Diffusion of hydrogen in metals at arbitrary concentration gradients

\ PERGAMON

International Journal of Hydrogen Energy 13 "0888# 702Ð706

Di}usion of hydrogen in metals at arbitrary concentration gradients L[I[ Smirnov Department of Physics\ Donetsk State Technical University\ 47 Artem Str[\ Donetsk 239999\ Ukraine

Abstract A nonlinear equation of hydrogen di}usion in metals for general case of all concentration gradients is derived[ The solution of linearized equation is presented[ It is shown that it describes the spinodal decomposition and modulated structures more adequately than Cahn|s theory and its modi_cations[ Þ 0888 International Association for Hydrogen Energy[ Published by Elsevier Science Ltd[ All rights reserved[

Nomenclature c local H concentration de_ned with respect to the number of interstitial positions which are occupied cm maximum hydrogen concentration in metal D di}usion coe.cient of hydrogen in dilute solu! tion j ~ux density of hydrogen atoms kB Boltzmann constant kc\ km wave number value that gives maximum b in Cahn theory and this work respectively k\ k wave vector\ wave number L kinetic coe.cient n¹\ c¹ mean H concentrations n"r# volume H concentration R0 radius of the _rst coordination sphere around given interstice Rc e}ective radius of interaction H atoms Tc critical temperature of spinodal decomposition Ts spinodal temperature T temperature U energetic parameter describing the interaction of H atoms in metal V"r?!r# interaction energy of two H atoms placed in interstices r and r? Greek symbols b kinetic ampli_cation factor "concentration increment# dc"r\ t# local concentration deviation from the mean value at instant of time t

9 Dm u

Hamilton operator excess chemical potential probability of interstice _lling by hydrogen atoms m chemical potential of hydrogen subsystem t  0:b distinctive time of the inhomogeneities growth tD di}usion relaxation time  Rc1:D V volume falling at one interstice

0[ Introduction The hydrogen treatment of metals is accompanied usu! ally by the occurrence of a high hydrogen concentration and its gradients or even necessitates such gradients ð0Ł[ In a concentrated metal!hydrogen system the H di}usion coe.cient depends on its concentration and may take even negative values ð1Ð3Ł[ This causes the occurrence and growth of the concentration inhomogeneities and* under appropriate conditions*the phase decomposition of the system ð4Ð8Ł[ It is obvious that for such systems the di}usion coe.cient\ de_ned from the _rst Fick|s law\ has no longer the original clear meaning[ Moreover\ in a medium with variable deformations\ the _rst Fick|s law by itself and the corresponding di}usion equation do not take place ð7\ 09Ł[ Nevertheless\ such equation is useful for interpretation of experimental data\ although there are yet more exact "but\ clearly\ more cumbersome# evol! ution equations for concentration as well as for structure factor ð7\ 8\ 00Ð03Ł[ The _rst nonlinear di}usion equation for a nonuniform

9259!2088:88:,19[99 Þ 0888 International Association for Hydrogen Energy[ Published by Elsevier Science Ltd[ All rights reserved PII] S 9 2 5 9 ! 2 0 8 8 " 8 7 # 9 9 0 4 0 ! 6

