Missing:
Joumalofthe
Less-Common
Metals, 167( 1991) 305-311
“UP-HILL” DIFFUSION AND HYDROGEN-HYDROGEN IN PALLADIUM-PLATINUM ALLOYS
305
INTERACTION
AARE MALL0 Institute
of
Theoretical Physics, Chalmers University of Technology, S-412 % Giiteborg (Sweden)
ANATOL KROZER Department of Physics, Chalmers University of Technology, S-412 % Giiteborg (Sweden)
(Received June 22, 1990)
Summary The reported anomalous “up-hill” diffusion of hydrogen in Pd-Pt and Pd-Ag alloys is explained in terms of attractive, long-range, non-local, elastic interactions between dissolved hydrogen atoms. A statistical model based on the continuum elasticity theory is presented, and model calculations for the PdX,PtlY system are shown to give results that are in semi-quantitative agreement with experimental data.
1. Introduction Metal-hydrogen systems have many economically and technologically important applications. Examples are hydrogen storage [ 1,2], catalysis and hydrogen embrittlement. From a more theoretical point of view, metal-hydrogen systems are also of great interest. Since the hydrogen atom is one of the most basic impurities in metals, metal-hydrogen systems often serve as model systems, inspiring new concepts and ideas. Hydrogen dissolves exothermally in many transition metals and occupies interstitial sites in the host lattice. In contrast to heavier elements such as oxygen and nitrogen, hydrogen has unusually high solubility and mobility. In many cases the maximum ratio of hydrogen to metal atoms is close to unity [l]. At low concentrations the dissolved hydrogen behaves essentially as a lattice gas [3]; this is the so called cc-phase. At higher concentrations a liquid-like (a’) phase [4, 51 and a solid, metal-hydride (/3) phase [6] can also occur. The factor that most dramatically influences the solubility of hydrogen in metals is the host valence electron density [7, 81, which is often characterized by the metal rs value. From model calculations [9] it is well known that the hydrogen atom favours an electron density around r, = 2.9 a.u. In the case of transition metals this simple picture is modified by covalent hybridization of the hydrogen orbital with the metal d-band [lo]. 0022-5088/91/$3.50
0 Elsevier Sequoia/Printed
in The Netherlands
306
Elastic interactions between the dissolved atoms are believed to play an important role in the a-at’ phase transition [ 1 l-131 and for the diffusion behaviour of hydrogen in the metal [ 141. An interstitial hydrogen atom in the metal lattice will induce a long-range strain field that affects all other dissolved atoms. The elastic hydrogen-hydrogen (HH) interaction is long range and non-local in the sense that it depends on the total hydrogen distribution and on the elastic boundary conditions. It has been shown that for a metal with a free outer surface, the eiastic HI-I interaction is effectively attractive while, with a clamped surface, it can also be repulsive [15]. A phenomenon closely related to the elastic HH interaction is the so-called Gorsky effect [ 161: hydrogen migration induced by an external strain field which can be created, for example, by bending the sample. Both the elastic HH interaction and the Gorsky effect are sometimes referred to by the common term (diffusion-) elastic effects. Quite recently, Lewis and coworkers have reported two examples, Pd,,Pti9 [17, 181 and Pd,,Ag,, [19], where elastic effects are believed to be responsible for a drastic modification of the diffusion behaviour of hydrogen in the metal, What makes their results somewhat unusual is the observation that the hydrogen actually migrates “up-hill”, meaning in this context migrating in the opposite direction of the overall gradient in the chemical potential. The’experimental set-up used by Lewis is very simple (Fig. 1). A small, thin-walled metal tube with an attached pressure gauge is enclosed in a larger vessel. Hydrogen gas inside and outside the tube is allowed to equilibrate with the metal until the pressures on both sides are equal to p,,. The outside pressure is then raised to a constant value pz >po, whereupon the pressure p, inside the tube is monitored as a function of time. After an initial rapid increase, p, decreases to a minimum value less than pcb,before finally increasing. The authors attributed this unusual pressure drop to the Gorsky effect [19] and a phenomenological model has been presented [ZO]. In this article we will investigate this hypothesis further by comparing it with a model calculation for the diffusion of hydrogen in a metal. Our basic assumption is that the domin~t interaction in the system under consideration is elastic in nature. a
b
metal tube 1 I
J
I
Time t
Fig. I. (a) The experimental set-up of Lewis et al. [I Sj. A small Pd,,Pt,, tube is enclosed within a larger chamber with voiume V, ii>Y,. H, gas in V, and Vz is allowed to eq~Iibrate with the metal until p, =pz =p(,. instantaneously the outside pressure p2 is raised to a constant value, 1.5-3 times po. The pressure p, inside the tube is then monitored as a function of time. (b) Schematic diagram showing the variation of p, VS.time f in the experiment of Lewis et ul. fnitially the pressure decreases and eventually reaches a minimum value that, somewhat surprisingly, is less than p,,. In the end a bre~through takes place and p, increases monotonically.
