Adsorption and Desorption Kinetics for Diffusion Controlled Systems with a Strongly Concentration Dependent Diffusivity

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1 The Open-Access Jounal fo the Basic Pinciples of Diffusion Theoy, Expeient and Application Adsoption and Desoption Kinetics fo Diffusion Contolled Systes with a Stongly Concentation Dependent Diffusivity Douglas M. Ruthven Depatent of Cheical Engineeing Univesity of Maine Oono, ME, 4469 USA Abstact The foal solution to the diffusion equation fo tansient adsoption in a seiinfinite ediu with a stongly concentation dependent diffusivity, oiginally deived by Fujita ( 3 ), has been extended to desoption. Pofiles fo adsoption and desoption ae copaed and siplified asyptotic expessions which ae useful when the diffusivity atio is lage ae deived. The effect of a stongly concentation dependent diffusivity on adsoption and desoption kinetics is biefly consideed.. Intoduction In contast to gas and liquid phase systes in which the vaiation of diffusivity with coposition is geneally odest, fo adsobed phases and gases dissolved in polyes the diffusivity ay vay vey stongly with concentation. Fo soption systes in which the soption ate is contolled by intapaticle diffusion this leads to a qualitatively diffeent patten of behavio copaed with a syste in which the diffusivity is constant o only odestly concentation dependent. In paticula, such behavio occus when the equilibiu isothe is highly favoable, appoaching the ectangula o ievesible fo. Since such systes ae quite coon and of soe pactical ipotance it is appopiate to conside thei behavio in soe detail. As an exaple, we conside a syste in which the equilibiu isothe is of Langui fo: q bp q = + b () p s and diffusive tanspot is diven by the gadient of cheical potential with a constant (concentation independent) obility. Fo such a syste the Fickian diffusivity is given by () : dlnp Do D = Do = lnq λq T () 7, D. Ruthven Diffusion Fundaentals 6 (7)

2 whee λ = q o /q s and Q = q/q o. The diffusivity atio (ove the concentation ange < Q < ) is given by = ( λ) -. If the equilibiu isothe is highly favoable (b ) can be vey lage.. Soption Kinetics Fo an isotheal paallel sided slab (thickness l) subjected at tie zeo to a step change in the suface concentation of sobate, the adsoption/desoption kinetics (assuing intenal diffusion contol) ae govened by the patial diffeential equation: Q f ( Q). Q = τ X X (3) whee f(q) = /(-λq) = D/D o, X = x/l *, conditions: and τ = D o t/l, with the initial and bounday τ <, Q = (ads.) o Q = (des.) fo all x τ, Q(, τ) = (ads.) o Q(, τ) = (des.) fo all τ (4) q x Q X x= = X= = Since Eq.3 is non-linea and the bounday conditions ae finite, an analytic solution is not pactical. Nueical solutions fo the coesponding poble in spheical coodinates wee pesented any yeas ago by Gag and Ruthven (). In the initial egion the concentation font has not penetated to the cente of the slab so the syste behaves as a sei-infinite ediu. In this situation the last bounday condition of Eq.4 ay be eplaced by: q x x = ; Q = ( ads. ) o. ( des. ) (5) x This allows the Boltzann tansfoation ( y x/ Dot) to the odinay diffeential equation: = to be used to educe Eq.3 d y = f ( Q) (6) * Note that distance x is easued fo the extenal suface at which x =.

3 with the initial bounday conditions: t < : Q(y) = (ads.) o. (des.) t : Q() =. (ads.) o (des.) fo all t (7) Q(y ) = (ads.) o. (des.) = y Integation of Eq.6 yields: yq d = f( Q) f ( Q). (8) y= y= y Since the gadient of concentation is zeo fo lage y this siplifies to: A= = = y Q= y= y= ( ads. ) ( ) des. (9) whee A is the aea unde the Q vs y pofile coesponding to the aount of ateial adsobed o desobed. Eq.9 povides a convenient check on the validity of nueical (o analytic) appoxiations to the solution fo the concentation pofile. 3. Adsoption Pofiles A foal analytic solution to this poble (fo adsoption with f(q) = (-λq) - ) has been given, in paaetic fo, by Fujita (3) (suaized by Cank (4) ): Q = e e (fo lage) y = f ( φ) φ e μ I( φ ) I( φ ) I( φ ) () whee: q dφ I ( φ) = ; f( Q) = φ μln φ; = () f λ o ( Q) The paaete μ is defined by the elationship: 3