703

L[I[ Smirnov : International Journal of Hydrogen Energy 13 "0888# 702Ð706

system\ describing the evolution of interacting con! centration inhomogeneities\ was derived by Cahn in his classical work on spinodal decomposition ð04Ł[ The basis for Cahn|s theory is the Helmholz free energy functional of the GinzburgÐLandau type\ in which the quadratic in gradient of composition term was taken into account\ whereas all highest!degree terms were neglected ð4\ 05Ł[ This approximation was retained in later works ð6Ð8\ 00\ 03Ł too[ The essential result of Cahn|s theory and its modi! _cations was the conclusion on the occurrence in the decomposed alloys of the macroperiodic concentration distribution*modulated structures[ Yet later some pec! uliarities were found\ which had de_ed explanation in the context of Cahn theory[ As it was shown in ð06Ð08Ł\ in the early stage of spinodal decomposition\ a system of the randomly arranged inhomogeneities appears\ which in speci_c cases are of a great mobility ð19Ł[ In addition\ it has become known ð6\ 8Ł that in some alloys the con! centration distributions are observed which can be inter! preted as a superposition of several sinusoidal structures with multiple wave vectors[ In short note ð10Ł we put forward the di}usion equation that takes into account the highest spatial derivatives of concentration and hence is a generalization of the Cahn theory and its modi! _cations for the highly nonuniform systems[ The objec! tive of the present work is to develop the approach ð10Ł for a metal!hydrogen system quantitatively[

the chemical potential can be neglected up to a maximum hydrogen concentration cm[ Neglecting possible cor! relation in the interstice _lling\ we can write u  c:cm  n:nm "c  nV#[ Furthermore\ it is easily shown that the concentration dependence of the vibrational con! tribution "through the local frequency# causes an additional interaction between hydrogen atoms[ For hydrogen in palladium\ for example\ this interaction comes close to 4) of the total[ Then\ expression "1# takes the form m"r#  kBT ln

c"r#:cm ¦ s V"r?−r#c"r?#[ 0−c"r#:cm r?

"2#

This expression is written within the mean _eld approxi! mation\ which works well only for low and high tem! peratures away from the critical area ð6Ł[ The more exact expression is derivable\ for example\ within the qua! sichemical approximation ð11Ł[ Taking into account that the concentration gradients in the nonuniform system may be su.ciently high\ let us write a formal expression] 0 c"r?#  c"r#¦"r?−r#9c"r#¦ "r?−r#191c"r#¦ [ [ [ 1 "r?−r#m m 9 c"r#[ m; m9 

 s

"3#

Then\ the excess chemical potential in eqn "2# is equal to] 1[ Generalized diffusion equation



Dm"r#  s g1n91nc"r#\

"4#

n9

It follows from nonequilibrium thermodynamics that di}usion ~ux density of hydrogen atoms in a metal can be expressed in terms of the gradient of its chemical potential j  −L9m[

u ¦Dm[ 0−u

g1n 

"0#

In the general case\ the chemical potential of the hydro! gen subsystem can be written as ð2\ 3\ 6\ 11Ł "it is assumed here that hydrogen atoms occupy the interstices of the same type#] m  kBT ln

where] "5#

"It is apparent that g1n¦0  9#[ All parameters g1n¦0 are constant\ as it follows from the translation symmetry of the system[ These parameters can be expressed in terms of a single energetic one U  g9[ Indeed\ with the mean value theorem it follows from eqn "5#]

"1#

The _rst term in eqn "1# describes the con_gurational "entropy# contribution to the chemical potential for the ideal "statistically random# distribution of hydrogen atoms over like interstices[ The second term\ which is called the excess chemical potential\ describes deviations from the ideal behavior[ In the absence of stresses\ this term consists of the interaction potential between hydro! gen atoms in metal\ vibrational\ and electronic con! tribution ð3\ 09\ 11Ł[ Within the maximal concentration approximation ð09\ 11\ 12Ł\ the electronic contribution to

0 s "r?−r#1nV"r?−r# "1n#; r?

g1n 

ðR1nŁ U\ "1n#;

"6#

where R  =r?−r=[ Substituting eqn "4# with eqns "5# and "6# in eqn "2#\ we obtain from eqn "0# an expression that in view of the relationship between L and D ð3\ 09\ 11\ 13Ł] L

0

1

c Dn 0− kBT cm

can be written as]

"7#

704

L[I[ Smirnov : International Journal of Hydrogen Energy 13 "0888# 702Ð706

j  −D9n−

DU c  ðR1nŁ 1n¦0 s n[ c 0− 9 kBT cm n9 "1n#;