The model that we construct, herein called the elastic interaction model, is largely based on the work of Wagner and Horner [ 12, 13, 151. In this model both the elastic HH interaction and the interaction between the dissolved hydrogen atoms and the externally induced strain field are treated within the same formalism. It has been argued in the literature ]3] that the elastic HH interaction can be modelled equally well and more easily by an attractive, nearest-neighbour (NN) interaction. At high hydrogen concentrations and for an almost homogeneous system in thermodynamic equilibrium this is probably true. However, it is doubtful whether the NN model can also describe the dynamics of a non-equilibrium situation with a spatially varying hydrogen distribution. In the latter case one can expect significantly different behaviour with the local NN model than with the non-local elastic model. For comparison we therefore present results from the NN model and also from a non-interacting particle model. The outline of this paper is as follows. In Section 2 we present the theoretical models. The equations of motion for the hydrogen density in the metal and for the gas pressure are derived in Section 2.1, and in Section 2.2 approximations for the hydrogen chemical potential are given. Finally, in Section 3 we present our results together with a short discussion of their implications. We find that, among the investigated models, it is only the elastic interaction model that can reproduce We also find that it is the elastic experimental results qualitatively. hydrogen-hydrogen interaction, rather than the Gorsky effect, that is responsible for the anomalous “up-hill” hydrogen diffusion in the experiment of Lewis and coworkers.
2. Model In this section we construct a model that enables us to solve the diffusion equation for dissolved atoms and, at the same time, calculate the pressure inside the metal tube. For a system with non-interacting particles the particle current is proportional to the gradient of the concentration, as assumed in the ordinary diffusion equation, Fick’s law [21]. When interactions are taken into account this is no longer true. Therefore, in order to describe the transport process we employ a mean-field model, where instead it is assumed that the driving force is proportional to the gradient of the chemical potential. To keep the arguments simple we have not included in our model any description of the kinetics of the a-a’ phase transition. This means that we here consider the concentration and temperature region where only a single (a) phase exists. Three different models will be considered for calculating the chemical potential: (i) the elastic interaction model, (ii) the nearest-neighbour interaction model and (iii) the non-interacting particle model. In the elastic interaction model, and for a given arbitrary hydrogen distribution, it is not possible to solve the elastic equilibrium equations analytically, except in two special coordinate systems: spherical coordinates and two-dimensional planar polar coordinates [22]. To utilize the analytical results, we therefore replace the cylindrical tube of the actual experiment
308
by a model geometry (Fig. 2) consisting of a spherical shell that has the same wall thickness as the tube used in the experiment. 2.1. Equations of motion for the hydrogen pressure and density At time t = 0 the gas pressure outside the vessel is raised from p. to a constant value p2, and the following happens. The internal volume V, of the vessel is then slightly reduced owing to mechanical compression of the metal shell, which gives rise to a small and immediate increase in the pressure p, of the enclosed gas. After that, and on a much longer time-scale, hydrogen atoms start to migrate across both the inner and the outer surfaces of the vessel. Let N, denote the number of H, molecules in volume V, and assume the ideal gas law to be valid. If the transport of hydrogen inside the metal is the rate-limiting step, the time evolution of pI can be written as
(1)
n, *j dS
where n, is the normal unit vector on surface S, and j is the particle current (i.e. the current of hydrogen atoms) in the metal. The factor 2 in the denominator on the right hand side of eqn. (1) enters because it takes two atoms to form a hydrogen molecule. The hydrogen number density p(r) is related to j by the equation of continuity: G(r) p= at
(2)
-V*j(r)
In the mean-field approximation the particle current j is proportional gradient of the chemical potential ,u [23], j(r)=
-o(r)B(r,
IPI) Vp(r, (~1)
to the (3)
where B is the particle mobility. In general, both B and ,u are functions of p(r), as indicated above. However, if the hydrogen density is low one may assume that B is only weakly dependent on p and that a local approximation can be made for B, B(r,{pJ)=B(p(r))=B,,(1-8(r))
(4)
metal shell,
Fig. 2. Model geometry used in the calculation. The tube used in the actual experiment replaced by a spherical shell with inner and outer radii R, and R, respectively.
