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Langmuir 1993, 9, 2676-2681
Theory of Sorption of Gases on Heterogeneous Solids-Polymeric Sorbents M. Pyda' and F. J. Lopez-Garzon Departamento de Quimica Inorganica, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain Received September 2, 1992. In Final Form: April 20, 199P A simple model of mono- and multilayer sorption of gases on a flexible linear chain of a polymer is investigated. The influence of degenerated conformational degrees of freedom of the polymer linkages on the adsorption isotherm GAB (Guggenheim, Anderson, de Boer) is studied. This description is an extension of the approaches presented in previous papers13 and involves changes in the internal structure of the polymer chain during the sorption process, e.g., different conformational states of linkages between monomers of the linear chain. To describe a conformation of the whole chain, the one-dimensional Ising model is used. The sorbed molecules of gas can be tightly bound to the sorption centers, or they are in an intermediate adsorption state. The moleculesinteract indirectly through the conformational subsystem of the polymer. This interaction leads to a cooperativeeffect in the mono- and multilayer sorption isotherm. The proposed isotherm equation for a heterogeneous solid (polymericsorbent) is a product of the isotherm equations for the adsorption of gases on a homogeneous solid surface and on the polymer's chain. In the case of a rigid chain of the polymer the proposed equation reduces to the equation for noncooperative heterogeneoussorption on two kinds of centers having different adsorptionenergies. The thermodynamics of the model is studied by the transfer matrix method. The approach presented can give, for example, a more realistic picture of the mechanism of water and alcohol vapor adsorption on starch and cellulose. A comparison of experimental results with theoretical calculations of the model is performed for the sorption isotherm of methanol on freeze-dried starch gel.
Introduction It is well known that the adsorption behavior of gases on polymers is quite different from that on a rigid solidstate surface; in particular, the sorption process on the former is more complicated than on the latter. Besides energetic and structural heterogeneity, swelling, plasticization, and dissolution effects- occur. Although many researcherss" have tried to describe the mechanism of polar gas sorption on polymeric sorbents, this problem is still recognized insufficiently. The majority of approaches to the sorption from a gaseous phase on the solid polymers are based upon the assumption that the sorbent is inert during the sorption process, and in the case of the polymeric sorbents (e.g., sorption of water or alcohol on starch or cellulose), it is not true. In recent years, it has become more and more acceptable that, in the case of polymeric sorbents, the sorption process is dependent on the conformational state of the macromolecules. This problem could be solved as soon as the problem of the conformational state of the polymer is known, but this, so far, is still insufficiently elaborated.'2J3
* To whom correspondenceshouldbe addressed at the Department of Chemistry, University of Tennessee, Knoxville, TN 37996-1501. 0 Abstract published in Advance ACS Abstracts, August 15,1993. (1) Pyda, M.; Kurzynski, M. Chem. Phys. 1982,67,7-16. (2) Pyda, M.; Jaroniec, M. Chem. Phys. Lett 1985,120, 416. (3) Pyda, M.; Jaroniec, M. Angew. Makromol. Chem. 1988,160,107. (4) Rudzinski, W.;Everett, D. M. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (5) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solida; Elsevier: Amsterdam, 1988. (6) Hatakeyama, T. Makromol. Chem. 1987,188,1875. (7) Zagrafi, G.;Kontny, M. J. Pharm. Res. 1986,3 (3), 187. (8) Van den Berg, C.; Kaper, F. S.;Weldring, J. A. G.; Wolters, I. J. Food Technol. 1975, 10, 589. (9) Van den Berg, C. Ph.D. Thesis, Agricultural University of Wageningen, Wageningen, 1981. (10) Kamiya, Y.; Naito, Y.; Hirose, T.; Mizoguchi, K. J. Polym. Sci., Part B Polym. Phys. 1990,28, 1308. (11) Kamiya, Y.; Mizoguchi, K.; Naito, Y.; Bourbon, D. J.Polym. Sei., Part B Polym. Phys. 1991,29, 225. (12) Kurzynski, M. J . Chem. Phys. 1990, 93 (9), 6793.
