Sorption and Permeation in Linear Laminated Media
TABLE IV: Calculated Changes of A t o m i c Charges (e) uuon Crvstal F o r m a t i o n a
a-glycine
NH3+H, H 2 H3
CH, H.) H 5
Cl
c2
N 0, 0 2
0.1024 0.1091 0.1053 0.0022 0.0031 0.0027 -0.0214 0.0098 -0,1543 -0.1595
0-glycine
y-glycine
0.1078
0.1059 0.1038 0.1028 0.0017 0.0026 0.0036 -0.0247 0.0069 -0.1507 -0.1519
0.1005 0.1032 0.0020 0.0029 0.0034 -0.0233 0.0078 -0.1511
-0.1531
a A positive value corresponds to a decrease in electron density .
to include the closed-shell overlap-dependent repulsion energy. For all three forms of glycine crystals, we observed relatively large inter- and intramolecular energy contributions. This has been observed for all hydrogen-bonded molecular crystals studied to date,11-16 and seems to be a characteristic feature of hydrogen-bonded crystals. In Table IV the calculated changes of the atomic charge upon crystal formation for all three forms of glycine are presented. The increase in polarity for the atoms participate directly in the hydrogen bond, e.g., oxygen atoms
The Journal of Physical Chemistry, Vol. 83, No. 27, 1979 2787
of the COO- groups and hydrogen atoms of the NH3+ groups have been observed. For the a-glycine crystal greater changes in atomic charges are observed than for the 0 and y forms.
Acknowledgment. The authors thank the Polish Academy of Sciences for support (MR-1-9).
References and Notes (1) K. Biemann, J. Seibl, and F. Gapp, J . Am. Chem. Soc., 83, 3795 (196 1). 12) G. Junk and H. Svec. J . Am. Chem. Soc.. 85, 839 (1963). (3) Y. Grenie, J.-C. Lassegues, and C. Garrigou-Lagrange, J . Chem. Phys., 53, 1827 (1972). (4) P.-G. Jonsson and A. Kvick, Acta Crysfallogr., Sect. 8 , 28, 1827 (1972). (5) R. E. Marsh, Acta Crysfallogr., 11, 654 (1958). (6) Y. Iitaka, Acta Crystallogr., 13, 35 (1960). (7) Y. Iitaka, Acta Crystallogr., 14, 1 (1961). (8) J. Bacon and D. P. Santry, J . Chem. Phys., 55, 3743 (1971). (9) J. Bacon and D. P. Santry, J. Chem. Phys., 56, 2011 (1972). (10) S.F. O'Shea and D. P. Santry, Theor. Chim. Acta, 37, 1 (1975). (11) R. W. Crowe and D. P. Santry, Chem. Phys. Lett., 45, 44 (1977). (12) D. P. Santry, J. Am. Chem. SOC.,94, 8311 (1972). (13) R. W. Crowe and D. P. Santry, Chem. Phys., 2, 304 (1973). (14) D. P. Santry, Chem. Phys. Lett., 27, 464 (1974). (15) K. Middlemiss and D. P. Santry, Chem. Phys., 1, 128 (1973). (16) D. P. Santry, Chem. Phys. Lett., 52, 500 (1977). (17) F.A. Momany, L. M. Carruthers, and H. A. Scheraga, J. Phys. Chem., 78, 1621 (1974). (18) J. L. Derissen, P. H. Smit, and J. Voogd, J. Phys. Chem., 81, 1474 (1977).
Time Moment Analysis of Sorption and Permeation in Linear Laminated Media H. L. Frisch," G. Forgacs,+ and S. T. C h d Department of Chemistry and Center for Biological Macromolecules, and Department of Physics, State University of New York at Albany, Albany, New York 12222 (Received January 22, 1979) Publicatlon costs assisted by the National Science Foundation and the US.Army Research Office
The time moments of the amount of penetrant (per unit area) in a desorbing membrane or the difference between the instantaneous and asymptotic flow (per unit area) up to a given time through a membrane in a permeation cell can be obtained by numerical integration of experimental data. We show that for membranes composed of a linear laminated medium these time moments can be exactly, recursively calculated without solving the diffusion equation. We then obtain novel, exact functional constraints on the local permeability and distribution coefficients which characterize such a linear laminated medium. We can relate the precursor functions of these time moments to the asymptotic long time behavior of penetrant sorption and permeation.