is in the model
309
where S(r) = p( r)/p, is the fraction of octahedral sites at r that are occupied by hydrogen atoms, ,L+,,being the density of such sites. This form for B can be derived from a hard-core model for the short-range hydrogen-hydrogen interaction and 8 is then the site blocking factor [24,25]. 2.2, Hydrogen chemical potential in the metal To calculate the hydrogen chemical potential in the metal we must first find the Helmholtz free energy F as a function of p_ By definition F= U- TS
(5)
where U is the total internal energy of the metal-hydrogen system, T is the temperature and S is the entropy. In the elastic interaction model the total energy is separated into two terms: U= U,)+ u,,
(6)
Here U,, is the energy of the unrelaxed system, i.e. the total energy of the metal-hydrogen system when the metal ions are kept in their original pure-metal equilibrium positions, and U,, is the elastic energy connected with the relaxation of the metal ions. We shall here take U,) to be a linear function of the total number of hydrogen atoms dissolved in the metal. In other words: we assume that the interstitial hydrogen atoms do not interact with each other when the metal ions are kept in fixed positions. It has been shown by Wagner and Horner [ 1 S] that, in a continuum model, the elastic energy can be written as
where E,/?=Ka,u,+QQJ C u&U>=%&N+!OojI~~v+ IL&)
(8) L$$J+
+%&Q--
L%J
= P~~~{r)
(9) (10)
Here E+ is the strain tensor, II is the displacement field and Cak,, are the elastic constants (isotropic material assumed). Paa is the so-called force-dipole tensor. The dissolved hydrogen occupies octahedral sites in the f.c.c. lattice, and for those sites tensor, PU, is diagonal [ 151, i.e. PUB= P6,,. The trace of the force-dipole Tr { Pap} = 3 P, can be related to the hydrogen-induced volume expansion A I/ by 1261 AV -.-G.= - pK7.p
where K,= l/( 1+ $,u) is the isothermal compressibility. Expe~mentally mined values of P are typically of the order of a few electronvolts [27].
(11) deter-
310
For a given arbitrary hydrogen distribution, the elastic energy can be written in the form (Appendix A, eqn. (Al 8)): u~,=-:IIp(r)W(r,r’)p(r’)drd;+IG(r)p(r)dr+~Ik’.u’(r)dS
(12)
where the kernels W(r, r’) and G(r) are independent of p(r). This expression for the elastic energy has a simple interpretation: the first term on the right-hand side of eqn. (11) describes the hydrogen-hydrogen interaction, the second term describes the interaction between hydrogen atoms and the externally applied forces, ie. the Gorsky effect, and the third term is the work done by the external forces on the hydrogen-free lattice. For a spherically symmetric hydrogen distribution, W and G are given by eqns. (A 19) and (A2 1). We now examine the entropy contribution to the free energy. For convenience the entropy is separated into two parts: (13)
S = SO+ SCO”f Here S,, is the entropy associated which we consider as independent [3, 281:
scad
with the lattice internal degrees of freedom, of p, and Sconfis the configurational entropy
= -k,p,I[(l-B(r))ln(l-O(r))+tI(r)lnO(r)]dr
(14)
Defining (15)
F,, = UC,- % the Helmholtz free energy can be written as F= F,, + u,, - TSconf
(16)
By taking the functional derivative of F with respect to the hydrogen density we obtain the chemical potential (17)
clcl(cb))=
-I
W(r,;),Q(;) d;+ G(r)
.wu,,,Mr))=k,T ln[e(rY(l- e(r))1
(18) (19)
where ,u()is now regarded as a constant, i.e. independent of p, for a given temperature. In the present work ,LQhas been determined from experimental data for the equilibrium concentration of dissolved hydrogen as a function of gas pressure [6]. An approximate method for deriving the chemical potential in the NN model can be found in the literature [29, 301, and we here only give the result. In the NN
311
model, pe, in eqn. ( 17) is replaced by 1+&y; h&(4)=
- W
~
In
l+&
[i
)I
where Y=
(21)
exp(- wdk, T 1
and &=
[(1-28)‘+4ye(l-e)]“‘-(1-28) 2y(1-
(22)
0)
Here z = 12 is the coordination number, i.e. the number of nearest-neighbour octahedral sites in the f.c.c. lattice, and wun is the pairwise hydrogen interaction energy. Finally, if the particles are regarded as non-interacting, only puo and pconf should be kept in the expression for the chemical potential, eqn. ( 17). This is equivalent to setting P = 0 in the elastic interaction model or setting wuu = 0 in the NN model.