The sorption process of polar gases on a polymer in its condensed state, as was shown in previous studies,lJ4 is dependent on the conformational degrees of freedom of the subsystem of primary linkages between the monomer units,' and on the subsystem of the secondary bonds between macromolecules of the p01ymer.l~ In this paper we investigate the influence of conformational degrees of freedom of a flexible linear chain of a polymer on the adsorption of gas. Moreover, the solids axe heterogeneous due to the changes in the conformational degrees of freedom of the linkages of the segments of the polymer. In this sense, in this paper one of the possibilities to create a heterogeneity effect on a polymeric sorbent will be demonstrated. Most approaches to describe the sorption process on polymers are phenomenological, and very often the classical adsorption theories of BET7J6J6or GAB7J6J6 or their modifications are used. Despite formal good agreement of the above adsorption isotherm equations with experimental results, the disadvantage for some of them is just the assumption that the sorbent does not change its internal structure during the adsorption process. It is a simple matter to fit experimental data into an adsorption equation, if enough parameters are introduced, but it is more difficult to develop a relationship which has physical meaning in the description of the molecular mechanism. Especially, it would be desirable to know the realistic microscopicpicture of the interaction mechanism between polar moleculesand the internal structure of the polymeric sorbent. The aim of this paper, which is based upon previous studies of gas sorption on polymers,'q isto further study the influence of degenerated conformational degrees of freedom of a polymer on the sorption process. An intermediate state, similar to that in the adsorption GAB7 model, is taken into account in this approach. It will also (13) Ikegami, A. Biophys. Chem. 1977,6,117. (14) Pyda, M.; Kurzynski, M. Chem. Phys. 1983, 79, 219. (15) Wolf, W.; Spiess, W. E. L.; Jung, G. J. Food Eng. 1984, 3, 51. (16) Young, D. M.; Crowell, A. D. Physical Adsorption of Gases; Butterworths: London, 1988.
0743-746319312409-2676$04.00/0 0 1993 American Chemical Society
Sorption of Gases on Heterogeneous Solids be shown that the heterogeneous nature of the polymeric adsorbent can be a consequence of its changes of conformational degrees of freedom between monomer units of the polymer linkages during the sorption process. In the present paper only the simplemodel is considered in which a relationship between changes in the conformational energy and energy of adsorption is described. Our approach is a complete statistical thermodynamic treatment, but for a gas-polymer system we have assumed a local equilibrium in the stastical physics sense. The present approach can give a more realistic picture of the mechanism of water and alcohol vapor adsorption on starch and cellulose polymers. A comparison of experimental data with theoretical results of the model is performed for the sorption isotherm of methanol on freeze-dried starch gel.
Theory Let us consider a sorbate-sorbent system consisting of multimolecular adsorbed gas, as in the GAB model, on a flexible linear chain of a polymer. Similarly as in the previous papers14 we assumed that the conformational states of linkages between monomers of the chain are discrete according to the rotational isomeric model,17and that their linkages can occur in two states only. However, an extension of the previous approach is made, that each of the conformationalstates can be degeneratedin multiple ways. We have neglected the excluded volume effects.17 To describe a conformation of the whole chain of the polymer, we have used the simple one-dimensionalIsing model.lg Also it is assumed that each sorption center can interact with the degenerated conformationof the nearest linkage. Consequently,the sorbed molecules can interact indirectly through the degenerated conformational subsystem of the polymer. This interaction leads to a cooperative effect in the mono- and multimolecular sorption of a gas on a flexible linear chain of a polymer and to deviation from Langmuir and GAB models of sorption on a homogeneous solid surface. According to the above assumptions, there are N identical sorption sites for adsorption of the gas molecules on a flexible linear chain of the polymer. Each Nlinkage can be found in one of two conformational states, and each state can be degenerated. The ground state, degenerated r times, is characterized by 0 energy and the excited state, degenerated u times, by the energy B > 0. The value of B can be modified by A, depending on the states of neighboring linkages, and the parameter A describes the cooperative behavior of degenerated conformational degrees of freedom. The molecules can be sorbed in the first monolayer with a sorption energy of E < 0, in the second and higher layers with energy K1, where K1 S 0. K1 is the energy of the intermediate sorption state as in the GAB model, and s is the maximum number of molecules in the cluster. Each sorption energy can be modified by parameter C if the nearest linkage of the chain is at the excited state, and C describes the coupling between degenerated conformational and sorptional degrees of freedom. If the linkage of the chain and its nearest sorption site is considered as one site, then we can present the Hamiltonian for the multimolecular sorption system on a degenerated conformational linear chain of an N site following (17) Flory, P. J. Staetistical Mechanics of Chain Molecules; Wiley:
New York, 1969. (18) Huang, K. Stastistical Mechanics; Wiley: New York, 1963.