4 When μ is sall f ( φ) dφ I ( ) = = ln () f ( Q) μlnφ φ and I( φ) I( ) + lnφ = ln( φ ) (3) φ With these appoxiations the liiting expession fo the concentation pofile (fo lage) ay be obtained in explicit fo: Q exp. μ y = (4) The liiting slope is given by: y= = (5) μ This appoxiation is valid only fo the egion whee Q. but the ange of validity inceases when is lage..e+8 Vaiation of with μ.e+7.e+6.e+5.e+4.e+3.e+.e+.e+.e-8.e-6.e-4.e-.e+.e+ μ Fig. Vaiation of diffusivity atio with paaete μ calculated fo Eq.. Anothe liiting explicit appoxiation which is valid as Q ay be obtained by noting that, fo φ 4

5 whee u φ φ π efc ( u / ) (6) μ d I( φ ) μ ln = lnφ. We thus obtain ( φ ) y π π efc( u); Q efc( ) μ μ y (7) as the asyptotic expessions fo the liit φ, u. The vaiation of the paaete μ with calculated by nueical integation of Eq., is shown in figue. Concentation pofiles fo vaious values of, calculated fo Eq. ae shown in figue. Also shown ae the liiting pofiles calculated fo the asyptotic expessions (Eqs.4 and 7) fo selected values of. When is lage Eq.4 povides a good appoxiation ove ost of the ange but Eq.7 is valid only at vey low values of Q. Since y = x/ ( Dot) = X / τ one ay plot diectly the pofiles of Q vs X fo a finite slab (fo ties less than the tie at which the penetating wave eaches the cente of the slab). Such a plot is shown in figue 3 fo which the fo of the penetating wave is appaent. Note that, in contast to the shockwave type of behavio obseved fo a acopoe diffusion contolled syste with ievesible adsoption (5) the concentation wave in the pesent syste dispeses as it penetates. Concentation Pofiles (Adsoption) Q =.5 = E+4.5E Fig y Concentation pofiles fo adsoption showing how the fo of the concentation wave changes with. Note that when is lage the asyptotic expession (Eq.4) povides a good appoxiation except when Q is sall. 5

6 Adsoption Pofiles (Finite Slab).9.8 = 3.5 E4.7.6 Q τ =. τ =.3 τ =. τ = X Fig. 3 Pofiles of figue eplotted in the coodinates Q vs X (fo a finite slab) fo vaious values of τ. Note the inceasing spead of the wave fo as τ inceases. 4. Desoption Pofiles Fujita pesented only the solution fo adsoption but by following his pocedue the coesponding desoption poble ay be solved to yield, fo the tansient pofile: { ( φ ) ( ) } Q = exp I I y = f ( φ) φ exp I( ) I( φ) μ (8) whee the sybols and integals have the sae eanings as fo the adsoption case. It is shown in the appendix that, when, Eqs and 8 both educe to the well known eo function fos fo a constant diffusivity syste. Following a pocedue siila to that used to deive Eq.4 the following explicit appoxiation (valid only fo sall C and lage ) can be obtained: y μ ln 8 ( Q) ( Q) (9) Concentation pofiles calculated fo Eqs.8 ae shown in figue 4. As is to be expected and in contast to a linea syste, the pofiles fo desoption and adsoption ae 6

7 quite diffeent. The desoption pofiles show no inflexion. The initial slope inceases only slightly with inceasing but the cuvatue inceases so that, when is lage, the gadient at soe distance fo the suface is sall. Physically this fo is to be expected since, when is lage, the desoption ate is contolled by diffusion though the suface egion in which D D o. Pofiles fo Desoption = = =.8 =.6 Q y Fig. 4 Pofiles fo desoption fo a sei-infinite ediu fo selected values of. 5. Uptake Rates The adsoption ate is given by: d dq Dq = D x= =. y= () dt dx Dt and, on integation: t = Dt q y= () When the diffusivity atio is lage (λ., ) the concentation pofile assues the fo of a penetating wave (see figue ) so the sei-infinite ediu appoxiation will be valid thoughout ost of the uptake. In this egie the factional appoach to equilibiu ( = q o l) will be given by : 7

8 t Dt = = y τ. y= y= d d y () This expession shows that the stongly concentation dependent. t law is still valid even when the diffusivity is Pediction of the uptake ate depends on estiating the concentation gadient at the suface ( d Q/d y) y =. When is lage we ay use Eq. 5 which yields: τ (3) μ t By copaison with a linea syste fo which the appoach to equilibiu is given by: D t = τ π π t (4) we see that the effective diffusivity (D e ) fo adsoption is given by: D D e π = (5) μ The poduct μ deceases with inceasing so the effective diffusivity will incease continuously (athe than appoaching an asyptotic liit) as the isothe appoaches the ectangula fo. Fo desoption the sei-infinite ediu appoxiation is valid only at shot ties since, as ay be seen fo figue 4, when is lage the concentation fa fo the suface deceases quite apidly. The expession coesponding to Eq. is: t = τ y= (6) but this expession is of liited utility since it applies only in the liit of vey shot ties. A oe useful appoxiation fo a finite slab ay be obtained by ecognizing that, when is lage, ost of the desobing ateial coes fo the inteio egion whee the concentation pofile is quite flat. As a siple odel we theefoe appoxiate the syste as a sei-infinite ediu in which the concentation (q o ) is essentially unifo (except in the suface egion) but deceases with tie: 8