1. Introduction Considerable interest exists in sorption and permeation studies of dilute penetrants in films of simple inhomogeneous materials which form a linear laminated medium.1-8 Such a medium is characterized by a partition coefficient, k, diffusion coefficient, D , and local permeability coefficient, P, whose spatial dependence arises solely from a dependence on the distance in the direction of penetrant flow, x k = k(x) D = D ( x ) P = Dk = P(x) (1.1) The penetrant activity must be sufficiently low so that both k (Henry's law) and D are independent of the penetrant activity. At ordinary pressures this restricts one Central Research Institute for Physics, 1525 Budapest, 114 P.O. Box 49, Hungary. t Departmentof Physics, State University of New York at Albany.
0022-365417912083-2787$0 1.0010
to gases or vapors whose critical temperatures are considerably lower than the ambient temperature of the transport measurement. Examples of such linear inhomogeneous media are (1)composite membranes which are laminates of films of different composition,4 (2) microporous powder compacts (e.g., graphite powder compacts),G8and (3) crystalline polymer films with effectively linear gradients in crystallinity or orientation of the spherulites as in the case of surface-nucleated transcrystalline films,g etc. The aim of the sorption and permeation studies in such films is t~ obtain some information concerning k , D, or P which characterize the medium and its diffusion properties; one is thus dealing with an inverse problem in diffusion theory. In this paper we obtain exact constraints, as functionals of k and P, on k and P from time moments of entities which can be expressed in the routinely measured weight change of the penetrant in sorption balance experiments3 and the penetrant flow up to time 0 1979 American
Chemical Society
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H. L. Frisch, G. Forgacs, and S. T. Chui
t in permeation cell measurement^.^ These constraints can be used to check the validity of the assumed functional forms for h and P as well as to evaluate the parameters which appear in these forms. Such information is ultimately useful in assessing a given linear medium as a potential barrier material or as a separation membrane. In the next section we define the time moments and relate them to the measured weight changes and flows. In section 3 we find the analytical form for these moments and provide some examples of their use. In the concluding section we provide some relationships between the precursors of these moments and the long time asymptotic representation of the desorption flux and permeation flow. Wherever possible we follow the notation introduced in ref 1. 2. Definition of the Time Moments and Their Relation to Experimental Measurements Diffusive dispersal of a penetrant in an inhomogeneous medium is conveniently described by the penetrant activity as a function of x and t. The activity is the local penetrant concentration c ( x , t ) divided by h ( x ) . In the case of the usual desorption experiment, we denote1 the penetrant activity by A(x,t). For a sufficiently large unit area of a film of thickness 1, the one-dimensional diffusion of the desorbing penetrant activity satisfies the boundary value problem
in the interval 0
< x < 1.
A(0,t) = 0 A(1,t) = 0 A(x,O) = (2.2) with a. the constant initial activity. The activity of the penetrant for the corresponding sorption problem, As(s,t), satisfies the same linear partial differential equation (2.1) with As(O,t) = a0 A,(l,t) = a0 As(x,O)= 0 (2.3) instead of (2.2). The two activities are simply related A,(x,t) = - A(x,t) (2.4) as can be verified by direct substitution. Permeation through a sufficiently large unit area of film of thickness 1 is described by the penetrant activity a(x,t) which satisfies
Figure 1. The function $ ( t ) . Note that area (0,-J&l)L, A') has the opposite sign of area (A, L, m). The line LA is obtained from line LA' by reflection in the taxis.