Langmuir, Vol. 9, No. 10, 1993 2677
H = Acmjmj+l + B 11 '
where K1= K - AK is the relationship between the energy of sorption at the intermediate state K1 and the condensation energy K of the sorbate. The quantity 6 is the Kronecker delta in eq 1. Each site of the model sorbatesorbent is described by the pair of numbers (mj, nj),where conformationalnumber mj = 0 corresponds to the ground state with 'Idegenerations, and mj = 1means the excited state with u degenerations of the jth linkage. The values rand u can be equal to 1,2,3, ...and 1,2,3, ...,respectively. Similarly, the sorption number nj = 0, 1,2, ...,s refers to an empty site and a site occupied by one, two, ..., and s molecules of the adsorbate with energy states of 0, E, E K1,...,E (s - 1)K1, respectively. Each site of the system can be in ('I + u)(s + 1) states. p in eq 1 is the chemical potential of the gas. We assume cyclic boundary conditions to obtain the thermodynamic limit as follows:'8
+
+
mN+j= mi; nN+j= nj
(2)
Results and Discussion The free energy F per site for a one-dimensionalsystem with Hamiltonian 1 and boundary conditions 2 can be calculated exactly in the thermodynamic limit, using the transfer matrix methodl8 from
F = -(ON)-'
In Tr exp(-@H) = -(@)-' In 0.5(U + (v2 - 4V1'2) (3)
where
U = 1+ ab'
- (kgx)' + k , k g ( l + ab'c) [ 11-kg ]
(1
+ cklkg[
(4a)
(4b)
The equilibrium constants kl, k2, a, b', and c are defined by the following relationships: a = exp(-@A);b' = I'b = r exp(-@B);c = exp(-@C) and
k, = exp(-@(E- Kl)); k2 = exp(-@(K- K,)) (4c) with @ = (kgn-l and r = u / r . Tr exp(-@H)in eq 3 means
the trace of a matrix. The sorbate activity x = exp(-@(K-p)),whichvariesfrom 0 to 1,correspondsto the relative pressure. The parameter I' describes degeneration of conformational linkages of the polymer chain and is equal to the ratio of the number of degenerated ground T , and excited, u, states. Then it is possible to find all the thermodynamic parameters, like the adsorption ratio, 8, and the ratio q of degenerated conformations of the excited states: 6 = ( n ) = - aF/dp; v = ( m ) = aF/aB
(5)
The above relationships lead to the following adsorption isotherm equation:
Pyda and Lopez-Garzon
2678 Langmuir, Vol. 9,No. 10, 1993
where
(7) Equation 6 of the adsorption isotherm combines two expressions. The first is identical with GAB’S isotherm equation for an inert homogeneous solid, and the second corresponds to the equation describing adsorption on a flexible linear chain of a polymer. In the absence of any coupling C = 0 (c = l),the sorption isotherm 6 is reduced to the GAB isotherm16and the coefficient vs given by eq 7 becomes an expression resulting from the one-dimensional degenerated Ising model, for r = 1 from the undegenerated one.l8 According to the above definition, in eqs 6 and 7 s describes the maximum number of adsorbed molecules in the cluster. If the number of adsorbed molecules, at intermediate states, is not limited by s a,eqs 6 and 7 give the expression
-
e, =
klk9 (1- kgx)[ I + ( k 1 - l I k 9 1
(c - 1)(1- kg) 11 + (k,c - 11k.91
I
(8) where
In the case of s = 1,from eqs 6 and 7 we obtain expressions for monomolecular adsorption on a single flexible linear chain of a polymer, which reads
where ql = 0.5
+ 0.5[kx1(ab’c - 1)+ ab’- 11/[[1+ ab’ +
kx1(l+ ab’c)12- 4(ab’ - b’)(l+ kx1)(l+ c ~ x , ) I ’ /(11) ~ and kxl = exp(-O(E - P ) ) (12) The isotherm equation (10) is a simple product of Langmuir’s isotherm equation and a function reflecting sorption on a polymer. It is interesting to note that in eqs 7,9,and 11describing the q coefficient, parameter b’ (which describes degeneration of the conformation of linkages between monomers) can be expressed in the following form: b’ = I’b = exp(-OB’)
where
(13)
E+2K,
Figure 1. Schematic diagram of the states of several sites in the model of multimolecular adsorption with intermediate states on a polymeric chain.