9 d q q D q D d dq = x= =. dt dx t y = (7) dq We assue that y= = κ eains appoxiately constant (tue fo lage ). With this appoxiation Eq.7 can be easily integated to yield: q ln κ τ q (8) Since q q this yields fo the desoption cuve: / t / t = e κ τ (9) This expession explains why, fo stongly adsobed species, desoption is vey slow and plots of ln / vs t which ae linea fo a constant diffusivity syste, ( ) coonly show a deceasing slope t (). 4 Vaiation of Soption Rates with. κ (des.) o -k (ads.) Adsoption Desoption Fig. 5 Vaiation of initial adsoption and desoption ate (sei-infinite ediu) with calculated fo the liiting slopes (κ) of the Q y pofiles accoding to Eq. (ads.) o Eq.6 (des.). 9

10 At shot ties Eq.9 educes to Eq. 6 (which is of the sae fo as Eq. fo adsoption) so the initial ates of adsoption and desoption fo a sei-infinite ediu ay be copaed siply on the basis of the liiting slopes and the diffusivity atio. Such a copaison is shown in figue 5. Fo both adsoption and desoption the ate inceases with but the incease is geate fo adsoption leading to an inceasing diffeence between initial adsoption and desoption ates with inceasing non-lineaity (6). 6. Conclusion Although the equations descibing tansient adsoption and desoption with a concentation dependent diffusivity can be solved nueically this task is not tivial when is lage and the equations ae stongly non-linea. The analytic solutions oiginally obtained by Fujita and extended hee to desoption yield geate undestanding and povide a basis fo siple asyptotic appoxiations which ae pactically useful fo odelling adsoption/desoption kinetics. Notation A defined by Eq.8 b Langui constant (Eq.) D Fickian diffusivity D o Liiting o coected diffusivity (Eq. ) D e effective diffusivity I ( φ ),I( ) Integals defined by Eqs. and l half-thickness (of paallel sided slab) t ass of sobate adsobed o desobed duing tie t value of t as t (i.e. at equilibiu) p sobate patial pessue q adsobed phase concentation q o value of q at equilibiu with extenal gas phase q s satuation liit (Eq.) Q diensionless concentation q/q s diffusivity atio D(Q)/D t tie u lnφ x distance (easued fo extenal suface) X diensionless distance x/l y diensionless distance x/ Dt κ liiting slope (/) y= fo desoption (Eq.8) μ constant defined by Eq. λ q o /q s τ diensionless tie D o t/l φ (Q) paaete defined by Eqs. and.

11 Refeences. D.M. Ruthven, Pinciples of Adsoption and Adsoption Pocesses pp. 5, 7. John Wiley, New Yok, D.R. Gag and D.M. Ruthven, Che.Eng.Sci. 7, (97) 3. H. Fujita, Textile Res. J., 757 and 83 (95) 4. J. Cank, Matheatics of Diffusion p. 67. Oxfod Univesity Pess, London, (956) 5. N.-K. Bä, B. Balco and D.M. Ruthven, I. and EC Reseach 4, 3-39 () Appendix: Reduction to Linea Syste fo. The linea liit (.) coesponds to μ (as ay be seen fo figue ). When μ is lage we have: f φ μ ln = μ. u whee u = -lnθ ( ) u φ dφ u π I( φ ) = = e du efc( u) φ μ = μ u π π μ I( ) = = ln, = exp, μ μ π π Both I( φ ) and I() ae sall so exp [- I( φ ) ] - I( φ ) = efc( u) μ Hence we have: Adsoption: I( φ ) Q = e efc( u) y I( φ) I( φ) f( φ) φ e ue μ u Q = efc( y) Desoption: Q = exp ( I( φ ) I( ) ) ef u y = π f ( φ) φ exp I( ) I( φ) uexp ef μ u u μ Q = ef(y) These ae the well known solutions fo a constant diffusivity syste. Acknowledgeent Financial suppot fo the National Science Foundation (Gant CTS-55386) is gatefully acknowledged.