseudoassociations exist and to calculate
the critical point exponent, we make a study in one dimension.
6 ADAMCHMAJ
The robust functional
1 1
(0.12) I(w)= [ ǫ2w′′(x)2+W(w′(x))+w2(x)]dx,
Z 2
0
canmodele.g.,thetwinnedmartensitephaseinNitinol[25]. Theelasticfoundation
third term in (0.12) is nonlocal. Namely, let us consider (0.12) with the boundary
conditions w(0)=w(1)=0. Let w =v′, v =(−D2)−1(u−m), where
1 1
−D2 :{v∈W2,2 :v′(0)=v′(1)=0, v =0}→{w∈L2 : w =0}.
Z Z
0 0
Then w′ =v′′ =m−u, w′′ =−u′ and
1 1 1 1 1
w2 = v′2 =− v′′v = (u−m)(−D2)−1(u−m)= u(−D2)−1u
Z Z Z Z Z
0 0 0 0 0
and (0.12) becomes
1 1 1 1
I(u)= [ ǫ2u′(x)2+W(m−u(x))]dx+ G(x,y)u(x)u(y)dxdy,
Z 2 Z Z
0 0 0
with G as in [32]. Using the improved van der Waals derivation, we obtain (0.7)
with J = J - the strong separation diblock copolymer functional. With G and
ǫ
small ǫ - asymptotically periodic local minimizers with sharp interfaces.
References
[1] G.AlbertiandG.Bellettini,Anon-localanisotropicmodelforphasetransitions: asymp-
toticbehaviour ofrescaledenergies,EuropeanJ.Appl.Math.9(1998), 261-284.
[2] S. Alexander and J. McTague,Shouldallcrystals bebcc? Landau theory ofsolidication
andcrystalnucleation,Phys.Rev.Lett.41(1978), 702.
[3] J. M. Ball and C. Mora-Corral, A variational model allowing both smooth and sharp
phaseboundariesinsolids,Commun.PureAppl.Anal.,inpress.
[4] P. W. Bates and A. Chmaj,Anintegrodifferential model forphase transitions: stationary
solutionsinhigherspacedimensions,J.Statist.Phys.95(1999), 1119-1139.
[5] P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model
forphasetransitions,Arch.Ration.Mech.Anal.138(1997), 105-136.
[6] F. Bethuel, G. Orlandi and D. Smets, Convergence of the parabolic Ginzburg-Landau
equationtomotionbymeancurvature, Ann.ofMath.(2) 163(2006), 37-163.
[7] M. Bode, Spin-polarized scanning tunneling microscopy, Rep. Prog. Phys. 66 (2003), 523-
582.
[8] D.BrandonandR.C.Rogers,Thecoercivityparadoxandnonlocalferromagnetism,Con-
tin.Mech.Thermodyn.4(1992), 1-21.
[9] J. Carr and R. Pego, Invariant manifolds for metastable patterns in ut = ǫ2uxx−f(u),
Proc.Roy.Soc.EdinburghSect.A116(1990), 133-160.
[10] X. Chen and Y. Oshita, An application of the modular function in nonlocal variational
problems,Arch.RationalMech.Anal.186(2007), 109-132.
[11] R.ChoksiandP.Sternberg,Onthefirstandsecondvariationsofanonlocalisoperimetric
problem,J.ReineAngew.Math.611(2007), 75-108.
[12] P.G.DeGennesandC.Taupin,Microemulsionsandtheflexibilityofoil/waterinterfaces,
J.Phys.Chem.86(1982), 2294-2304.
[13] J.delaFiguera,F.Leonard,N.C.Bartelt,R.StumpfandK.F.McCarty,Nanoscale
periodicityinstripe-formingsystemsathightemperature: Au/W(110),Phys.Rev.Lett.100,
186102(2008).
[14] M.delPino, M.Kowalczyk andJ.Wei,OnDeGiorgiconjectureindimensionN ≥9.
[15] H. F. Ding, W. Wulfhekel and J. Kirschner, Ultra sharp domain walls in the closure
domainpatternofCo(0001), Europhys.Lett.57(2002), 100-106.
[16] Y.-F.Duan,H.-P.Liu,L.Yi,AtomicallysharpdomainwallsofRb2Cd2(SO4)3langbeinites:
4delectroneffects, PhysicsLettersA350(2006), 278-282.
PERIODIC MICROSTRUCTURE 7
[17] J.L.Ericksen,Equilibriumofbars,J.Elasticity5(1975), 191-201.
[18] R.L.FosdickandD.E.Mason,Onamodelofnonlocalcontinuummechanics.II.Structure,
asymptotics,andcomputations, J.Elasticity48(1997), 51-100.