B’ = B
+ k,T
In = B + Ta
(13a)
It means that the conformational parameter B’ can be split into energetic (B) and entropic (a)terms. As it was assumed above the first term describes the difference of energy between the ground and excited states of the conformational linkage between monomers of the linear chain. The second,entropic part, reflects the degeneration of states of conformational effects. Generally, introducing a degeneration to the conformation, which is described by the one-dimensional Ising model, leads to readjustment of the sorption isotherm on the linear chain of the polymer, and it will have entropic character. Furthermore, introduction of the intermediate states into the adsorption degrees of freedom, instead of condensation states in the BET model, leadsto reevalution of the isotherm sorption in the energetic sense and gives the GAB equation. All the equations of the sorption isotherm presented in this paper depend on the sorbate activity, both directly and indirectly by the coefficientof excited conformational states q ( x ) . The plots of the sorption isotherm for several values of the model parameters for I’ = 1and k2 = 1were presented in a previous paper.l Of course, in the case of I’ # 1 and k2 # 1, the sigmoid shape of the sorption isotherm, eq 6, is originated by the cooperative effect and the second inflection point by the multimolecularsorption. In general, the presented eqs 6,8, and 10of the adsorption isotherm, for heterogeneous solid-polymeric sorbent,result from the isotherm equation of gases on a homogeneous solid (the first term), and the equation describingsorption on a polymer, which gives the contribution from the heterogeneity of the solid. Equation 6 is examined numerically. Figures 2 and 3 present the plots of sorption isotherms calculatedaccording to eq 6 for fixed PA, OB, OC,kl, and s and varying k2 and I’, respectively. Figure 2 shows the influence of the parameter k2 on the adsorption isotherm. For greater values of kz (k2 I 1)sorption is greater until kz = 1,where it corresponds to the case when the intermediate energy is equal to the condensation energy as in a previous paper.’ The parameter k2 reflects the difference between BET and GAB multimolecular adsorption. Figure 3 displays the influence of the parameter I’ on the adsorption isotherm. For greater values of I’sorption is lower. The parameter I’reflects degenerationeffectsof conformational linkages of the polymeric chain. In the case of the rigid chain (that is, instead of parameters A and B, q should be considered as the fixed one), and multimolecular adsorption, eq 8 becomes the noncooperative GAB equation of the adsorption isotherm
Sorption of Gapes on Heterogeneous Solids
Langmuir, VoE. 9, No. 10, 1993 2679
5
3
14-50;
$4C1;
E A d : EB.1:
k,-1
EC.5
/
4
3
0c.-2
2
0,
0s 2
1
4
0
c
0.2
0.6
0.4
0.8
1 .o
X
X
Figure2. Multimolecularsorptionratio4 vrsus sorbate activity x for fixed ,!?A,flB, ,!?C, kl, and s and varying kz.