finite time. For constant k = ho,P = Po, Do = Po/ko, one finds that the truncation for a time t 1 l2/D0produces less than 1% error in the first few moments. Random experimental error can be considerably amplified in the higher moments hence judgement should be employed in using these relationships. In a permeation experiment (6.(2.5)) one measures Q(t), the total flow per unit film area up to time t. Figure 1 shows schematically the behavior of Q(t)
After an initial transient response the flux per unit film area at x: = 1, J(l,t), approaches its steady state value
J&1) = a,/[ J'dr/P(r)]
I%).([-
(2.10)
=
x=1
Equation 2.10 for J,(l) is obtained from the steady-state permeation activity a,(x) (aa,/at = 0) by quadrature of the steady-state version of (2.5) which yields3
(2.5) a(0,t) = a0
a(1,t) = 0
The time intercept, L, of the asymptotic line of slope J,(O to Q(t),the so-called permeation time lag, is given by3-6
a(x,O) = 0
In desorption (cf. (2.1) and (2.2)) one measures the amount of penetrant per unit area remaining in the film up to time t , W ( t ) where ,
W ( t )= Jik(x)A(x,t) 0 dx
(2.6)
and in the sorption experiment (cf. (2.3) and (2.4)) one measures the amount of penetrant per unit area taken up by the film up to time t , Ws(t),where W s ( t )= W(0)- W ( t ) (2.7) with W(0)= aoSJh(x)dx. By numerically integrating the experimental W ( t )curve, one can obtain estimates of the time moments
W,, = 1 - t " W ( t )dt 0
n = 0, 1, 2,
(2.8)
...
Experimental data for W ( t )is always truncated a t some
Since J,(1)and L are obtainable from the experimentally measured Q ( t )one can form (2.13) $(t) = Q(t)- Qasymptotic(t1 = Q ( t )- Js(Z)(t- L )
$(t)is the cross-hatched area in Figure 1. The permeation time moments are $n
= J m t n $ ( t )dt
n = 0, 1, 2, ...
(2.14)
Sorption and Permeation in Linear Laminated Media
The Journal of Physical Chemistry, Vol. 83, No. 21, 1979 2789
Again, for constant k = ko, P = Po, Do = Po/ko, the error in truncating the infinite integral in (2.14) a t a time t 2 k 2 / D o produces negligible errors in the first few time moments. In certain cases 4, can be negative.
3. Calculation of Wo, W1, go, and the Recurrence Relation for the Higher Moments We focus on certain natural precursors of W , and 4,. These will actually be the functions of x which can be recursively computed. Consider the usual solution of the boundary value problem (2.1) and (2.2) by the method of separation of variables. Providing P(x) is continuously differentiable and P(x) and k(x) do not vanish in the interval 0 5 x I 1, which are the physically plausible cases (no phase separation or uphill diffusion), one can write the solution as an infinite series
The left-hand sides of (3.6) and (3.8) follow on noting that
and
The solution by direct quadrature of the ordinary differential equation
-[ P(x)G] d dx
m
(3.1)
dB
= -f(x)
B(0) = B(2) = 0
(3.10)
is
where & ( x ) and A, are the eigenfunctions and eigenvalues of the regular Sturm-Liouville problem'O
in 0
< x < 1 with &(O) = #,(I)
and 0
< A0 < A, < A2 ...
(3.3)
=0
(A,
- 00,
n
m)
(3.4)
The coefficients c, are the expansion coefficients of the initial value A(x,O) = a. in the orthonormal functions 4, with respect to the weighting function h(x). Unfortunately, 6, and A, are not exactly known for arbitrary h(x) and P(x) satisfying the above conditions. For our immediate purposes though, we note that by virtue of (3.1) and (3.4) A(x,t) vanishes asymptotically exponentially fast. Returning to (2.6) and (2.8), we can write
W, = j ' 0k ( x ) B , ( x ) dx n = 0, 1, 2,
Setting B(x) = Bo(x)with f ( x ) = h(x)ao,and B ( x ) = B,(x) with f ( x ) = n k ( ~ ) B , - ~ (one x ) , obtains
and for n 1 1
(3.5)
...