[19] G.FuscoandJ.K.Hale,Slow-motionmanifolds,dormantinstability,andsingularpertur-
bations,J.Dynam.DifferentialEquations 1(1989), 75-94.
[20] S.Janeczko,Anoteonsingularsubmanifolds,J.Geom.Phys.2(1985), 33-59.
[21] S. Janeczko, T. Mostowski and J. Komorowski, Phase transitions in ferromagnets and
singularities,Rep.Math.Phys.,21(1985), 357-381.
[22] G.D.Kahl,GeneralizationoftheMaxwellcriterionforvanderWaalsequation,Phys.Rev.
155(1967), 78.
[23] R. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Roy.
Soc.EdinburghSect.A111(1989), 69-84.
[24] E.S.MatsuoandT.Tanaka,Kineticsofdiscontinuousvolume-phasetransitionofgels,J.
Chem.Phys.89(1988), 1695.
[25] S.Mu¨ller,Singularperturbationsasaselectioncriterionforperiodicminimizingsequences,
Calc.Var.PartialDifferentialEquations1(1993), 169-204.
[26] K. Nakamura, Y. Takeda, T. Akiyama and T. Ito, A. J. Freeman, Atomically sharp
magneticsdomainwallinthinfilmFe(110): afirstprinciplesnoncollinearmagnetismstudy,
Phys.Rev.Lett.93,057202(2004),
[27] P.D.OlmstedandS.T.Milner,Strong-segregationtheoryofbicontinuousphasesinblock
copolymers,Phys.Rev.Lett. 72(1994), 936.
[28] A. Parola and L. Reatto, Hierarchical reference theory of fluids and the critical point,
Phys.Rev.A31,(1985), 3309.
[29] G. B. Partridge, Wenhui Li, Y. A. Liao and R. G. Hulet, M. Haque and H. T. C.
Stoof,Deformationofatrappedfermigaswithunequal spinpopulations,Phys.Rev.Lett.
97,190407(2006).
[30] M. Pratzer and H. J. Elmers, M. Bode, O. Pietzsch, A. Kubetzka and R. Wiesen-
danger,Atomic-scalemagneticdomainwallsinquasi-one-dimensionalFeNanostripes,Phys.
Rev.Lett.,127201 87(2001).
[31] X.Ren andL.Truskinovsky,Finescalemicrostructuresin1-Delasticity,JournalofElas-
ticity59(2000), 319-355.
[32] X.RenandJ.Wei,Wriggledlamellarsolutionsandtheirstabilityinthediblockcopolymer
problem,SIAMJ.Math.Anal.37,455-489.
[33] P. Rosenau, Free-energy functionals at the high-gradient limit, Phys. Rev. A 41 (1990),
2227.
[34] O.Savin,Regularityofflatlevelsetsinphasetransitions,Ann.ofMath.169(2009).
[35] S. Serfatyand I. Tice, Lorentz space estimates forthe Ginzburg-Landau energy, J. Func.
Anal.254(2008), 773-825.
[36] J.Serrin,Thearearuleforsimplefluidphasetransitions,J.Elasticity90(2008), 129-159.
[37] L. Truskinovsky and A. Vainchtein, Quasicontinuum models of dynamics phase transi-
tions,Contin.Mech.Thermodyn.18(2006), 1-21.
[38] M.S.Turner,M.RubinsteinandC.M.Marques,Surface-inducedlamellarorderingina
hexagonal phaseofdiblockcopolymers,Macromolecules27(1994), 4986-4992.
[39] R. van Gastel, N. C. Bartelt, P. J. Feibelman, F. L´eonard and G. L. Kellogg, Re-
lationshipbetween domain-boundaryfreeenergyandthetemperature dependence ofstress-
domainpatternsofPbonCu(111), Phys.Rev.B70,245413(2004).
[40] G. M. Whitesides and M. Boncheva,Beyond molecules: Self-assemblyof mesoscopic and
macroscopiccomponents, Proc.Natl.Acad.Sci,USA99(2002), 4769-4774.
[41] W.Windl,T.Liang,S.LopatinandG.Duscher,Modelingandcharacterizationofatom-
ically sharp “perfect” Ge/SiO2 interfaces. Materials Science and Engineering B 114-115
(2004), 156-161.
[42] R. A. Young, Theory of quantum-mechanical effects on the thermodynamic properties of
Lennard-Jonesfluids,Phys.Rev.A23(1981), 1498.
Division of Integral Equations, Faculty of Mathematics and Information Science,
Warsaw University of Technology,Pl. Politechniki 1,00-661Warsaw, Poland
E-mail address: [email protected]