Figure 4. Multimolecularsorption ratio 8, versus sorbate activity x for a rigid chain for fixed kl, k2, and q- and varying BC.
sorptibn on two kinds of adsorption centers with energies E andE + C, and concentrations1- 91and,1' respectively:'
.t
r.o.2
I
X
Figure 3. AB in Figure 2 for fixed kz and varying r.
on two kinds of sorption centers with two concentrations (1- q.. and 7-1 in the formula
k,k,cx (14) (1 - k . p ) [ l + ( k , - l)k,cx] For the monomolecular sorption,with the same assumption as above, eq 10is reduced to the noncooperative Langmuir m'
As it can be seen from eqs 14 and 15, conformational changes of the polymer cause an energetic heterogeneity effect. In this sense a relationship between the heterogeneous nature of the solid-polymeric sorbent and the energy of adsorption is described." Equations 14 and 15 refer to adsorptionon heterogeneouspolymeric ahorbents for a rigid chain and are examined also numerically for several values of the parameters in Figures 4-7. Finally, we would like to illustrate the above theoretical model by a comparison to some experimental data. We have selected experimental data of methanol vapor adsorption on freeze-driedstarch gel which were presented in ref 19. It appears that this adsorption process can be described by eq 10for monomolecular sorption on a linear chain of the polymer. The values of parameters of the model, A, B, C, and k , are obtained from the fitting of the experimental data of the adsorption isotherm to eq 10 with r = 1. These results are shown in Table I. The values of the parameters were determined by an optimization computer program usingthe experimental data and theoretical equation for the confidence level equal to 93% in a range of sorbate activity XI = pIpo of 0-0.95; see Figure 8. Generally thus obtained values of the parameters can be considered as those having physical meaning. In order to make more clear the physical significance of the fitting parameters, we compare the values of some parameters of our model with the results from the literature. According to the physical picture presented in a previous paper,' to understand the mechanism of the equilibrium sorption of polar molecules on the linear macromolecules (19)Klimek, D.;Poliazko, St.; F'ydn, M.Matariala of the VI Internntionnl Colloquium of Chemistry and Technology of Starch, KrnkowKarnowice, June, 443,1990.
Pyda and Lopez-Garzon
2680 Langmuir, Vol. 9, No. 10, 1993 1.o
I
II.-
0
''
k p 5 0 k,-0.1: oc--2 0.8
7-=1/
/
0.25
/
0.4
0.6
0.6
0,
Qe
0.4
0.2
I
0.0
0 0.0
0.2
0.8
0.4
0.2
C
1
0.8
0.8
1 .o
x,
X
Figure 1. As in Figure 4 for fmed 4C and varying +.
Figure 7. As in Figure 6 for fixed BC and varying 71. Table I. Parameters Obtained from Fitting of the Experimental Data of the Methanol Adsorption on Freeze-Dried Starch Gel Reference 19 into Equation 1V A (kcal/mol) B (kcal/mol) C (k&mol) kb -1.14 @A -1.90) 0.30 (@B= 0.6) 1.73 (@C 2.89) 10.69 'Calculation was performed for 1/8 = kBT = 0.6 kcal/mol. b Dimensionless. El OA-.l .QO BB- 0.50 DC=2.89 k-10.59
0.0 0.0
0.2
0.4
0.8
0.8
1 .o
XI
Figure 6. Multimolecularsorption ratio 91 vmuseorbate activity rl for a rigid chain for fixed 71 and varying OC.
of the cellulose and starch, it is necessary to take into account the behavior of the conformation of primary linkages between glucose units. Amylose, which is one of two component polysaccharides of starch, is based on a-1,4-D-glUCOpyranOSe units, and residues of cellulose are joined together by 8-1,4-~glycosidic linkages. To determine a conformation of the two a-D-glucoses or 8-D-glUCOSeS bonded with a-1,4 or @1,4 glycosidiclinkagesresearchersmusually w e two rotational angles (a,*) to describe the energy of the conformation
-f
0.0
0.2
0.4
0.6
0.8
1 .o
x=p/po
Figure 8. Fit of theoretical eq 10 with r = 1 to methanol vapor sorption for freeze-dried starch gel at 23 OC: O, experimental; +, theoretical. Experimental data taken from ref 19.