3. Results and discussion The equations of motion for the pressure and the hydrogen density, eqns. (1) and (2), are solved numerically using the Runge-Kutta method. The initial condition at t = 0 is a uniform hydrogen distribution in the metal shell, P(C t=O)=p,
(23)
and the boundary conditions for all times tare i=1,2
/+-=R,)=tP,(P,)
(24)
i.e. we require that the chemical potentials on the metal surfaces S, and Sz equal the corresponding gas phase values, as calculated per atom in the gas. TABLE
1
Parameter
values used in the numerical
zrn]
HZ (mmi
k
4.00
4.25
4 x 10”‘”
solution
of the diffusion
equation
Y $p
m ~?)
,‘Ref.31. hPalladium data from ref. 32. ‘Ref. 33. ‘Refs. 26 and 34.
0.31h
smI)
3.4 x IO”
I’
g-z1
WHH (mev)
iev)
6.8 x 10’”
- 26’
- 1.1”
312
:
-
- N-N model
Fig. 3. Calculated Hz gas pressure p, inside the vessel plotted as a function of time for three different ), (ii) the nearest-neighbour interaction model (- -- -) models: (i) the elastic interaction model (and (iii) the non-interacting particles model (.....). Circles represent experimental data taken from ref. 18. For f < 0 the dissolved hydrogen is in equilibrium with the surrounding HZ gas at 604.6 Torr and at t = 0 the outside pressure is raised to a constant value pz = 8 15 Torr. Only the elastic model for the hydrogen-hydrogen interaction can qualitatively reproduce the characteristic pressure drop that is seen in the experimental data. In this diagram no significant difference can be seen in the results from the non-interacting particles model and from the nearest-neighbour model. Fig. 4. Variation in hydrogen chemical potential across the metal sphere wall calculated within the elastic interaction model. Time t = 0- 1000 s in steps of 200 s. The inner surface is at x = 0, the outer surface at x = 1 and the energy scale is chosen so that y( r, I = 0) = 0. Hydrogen entering the wall at the outer surface expands the metal and initially lowers the chemical potential in the interior of the wall. This creates a gradient in the chemical potential at the inner surface which will drive a current of particles, from the gas phase inside the vessel and into the metal wall.
The parameter values used in the calculation are given in Table 1. The inner and outer radii of the sphere are chosen so that the wall thickness is the same as that of the cylinder used in the actual experiment. The mobility B,, is determined from the experimental data [ 181 and the density of octahedral sites ,o,,, is equal to the number-density of metal ions in the f.c.c. host lattice. In Fig. 3 the calculated gas pressure inside the tube is shown as a function of time for the three different models: (i) the elastic interaction model, (ii) the nearestneighbour interaction model and (iii) the non-interacting particle model. For comparison, experimental data taken from ref. 18 are also included. The nearest-neighbour model yields essentially the same pressure against time behaviour as obtained for non-interacting particles and the two curves can hardly be distinguished in this figure. It is evident from Fig. 3 that among the three models investigated here only the elastic interaction model can qualitatively reproduce the experimental data. What actually happens in this model is best understood by observing how the chemical potential varies in space as a function of time. In Fig. 4 we have plotted the variation in hydrogen chemical potential across the metal wall for different times rB 0. As the outside gas pressure is raised, hydrogen
313
Fig. 5. Change in relative within the elastic interaction
hydrogen concentration (O(r)-@,)/O,, across model for I = Ok I500 s in steps of 100 s.