0

1

"8#

From the continuity equation it follows that to the ~ux density eqn "8# corresponds to a nonlinear di}usion equation] DU c  ðR1nŁ 1n¦0 1c s c [  D91c¦ 9 c 0− 9 1t kBT cm n9 "1n#;

$0

1

%

"09#

As can be seen from the preceding\ this equation\ within assumed approximations\ is applicable to the description for an evolution of any inhomogeneities with arbitrary _nite concentration gradients[ The in_nite series in eqn "09# takes into account the deviations from the classical linear di}usion equations\ that is from Fick|s law[ The inclusion of the _rst term only "n  9# gives a nonlinear di}usion equation 1c  9ðD"c#9cŁ 1t

"00#

with e}ective di}usion coe.cient

$

D"c#  D 0¦

0

1%

U c c 0− kBT cm

Tc [ cm

"01#

"02#

Taking into account the second term of series in eqn "09# "n  0# too we arrive at the nonlinear Cahn|s equa! tion of his theory on spinodal decomposition\ which describes an interplay of concentration inhomogeneities[ It is easy to verify this\ calculating from eqn "4# and "6# with integration by parts the Helmholz free energy\ then a member with n  0 in eqn "4# gives the gradient and a surface contribution to the free energy[ Furthermore\ to analyze the contribution of the highest derivatives in comparison with the Cahn theory\ we will linearize eqn "09#\ since the linear equation only was considered by Cahn[ It turns out that in linear approxi! mation it has been possible to complete the summation of the in_nite series in eqn "09#[

2[ Analysis of linearized equation We will express concentration c"r\ t# in terms of the small deviations dc"r\ t# from its mean value c¹ as] dc"r\ t#  c"r\ t#−c¹\

0

DUc¹ 0− 1dc  D91dc¦ 1t

1

c¹ cm

kBT

ðR1nŁ 1n¦1 dc\ 9 n9 "1n#; 

s

"03#

"04#

that in view of the spinodal curve equation D"c\ Ts#  9 "see eqn "01##\ can be written in a more convenient form] Ts  ðR1nŁ 1n¦1 1dc dc[  D91dc−D s 9 1t T n9 "1n#;

"05#

We will seek a solution to this equation as] dc"r\ t#  u"t# cos kr[

"06#

Substituting eqn "06# into eqn "05#\ one can readily see that the series in eqn "05# evolves to the Taylor series for ðcos kRŁ\ and we obtain for the amplitude u"t# a time dependence] u"t#  u9 exp bt\

\

which at U ³ 9 describes the occurrence and growth of a single concentration inhomogeneity ð5\ 6\ 09\ 12\ 14Ð16Ł[ In this case\ equating eqn "01# to zero\ we obtain the equation of the spinodal curve in metal!hydrogen system ð2\ 6\ 09\ 12Ł\ whence it follows that U  −3kB

"Notice that smallness of dc yet implies no smallness of its spatial derivatives#[ Substituting eqn "03# into eqn "09# and using a linear approximation with respect to dc\ we come to the linear equation]

"07#

where the increment b is de_ned by expression]

0

b"k#  −Dk1 0−

1

Ts ðcos kRŁ [ T

"08#

Now\ it is appropriate to take a fresh look at the _rst members of Taylor series for ðcos kRŁ[ But notice that at high temperature or small concentration "T Ł Ts# b"k#  −Dk1[ This corresponds to the dissolution of con! centration inhomogeneities in ideal system in accordance with the classical linear di}usion equation[ Furthermore\ the _rst term of Taylor series gives

0 1

b"k#  −Dk1 0−

Ts  D"c¹#k1\ T

"19#

This case corresponds to the long!wavelength limit "k : 9#\ that is\ to the linear approximation[ Taking into account the second term of Taylor series\ we obtain