and where the precursor functions of the W, satisfy
where we have used (3.1) and tacitly assume that the sum in (3.5a) converges. To obtain the ordinary differential equations satisfied by BO(x)and B,(x),, n 2 1,we multiply (2.1) by 1and t , respectively, and integrate over t from 0 to m. This yields, using (2.2)
(3.13) respectively. By substituting (3.12) into (3.13), one finds B,(x) and substituting this result again into (3.13) one obtains B2(x),etc. By integration by parts, one can write the double integrals in (3.12) and (3.13) as single integrals, viz.
subject to (3.14)
and for n 2 1
[ dRd;(r)]
-nh(x)B,-,(x) = - P(x)dx
(3.8)
subject to with
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H. L. Frisch, G. Forgacs, and S.T. Chui
A(0,t) = A(l,t) = 0 A(x,O) = a&) Subject to our previous assumptions on k and P the solution of (3.21) can be written as
Thus
m
A(x,t) = C E n 4 , ( x ) e - A n t
(3.22)
n=O
where &(x) and A, satisfy (3.2)-(3.4), and the Zn are the expansion coefficients of a&) in the orthonormal functions &(x) with weighting function h(x). Thus, A(x,t)vanishes asymptotically exponentially fast. Noting that by virtue of (3.20), (2.9), and (2.10)
we find on integrating (3.21) over x from 1 to x
Dividing both sides of this expression by P(x) and integrating over x from 0 to 1 and using (3.21), we obtain etc. In general B,(x) will be made up of (3)n+1 terms consisting of ( n 1)-fold (iterated) integrals of h, K , or
+
P.
Using (3.14) and (3.17) in (3.5), we obtain explicitly the exact form of the first two time moments: I'tegrating (3.23) over t from 0 to t and using the fact that A(z,O) = a&), one finds that (9.;), (2.10), and (2.12) yield the desired relation between A(x,t) and $ ( t ) defined by (2.13): (3.18)
with $40) = J,(l)L. We can now rewrite (2.14) in terms of the permeation time moment precursor functions B,(x) as follows
n = 0, 1, 2,
,
. . with (cf. (3.22))
(3.19)
Equations 3.5,3.14, and 3.15 taken together constitute the desired recurrence relations for the desorption time moments. To obtain the corresponding permeation time moments we consider a desorption problem associated with permeation. We represent the solution a(x,t) of (2.5) as a ( x , t ) = a&) - A(x,t) (3.20) where a,(x)-is the steady-state solution of (2.5) given by (2.11) and A ( x , t )satisfies the desorption boundary value problem (3.21)
Multiplying (3.21) again by 1and tn,n I1, and integrating over t from 0 to a,we find, as bsfore, theprdinary differential equations satisfied by B&) and B , ( x ) , viz.