the sphere
wall calculated
atoms enter the metal at the outer surface and induce a lattice expansion which is propagated across the wall at the speed of sound. The hydrogen chemical potential therefore decreases in the interior of the metal. In practice, as well as in theory, this effect can be pictured as a non-local elastic interaction between the dissolved atoms. (See the discussion in connection with eqn. ( 12). The change in the chemical potential is almost entirely caused by the elastic HH interaction, the contribution from the Gorsky effect being typically two orders of magnitude smaller.) A gradient in the chemical potential is thus established at the inner surface of the sphere, which will cause hydrogen to enter the metal also from the gas phase across surface S,, i.e. from the interior of the vessel into the wall. This is shown in Fig. 5 where we have plotted the change in hydrogen concentration across the metal for different times. The gas pressure p, inside the sphere now decreases, eventually below its initial value P,,. Since the vessel at the same time expands, its internal volume increases which further reduces the pressure p,. However, that is only a minor secondary effect. Finally, as hydrogen atoms coming from the outer surface migrate inwards and reach the inner surface, the chemical potential in the latter region increases. After a certain time the gradient at the inner surface is reversed and the net particle current across S, then goes from the metal to the gas phase, i.e. p, increases. It is interesting to compare our results with those obtained in ref. 20. In that work a flat plate served as a model geometry to represent the tube used in the actual experiment. For short times the internal pressure is shown to be time dependent p,(t) = - at”‘. The t”i7 behaviour is apparently in good agreement with experimental data. An expression for the constant Q is derived, but no attempt is made to compare it with the experimentally determined value. Identifying I/, in ref. 20 with - K,P and using the definition for a in their eqn. (47), we have = 10 - ?- 10 - j. We believe that this large discrepancy estimated that atheory/acxper,mmt
314
between theory and experiment is mainly due to an unfortunate choice of model geometry (the volume change and the corresponding change in chemical potential that hydrogen induces on the far side will be much smaller for a flat plate than for a cylinder or for a sphere). This illustrates how sensitive the elastic HH interaction is to the geometry and to the elastic boundary conditions. In summary, we have shown that the anomalous “up-hill” diffusion reported by Lewis and coworkers can be explained semiquantitatively in terms of non-local, elastic interactions between the dissolved hydrogen atoms. Our results also indicate that the Gorsky effect is only of minor importance compared with the hydrogen-hydrogen interaction. We have further demonstrated that, for a nonequilibrium hydrogen distribution and as far as the kinetics of the system is concerned, the elastic hydrogen-hydrogen interaction can not be modelled by a local nearest-neighbour interaction.
Acknowledgment
Financial support from the Swedish Natural Research Council is gratefully acknowledged.
References 1 R. Wiswall, Hydrogen in Metals II, Vol. 29, Topics in Applied Physics, Springer-Verlag, Berlin, 1978, chapter 5, p. 201. 2 R. Povel, W. G. Feucht and G. Withalm, Interdiscip. Sci. Rev., 14 (1989) 365. 3 E. Wicke and H. Brodowsky, Hydrogen in Metals II, Vol. 29, Topics in Applied Physics, SpringerVerlag, Berlin, 1978, p. 73. 4 A. Magerl, R. R. Rush and J. M. Rowe, Phys. Rev. B, 33 (1986) 2093. 5 0. Hartmann, Hyperfine Interact., 31(1986) 241. 6 T. B. Flanagan, in B. Bambakidis (ed.), Metal Hydrides, Plenum, New York, 1981, p. 36 1. 7 J. K. Nsrskov and N. D. Lang, Phys. Rev. R, 21(1980) 2 13 1. 8 M. J. Stott and E. Zaremba, Phys. Rev. B, 22 (1980) 1564. 9 M. J. Puska, R. M. Nieminen and M. Manninen, Phys. Rev. R, 24 (1981) 3037. 10 P. Nordlander, S. Holloway and J. K. Nsrskov, Su$ Sci., 136 (1984) 59. 11 G. Alefeld, Phys. Status Solidi, 32 (1969) 67. 12 H. Horner and H. Wagner, .I. Phys. C, 7( 1974) 3305. 13 H. Wagner, Hydrogen in Metals II, Vol. 29, Topics in Applied Physics, Springer-Verlag, Berlin, 1978, p. 5. 14 H. Wipf, J. Volkl and G. Alefeld, Z. Phys. B, 76 (1989) 353. 15 H. Wagner and H. Horner, Adv. Phy., 23 ( 1974) 587. 16 W. S. Gorsky, Phys. Z. Sovjetunion, 8 (1935) 457. 17 F. A. Lewis, J. P. Magennis, S. G. McKee and P. J. M. Ssebuwufu, Nature, 306 (1983) 673. 18 F. A. Lewis, B. Baranowski and K. Kandasamy, J. Less-Common Met., 134 (1987) L27. 19 F. A. Lewis, K. Kandasamy and B. Baranowski, Platinum Metals Rev., 32 (1988) 22. 20 B. Baranowski, J. Less-Common Met., 154 ( 1989) 329. 21 J. Crank, The Mathematics ofDiffUSion, Oxford University Press, London, 1970. 22 P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part ZZ,McGraw-Hill, New York, 1953, Chapter 13.2. 23 H. K. Janssen, Z. Physik B, 23 ( 1976) 245.