$

b"k#  −Dk1 0−

Ts k1ðR1Ł 0− T 1

0

1%

[

"10#

This approximation leads to the Cahn theory ð4\ 6\ 8\ 04Ł[ If k ³ k0\ where k0 is the nontrivial solution of the equation b"k#  9\ 0:1

$ 0 1> %

k0  1 0−

T Ts

ðR1Ł

\

"11#

then increment b"k# is positive\ that is\ concentration inhomogeneities increase with time\ forming a modulated

705

L[I[ Smirnov : International Journal of Hydrogen Energy 13 "0888# 702Ð706

structure\ eqn "06#[ From eqn "10# it follows that the maximum growth rate is accomplished at k  kc  k0:z1[ The temperature dependence of ke is given by formula kc"T#  kc"9#

X

0−

T \ kc"9#  0:ðR1Ł[ Tg

$

Ts k1ðR1Ł k3ðR3Ł 0− ¦ T 1 3;

0

1%

\

"13#

and equation b"k#  9 has now two real positive roots k0 and k1[ At 9 ³ k ³ k0 increment b"k# is positive\ and the growth of inhomogeneities takes place\ as in Cahn theory[ At k0 ³ k ³ k1 inhomogeneities are smoothed as well\ although T ³ Ts[ However\ at k × k1 comes the growth of inhomogeneities again\ that is\ a new modulated structure with less wavelength evolves into existence[ Obviously\ in the general case all terms of Taylor series must be taken into account in accordance with the more exact formula of eqn "08#[ We consider here two limiting cases of short! and long!range interaction[ In the nearest neighbors approximation "ðcos kRŁ  cos kR0# the modulated structures with k  9 are all impossible in view of the condition k ¾ p:R0\ and some single concentration inhomogeneities may occur only at "k : 9#[ In the case of the long!range interaction "Ra Ł R0#\ which is most probable for hydrogen in metals ð17Ł\ it may be written within the uniform isotropic continuum approximation

g

Rc

R1 cos kR d R

9

ðcos kRŁ 

"3:2#pRc2

0

2

"12#

When k × k0\ the increment b"k# becomes negative\ and such inhomogeneities level o}\ although at T ³ Ts the system is kept in the two!phase area[ Thus\ all short wavelength ~uctuations in the Cahn theory are damped out\ that makes possible to form monoperiodic sinusoidal structures[ This view turns out to be too simpli_ed[ Indeed\ a consideration in Taylor series of the member with k3 implies instead eqn "10# a dependence b"k#  −Dk1 0−

3p

1

sin8 1cos8 1sin8 − \ 8  kRc ¦ 8 81 82

"14#

Then\ it follows from eqn "08# that at T ³ Ts increment b"k# has a set of the positive maximum values instead of one\ as in Cahn theory[ The number of such maximum values and respectively of regions where b"k# × 9\ as well as the maximum values of the dimensionless increment btD  bRc1:D by itself\ increases with decreasing tem! perature[ In other words\ the distinctive time of the inhomogeneities growth t  0:b reduces progressively in comparison with di}usion relaxation time tD  bRc1:D[ This result is manifestation of the general tendency for the suppression of chaotic movement by the HÐH interaction with decreasing temperature[ As a result\ a possibility of the more ordered\ correlated\ collected movements in hydrogen subsystem appears ð09\ 14Ð16Ł[ The results of numerical estimations for palladium! hydrogen system are tabulated in Table 0[ We put D  1[8×09−2 exp "−9[12 eV:kBT# sm1:s ð18Ł\ Ts  Tc  454 K ð11Ł\ Rc  09 mm[ "Setting Rc\ for exam! ple\ 09 times less\ we obtain for b values 099 times larger and for tD\ t respectively 099 times less#[ In Table 0 "in the third column# the dimensionless wave number values kmRG:p are shown\ which gives the positive maximum to b "see the next to last column#[ As is seen\ at elevated temperatures "499\ 399 and 299 K# there is only one such value km ¼ kc for each T and\ respectively\ only one wavelength l "period of the modu! lated structure#[ However at low temperatures there are yet two "at 199 K# or three "at 099 K# such values of the wave number km[ It is likely that for systems with a higher critical tem! perature Tc or at small Rc such structures with several km and l may occur at more high tempertures[