for n I 1, subject to &(O) = Bn(2)= 0 for all n. (3.28) is identical with (3.8) and (3.27) differs from (3.6) by the replacement of a. by a&) = ao[l- ~ ( x ; l )(cf. ] (2.11)). The
The Journal of Physical Chemistry, Vol. 83, No. 21, 1979 2791
Sorption and Permeation in Linear Laminated Media
solutions of (3.27) and (3.28) can be written using (3.10) and (3.11), as
We note in case I that for k(x) = ko/b(x),P(x) = Pob(x), and Do = Po/ko,we can rewrite (2.1) and (2.2) as
aA -- D oa2A _ 2
at A(0,t) = A(2,t) = 0
OCZCZ A(2,O)
(3.32)
= a,
on making the substitutions:
A(x,t) = A(z,t) (3.29) and
This boundary value problem is identical with the homogeneous medium problem described by the constant diffusion coefficient, Do, and constant partition coefficient, ko, if z is replaced by x and 2 by 1. The Fourier series solution
A(z,t) =
Equation 3.30 is just (3.15) with B,'s replacing B,'s. In particular by substituting (3.29) into (3.30) one obtains
4ao
+ (2n + 1 ) r z
m
-C (2n
r
n=O
is not very useful for calculations because infinite series have to be summed. We employ instead the Laplace transform
A(z,s) = l m e - 8 t A ( z , td) t
"[ S
=
sinh (42) - sinh ( 4 2 ) - sinh [q(2- z)] sinh ( 4 2 )
(3.35)
with q = (S/D~)~/~
Noting that if the Laplace transform of A(x,t),A(x,s), possesses the indicated derivatives then, by virtue of (3.5a), we obtain
Bn(x)= [(-l)na"A(z,s)/asn],=o
(3.36)
In our case from (3.35) &(z) = S m A ( z , t )dt =
Substitution of (3.29) and (3.31) in (3.25) yields q0and $,, respectively, which we do not exhibit explicitly since there is no simplification of the formulas. Thus (3.29), (3.30), and (3.25) constitute the desired recurrence relations for the $,'sa These relationships can be directly tested in the few instances of some physical interest in which solutions for the activities of (2.1)-(2.5), or rather their Laplace transforms, are available. These correspond to the following choices:ll (I) h ( x ) = ho,P(x) = Po,Polko= Do or more generally k ( x ) = h o / b ( x ) ,P ( x ) = Pob(x)where b ( x ) is a dimensionless once differentiable function of x. (11) k ( x ) = ho or ko(l + a x ) , P ( x ) = Po(l a x ) . (111) h ( x ) = ho(x/Z)", P(x) = Po(x/l)m. Cases I1 and I11 are discussed in ref 11,where it is shown that the Laplace transforms of the activities can be written as linear combinations of modified Bessel functions. Since case 11can be studied by the same technique which we will apply to case I, we will not study it further here. Case I11 is not physically interesting for diffusion. We will only show explicitly here tests of the desorption time moments.
a0 A(z,O) = -(zZ
-
2)
(3.37)
200
0
which is in accord with the direct evaluation of (3.12) or (3.14) for case I. Similarly
Bl(z) = JmtA(z,t)dt = -(aA(z,s)/as),=o= a0 ----(2z3
2400~
-
22z3 + 2 3 (3.38)
by using (3.35) which agrees with the direct evaluation of (3.17) for case I. Denoting by co = aokO,one finds from (3.37) and (3.38) (or (3.18) and (3.19)) that
+
wo = and
where L = P/6Do is the permeation time lag (2.12) for case
2792
The Journal of Physical Chemistry, Vol. 83, No. 21, 1979
H. L. Frlsch, G. Forgacs, and S.T. Chui
with
'OI
In W ( t ) In W) = lim t-m t (4.2) t is the lowest eigenvalue of (3.2)-(3.4). The constants wo and K~ satisfy -Ao = lim
~
t--
15
mztb ol
COL 1
B
wo =
5
I
2.5
2.0
3.0
k ( x ) @ o ( x )dx
I I
35
The fact that the asymptotic time dependence of the permeation and desorption (or sorption) problem is the same should provide a useful check on the consistency of experimental desorption and permeation data. If f ( x ) is any continuously differentiable function of x , satisfying f ( 0 ) = f(1) = 0, then
L'Pb)[df(x)ldx12 dx A0
(4.4)
I
L ' k ( x ) [ f b ) l zdx Flgure 2. Plot of the lower time moments W , and q0 for case 11, k = ko(l ax), P = P,(1 PI) as function of the reduced thickness p l , for several values of (alp).
+
+
I. These results are of some interest for the physically realized homogeneous version of case I, 12 = ko, P = Po, where Wo _ - - i~ =co
wi
- = -31L2 =-
13
12Do $0 - - 713 Co 36000 2
co
10
$1 -
Co
15
120D02
(3.41)
15
=336D02
P.