(see Figure 10in ref 1). Recently, anumber of researchers, Rees et Rao et al.122 and Pizzi,2s have obtained the steric map for a-1,4 and energy of conformation on a al.21126
@-*
(20) Walton, A. G.; Blackwell, J. (with a contribution by Cam,St. H.) Biopolymers; Academic Press: New York, London, 1973. (21) Rem, D. A.; Skerret,R. J. J. Chem. Soc., B 1970, 189. (22) Rao,V. 5.R.;Sudarar@n, P.R.;Ramaleriehnan, C.; Rnmachsn&an, G. N. Conformation of Biopolymers; Academic Press: New York, 1967; Vol. 2.
Sorption of Gases on Heterogeneous Solids P-1,4 glycosidic linkages with two potential wells separated by a rather high potential barrier (H-Laround 5-10 kcal/
mo1).1*20*uFor instance in the cellulose21and amylose20 these values correspond to the change of the helix-loop conformation. However, more detailed calculations20~24 indicate that the well H is split into two subwells, corresponding to two conformations, HLand H R . ~The values of the difference in potential energy between the two conformations (HL-HR) obtained by P i ~ z for i~~ cellobiose for a 8-1,4 glycosidic linkage are around 0.8 kcal/ mol and for cellobioside around 0.6 kcal/mol. Gracomini et aL20show a similar result for maltose with a difference of energy between the two states around 0.2 kcal/mol. The latter value is very close to the value of parameter B which is equal to 0.3 kcal/mol in our model and corresponds to the difference of energy betwen the two states of conformations HL-HR for amylose. The potential energy for maltose, V-amylose, has been calculated independnetly by a number of groups; the results are similar.20*22 According to Walton et al.,2O conversion of form Vamylose to form B-amylose is effected by humidification and the mechanism probably involves breaking the intermolecular hydrogen bond as the water is adsorbed. It seems that the conformation of amylose involves the cooperative interactions within the chain. Each of the primary bonds supplies individually only a small amount of energy to the macromolecule, and in the case of amylose cooperative properties of the conformational subsystem are well confirmed, as well as for cellulose.' This indicates that one of the major effects which play an important role, (23) P h i , A. In Cellulose and Its Deriuaties; Kennedy, J. F., et al., E&.; Ellis H o r w d New York, 1985; p 59. (24) Rees, D. A.; Skenet, R. J. Carboohydr. Res. 1968, 7, 334. (25)Banks, W.; Greenwood, C. T. Starch and its Components; Edinburgh University Press: Edinburgh, 1975. (26) Rws, D. A. Polysaccharide Shapes; Chapinan and Hall: London, 1977.
Langmuir, Vol. 9, No. 10, 1993 2681 and can dominate during the gas sorption process on the tested polymer, is the effect of sorptional-conformational coupling.
Conclusions A theoretical model of mono- and multimolecular sorption of gases with the intermediate adsorption states on a flexible linear chain of a polymer with degenerated conformational degrees of freedom was formulated, and the model has been illustrated for the methanol-starch gel system. Introduction of the intermediate adsorption state of the gas molecules on the polymer, instead of the condensation state, leads to readjustment of the BET to GAB isotherm adsorption equation; this is termed "energetic rescaling". If we introduce the degeneration of conformational degrees of freedom into the sorption isotherm, it means rescaling of entropic meaning and introducing a new parameter (B') which describes the conformation of the polymer, and which can be split into energetic and entropic terms. It appears that sorption on polymeric sorbent has heterogeneous character. In the case of a rigid chain of the polymer the proposed equation of the adsorption isotherm can be reduced to the simple equation for heterogeneous sorption on two kinds of centers having different adsorption energies. Our approach gives a more realistic picture of the mechanism for polar vapor sorption on starch and cellulose. The values of the parameters of our model, obtained from fitting experimental data of the adsorption isotherm to the theoretical equation, have physical sense and correspond to the predicted picture of sorption of methanol on freeze-dried starch gel. Acknowledgment. The authors wish to thank Professor E. Chibowski for discussions and critical reading of the paper.