Table 0 Results of numerical estimations for palladium!hydrogen system T K

tD s

km Rc:p

l:Rc

bt   tD:t

b s−0

t s

499 399 299 199

9[96 9[16 1[41 104

099

0[24×097 "×3 years#

9[04 9[19 9[29 9[24 1[25 9[39 1[39 3[39

02[2 09[9 5[56 4[60 9[74 4[99 9[72 9[34

9[901 9[987 9[251 0[943 6[922 2[49 69[89 28[89

9[06 9[25 9[03 9[994 9[922 1[5×09−7 4[2×09−6 2[9×09−6

4[7 1[7 5[8 193[9 29[5 2[7×096 "0[1 years# 0[8×095 "11 days# 2[3×095 "28 days#

L[I[ Smirnov : International Journal of Hydrogen Energy 13 "0888# 702Ð706

Notice that the values Km are near to kG¦1pn:Rc\ where n are natural numbers[ At large kRc these wave numbers km become equal to the roots of equation sin kRc  0[

ð03Ł ð04Ł

3[ Conclusion ð05Ł

It is shown that in metal!hydrogen systems away from the spinodal curve instead of the Cahn periodic structures the complicated structures may occur\ which are the superposition of inhomogeneities\ eqn "06#\ with multiple wave vectors[ It seems likely that the experimental results mentioned in the introduction become more under! standable[ However\ the more rigorous treatment is required that must be based on the general nonlinear eqn "09# with regard to the concentration "hydrogen# stresses ð6Ð8\ 29Ł as well as to the interplay between both metal and hydrogen subsystem ð09\ 20Ł[

ð06Ł ð07Ł

ð08Ł

ð19Ł

ð10Ł

References ð0Ł Gol|tsov VA[ Fundamentals of hydrogen treatment of materials and its classi_cation[ International Journal of Hydrogen Energy 0886^11"1:2#]008Ð13[ ð1Ł Geld PV\ Ryabov RA[ Hydrogen in metals and alloys[ Moscow] Metallurgia\ 0863[ ð2Ł Alefeld G\ Volkl J\ editors[ Hydrogen in metals\ vols[ 0 and 1[ Berlin] Springer\ 0867[ ð3Ł Smirnov LI\ Phylonenko SS[ On the concentration depen! dence of the di}usion coe.cient of hydrogen in palladium[ Fizika Metallov i Metallovedenie 0878^56"1#]139Ð2[ ð4Ł Cahn JW\ Spinodal decomposition[ Trans[ Met[ Soc[ AIME 0857^131"1#]055Ð68[ ð5Ł L|ubov B Ya[ Kinetic theory of phase transformations[ Moskow] Metallurgia\ 0858[ ð6Ł Khachaturyan AG[ Theory of phase transformations and strukture of solid solutions[ Moskow] Nauka\ 0877[ ð7Ł Katsnel|son AA\ Olemskoy AI[ Mikroskopic theory of inhomogeneous struktures[ Moskow] Moskow University\ 0876[ ð8Ł Ustinovshchikov Yu I[ Precipitation of another phase in solid solutions[ Moskow] Nauka\ 0877[ ð09Ł Smirnov LI\ Gol|tsov VA[ Dynamics of metal!hydrogen systems in a continuum approximation and some hydro! gen!induced elastic e}ects[ The Physics of metals and Met! allography 0886^73"5#]488Ð594[ ð00Ł Langer JS[ Theory of spinodal decomposition in alloys[ Ann[ Phys[ 0860^54]42Ð75[ ð01Ł Binder K\ Stau}er D[ Statistical theory of nucleation\ con! densation and coagulation[ Adv[ Phys[ 0865^14]232Ð85[ ð02Ł Binder K[ Kinetics of phase separation[ In] Arnold L\ Lef!