(generalized case I is obtained by replacing 1 by 2 of this text). Thus the permeation time lag can in these special cases be calculated from sorption data which could be used to test the consistency of sorption and permeation data. Equations 3.39 and 3.40 are not true in general (2 = S 2 [ k ( x ) / k 0 ]dx). This is shown easily by the counterexample P ( x ) = Po, k ( x ) = ko(l + a x ) for which L = ( 1 2 / 6D0)(l 1 / 2 4 and
+
w,
col(
&)(r 12
which is not equal to (2 = l(1
+ E! + 12
e) 45
+ 1/2al))
12
12
48
+
4. Relationship to Long Time Sorption and Permeation Behavior From the definitions (2.6) and (3.24) and the properties of the boundary value problem (3.2)-(3.4), we can expect that as t 00 In W ( t )5 In coo - Xot -+
K~
-
lot
We have shown in this paper that proper numerical treatment of experimental sorption and permeation data for a linear laminated medium can yield, new, exact constraints on k and P and we have attempted to relate these to other means of interpreting diffusion data.
Acknowledgment. H.L.F. was supported by the U.S. Army Office of Research, G.F. by the N S F CHE7682583A01, and T.C. by the NSF DMR7611338. Supplementary Material Available: Analytic expressions for case 11, k ( x ) = ko(l + a x ) , P ( x ) = Po(l + P x ) (2 pages). Ordering information is available on any current masthead page. References and Notes
Figure 2 shows the behavior of Woand $o for case 11, h = ko(l a x ) , P = Po(l+ px); the analytic expressions for this case are available as supplementary material (see paragraph at end of text regarding supplementary material).
In $ ( t ) 5 In
with equality if the trial function f ( x ) is directly proportional to @ o ( x ) . We remark here that the functions B,(x) (cf. (3.5a)) are potentially useful candidates as trial functions f ( x ) (cf. (3.9)). For example, for case I of the previous section, Xo = r2D0/l2(or r 2 D 0 / Z 2 and ) by using B(x) = (ao/2Do)(1x- x2) in (4.4), one obtains Xo I10Do/12, an error of +1.3%. Using Bl(x) = (ao/12Do)(x13- 21x3 + x4) in (4.4), one obtains Xo I3O6D/31l2,with an error of +0.01%. Unfortunately, we cannot give exact error bounds in the general case. Reliable estimates via (4.4) require strictly knowledge of the correct functional form of k and
(4.1)
(1) H. L. Frisch, J . Phys. Chem., 82, 1559 (1978), and papers cited therein. (2) H. L. Frisch, J . Membr. Sci., 3, 149 (1978). (3) J. Crank, "The Mathematics of Diffusion", 2nd ed.,Clarendon Press, Oxford, 1975. (4) R. Ash, R. M. Barrer, and J. H. Petropoulos, Jr., J. Appl. Phys., 14, 854 (1963): R. Ash, R. M. Barrer, and D. G. Palmer, ibid., 16, 873 (1965). ' (5) H. L. Frisch and S. Prager, J . Chem. Phys., 54, 1451 (1971); H. L. Frisch and J. Bdzil, ibid., 62, 4804 (1975). (6) R. M. Barrer and T. Gavirm Oric, Proc. R . SOC.London, Ser. A , 251, 353 (1959); 258, 267 (1960); R. Ash and R. M. Barrer, ibid., 304, 407 (1968); R. Ash, R. M. Barrer, J. W. Clint, R. J. Dolphin, and C . L. Murray, Phil. Trans., 275, 255 (1973). (7) P. P. Roussis and J. H. Petropoulos, J . Chem. Soc., Faraday Trans. 2, 72, 737 (1976); 73, 1025 (1977). (8) K. Tsimiilis and J. H. Petropoulos, J . Phys. Chem., 81, 2185 (1977). (9) R. K. Eby, J . Appl. Phys., 35, 2720 (1964). (10) E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations", McGraw-Hili, New York, 1955, Theorem 2.1, p 212. (11) H. S. Carslaw and J. C. Jaeger, "Conduction of Heat in Solids", 2nd ed., Oxford at the Ciarendon Press, 1959, p 412 ff.