315 24 C. T. Chudley and R. J. Elliott, Proc. Phys. Sot., 77( 196 1). 25 D. K. Ross, Z. Phys. Gem. Neue Folge, 164 (1989) 897. 26 H. Peisl, Hydrogen in Metals I, Vol. 28, Topics in Applied Physics, Springer-Verlag, Berlin, 1978. p. 53. 27 G. Schaumann, J. Volkl and G. Alefeld, Phys. Status Solidi, 42 (1970) 40 1. 28 R. H. Fowler, Statistical Mechanics, Cambridge University Press, London. 1955,2nd edn. 29 R. E. Peierls, Proc. Cambridge Philos. Sot., ( 1936) 47 I. 30 D. L. Adams, Surf: Sci., 42 (1974) 12. 3 1 M. Yoshihara and R. B. McLellan, J. Less-Common Met., 107( 1985) 267. 32 C. Nordling and J. osterman, Physics Handbook, Chartwell-Bratt, Bromley, 1982,2nd edn. 33 J. D. Clewley, J. F. Lynch and T. B. Flanagan, J. Chem. Sot. Faraday Trans. I, 73 ( 1977) 494. 34 B. Baranowski, S. Majchrzak and T. B. Flanagan, .I. Phys. F, l(l971) 258.
Appendix A: Solution of the elastic equations
for a spherical shell
In the continuum approximation, the elastic contribution to the internal energy of a metal loaded with hydrogen has been derived by Wagner and Horner [Al]:
(Al) The strain field caS and hence the elastic energy rl-,, depend both on the hydrogen distribution in the metal, p(r), and on the external forces k’ that are applied to the system. Linear elasticity theory allows us to superimpose the internal strain (due to hydrogen) and the externally induced strain. We therefore write U=Ui+Ue
(AZ)
&US= &$ + E,s’
(A3)
This naturally leads to a similar division of the stress tensor (I into an internal and an external part, (7c$ = G/J&l
+ “c$
(A4)
o,B’= C+&lcve
(A9
The equations become
of equilibrium
and the boundary
a,C$
a,a$
rin V
(A6)
ronS
(A7)
=0
nsa,~=O
=0
nBa,ge = k,’
conditions
for a’ and CF then
In this notation the elastic energy, eqn. (Al), can be rewritten as
(A81
316
Partial integration of the first term on the right-hand side of eqn. (A8), together with eqns. (A6) and (A7), leads to UC,=; J K&)k+!W+
cab“(r)] dr+;
j- k’.u’(r) dS+;
j- k%‘(r)
Solving the equations of equilibrium (A6) in spherical u(r) = u,( r)*Fwith the appropriate boundary conditions
dS
(A9)
polar coordinates
for
a),(r) = 0
ronS,,
(AlO)
C(r) = -h2
r on S,,,
(All)
one arrives at Il(+-r2
dr’+Ar+!
r2
urc(r)=cr+g r2
(Al2) (A13)
where R, and R, are the inner and outer radii of the shell respectively, ;Zand ,u are the Lame constants and A, B, C and D are given by R2
4fi A= -(~+2~)(31+2#2’-R,‘)R,
I TL(y’)y’2dY’
(A14)
R2
R,” B= -(n + 2p)(R23_ RI”) Rl I NY’)+’
dr
D = _ R,“Rz3(p, -PI)
(A15)
L417)
414R2~ - R13) The elastic energy can then finally be written in the form
L418) where P2 W(r, r’)= ~ (A+%)
6(r-r’)+M(r,
r’)
(A19)
317
M(r,
r’)=M=
3pP? JC(I+~/L)(~L+~~)(&~-~)
(A20)
(A21)