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706

ever R\ editors[ Stochastic nonlinear systems in physics\ chemistry\ and biology[ Proceedings of the Workshop\ Bielefeld\ Federal Republic of Germany\ 4Ð00 October\ 0879[ Berlin] Springer[ pp[ 51Ð60[ Mazenko GF\ Valls OT\ Zannetty M[ Field theory for growth kinetics[ Physical review B 0877^20"0#]419Ð31[ Cahn JW[ On spinodal decomposition[ Acta Metallurgica 0850^8"8#]684Ð797[ Cahn JW\ Hilliard JE[ Free energy of a nonuniform sys! tem*0] interfacial free energy[ J[ Chem[ Phys[ 0847^17"1#]147Ð56[ Ardel AI\ Nicholson RB[ On the modulated structure of aged NiÐAl alloys[ Ibid 0855^03]0184Ð298[ Hornbogen E\ Roth M[ Die Verteilung koherenter Teilchen in Nickellegierungen[ Zeitschrift Metalkunde 0856^47"01#]731Ð44[ Miyazaki T\ Takagishe S\ Mory H[ Kozakai T[ The phase dekomposition of FeÐMo binary alloys by spinodal mech! anism[ Acta Metallurgica 0879^17]0032Ð42[ Artemenko Yu A\ Gol|tsova MV[ Decomposition of solid solution of hydrogen in palladium at rapid cooling[ Fizika Metallov i Metallovedenie 0884^68"1#]50Ð3[ Smirnov LI[ Di}usion equation of interstitial atoms in met! als at arbitrary concentration gradients[ The Physics of Metals and Metallography "in press#[ Wicke E\ Brodowsky H[ Hydrogen in palladium and pal! ladium alloys[ In] Alefeld G\ Volkl J\ editors[ Hydrogen in Metals[ Berlin] Springer\ 0867^1"2#[ Smirnov LI\ Gol|tsov VA\ Lobanov BA\ Ruzin EV[ E}ect of chemical and deformation interaction of hydrogen atoms on their di}usion in metals[ Fizika Metallov i Met! allovedenie 0874^59"3#]669Ð4[ Smirnov AA[ Theory of di}usion in interstitinal alloys[ Kiev] Naukova dumka\ 0871[ Smirnov LI[ Wave propagation of concentration dis! turbances in hydride!forming metals and kinetics of hydride phase precipitation[ Fizika Metallov i Metallovedenie 0889^09]03Ð19[ Smirnov LI\ Nosenko V Yu[ On kinetics of phase sep! aration in solid solutions[ Ibid 0882^64"3#]040Ð5[ Smirnov LI\ Gol|tsov VA[ Dynamics of the concentration inhomogeneities of hydrogen in metals[ The Physics of Met! als and Metallography 0887^74"1#]08Ð12[ Wagner H[ Elastic interaction and phase transitions in coherent metal!hydrogen alloys[ In] Alefeld G\ Volkl J\ editors[ Hydrogen in Metals[ Berlin] Springer\ 0867^0"1#[ Volkl J\ Alefeld G[ Di}usions of hydrogen in metals[ In] Alefeld G\ Volkl J\ editors[ Hydrogen in metals[ Berlin] Springer\ 0867^0"01#]210Ð37[ Smirnov LI[ E}ects of concentration stresses on growth kinetics of hydride phase in open palladium!hydrogen system[ Fizika Metallov i Metallovedenie\ 0885^71"2#]64Ð 71[ Gol|tsov VA\ Glukhova ZhL\ Redko AL[ Hydrogen elas! ticity e}ect and its importance in di}usion dissolution on concentration inhomogeneities in metals[ International Journal of Hydrogen Energy 0886^11"1:2#]068Ð72[