Mass transfer in the system cellulose nitrate-ethyl acetate

2424

YE. D. POPOVA and A. YE. CHALYKH

7. A. D. POMOGAILO, A. I. KUZAYEV, F. S. D'YACHKOVSKII and N. S. YENIKOLOPYAN, Dokl,

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Akad, Nauk SSR 256: 132, 1981. W. L. GARRICK, J. Polymer Sci. 8: 215, 1970. J. P. KENNEDY and R. R. PHILLIPS, J. Macromolec. Sci. Chem. 4: 1759, 1970. E. PAPIRER, J. C. MORAWSKI and A. VIDAL, Angew. Makromolek. Chemie 42: 91, 1975. J. K. MO, J. Polymer Sci. Polymer Lett. 13: 651, 1975. B. PUKANSZKY, J. P. KENNEDY, T. KELEN and F. T(J'DOS, Polymer Bull. No. 5,469, 1981. ldem, Ibid. No. 6,327, 1982. ldem, Ibid. No. 6,335, 1982. F. R. KHALAFOV, F. A. NOVRUZOVA, E. R. ISMAILOV and B. A. KRENTSEL, Vysokomol. soyed. B32: 165, 1990 (not translated in Polymer Sci. U.S.S.R.). ldem, Acta Polymerica 41: 226, 1990. G. OLAH, Makromolek. Chem. Macromolec. Symp. 13/14: 1, 1988. Kvantovo-khimicheskiye metody rascheta molekul (Quantochemical Methods of Calculating Molecules) (Edited by Yu. A. Ustynyuk). Moscow, 1980. A. I. GYUL'MALIYEV, S. G. GAGARIN and A. A. KRICHKO, Kinetika i kataliz 28: 1079, 1978.

PolymerScienceVol. 33, No. 12, pp. 2424-2434,1991 Printed in Great Britain.

0965-545X/91$15.00+.00 © 1992PergamonPressLtd

MASS TRANSFER IN THE SYSTEM CELLULOSE NITRATEETHYL ACETATE* Y E . D . POPOVA a n d A . Y E . CHALYKH Institute of Physical Chemistry, U.S.S.R. Academy of Sciences

(Received 30 January 1991) A qualitative model of mass transfer was formulated, explaining the appearance of anomalies on kinetic curves of sorption in complex microheterogeneous nitrocellulose (NC)-solvent systems. The model is based on the assumption that these systems are of fractal nature. Consequently it is possible to apply the Kohlrausch equation for describing the kinetic curves of desorption. Traditional criteria for assessing the mechanism of mass transfer in NC-solvent systems, as well as the determination of effective mass transfer constants in the glassy and highly elastic states of these systems by the classical sorption method are shown to be inadequate. IN REFERENCE [1] it has b e e n s h o w n that the practically a m o r p h o u s N C films p r e p a r e d f r o m solution on a solid s u p p o r t exhibit a hierarchical structure which can be m o d e l l e d by a self-similar object of fractal d i m e n s i o n 2.5. In this c o n n e c t i o n it was possible to explain the substantial difference in the sorption capacity of two films of the same chemical composition, but different thickness, by a difference in the e l e m e n t s of their fractal structures, and also by a difference in the stability of these structures u n d e r the influence of a solvent. Evidently the hierarchical organization of N C films

*Vysokomol. soyed. A33: No. 12, 2574-2584, 1991.

Mass transfer in the system NC-ethyl acetate

2425

CEA, m a s s %

Zq

CEA, m a s s %

8

;b

~

]

qz

Z

!

J~

Z6 I

q

I

8

I

t2'

Z

I

6

I

IO

t 7/2, rain 7/2

Fio. 1. Kinetic curves of stepwise sorption of ethyl acetate by NC film (100/zm). Intervals of change in p/p~ are 0.23-0.37 (1); 0.37-0.47 (2); 0.47-0.50 (3); 0.58-0.65 (4); 0.65-0.72 (5); 0.72-0.79 (6); 0.79-0.84 (7); 0.84--0.93 (8).

should be manifested not only in their equilibrium properties, but also in the characteristic course of kinetic processes. The kinetics of mass transfer was studied in the glassy and highly elastic states of the NC-ethyl acetate system at 20°C. The method of measuring stepwise and integral sorption and desorption by means of a standard McBain apparatus was the same as that described in the monograph [2]. The same NC films containing 13.3% nitrogen and 100 and 25/zm thick were used both for the kinetic measurements, and for the structural and X-ray sorption studies [1, 3, 4]. In Fig. 1 are shown typical kinetic curves of stepwise sorption of ethylacetate. The shape of these curves is similar to that of the previously studied NC films with a low nitrogen content [5]. With increasing relative pressure of ethyl acetate vapour P/Ps, the shape of the kinetic curves usually changes in the following order: S-shaped--~pseudonormal-,Fickian-~two-stage--~extremal. In some cases, determination of the shape is difficult, as in the curves a number of features may combine, for example pseudonormality with extremal character. The kinetic curves of integral sorption are S-shaped and exhibit an extremum, and this is typical not only of ethyl acetate, but also of other solvents [5]. During desorption, the curve types are less varied than during sorption. A typical set of integral desorption curves is shown in Fig. 2. In both films, on reducing the value of p/ps, the linear section in the coordinates t = In (1 -Mr~Moo) is shortened, but the rate of solvent removal increases. Here Mt is the amount of solvent desorbed at time t; Moo is the total amount of solvent desorbed in the given interval ofp/ps. Enhancement of the rate of mass transfer is indicated by an increased slope of the initial linear section of the corresponding curves. The slope reaches its maximum in the intervals P/Ps = 0.65-0.55 for the thick film, and p/p~ = 0.61-0.49 for the thin film. On further lowering p/p.~, the linear sections degenerate, and the rate of desorption decreases.

2426

YE. D. PoPOVA and A. YE. CHALYKH t, min

Z,5

7,~

I

1

4-

8 X

q

--z l-

\

,",,J

Lnfd-Mt/M~ Fio. 2. Caption opposite.

During desorption in the integral variant the curves do not exhibit the linear section, and the rate of solvent loss decreases continuously, starting from the very first moments, i.e. qualitatively the curves have the same appearance as during stepwise desorption at P/Ps < 0.5. Thus during sorption, a collection is observed of all the anomalies in sorption kinetics that had been described in the literature, such as the S-shaped course, pseudonormality, two-stage [2] and extremal character [6]. Except for the latter, all these have been observed in the system NC-acetone [7]. The characteristics of stepwise desorption have not been described in the literature, but the appearance of the curves of integral desorption of acetone, as described for example in reference [8], fully corresponds to that described above. Thus it may be assumed that the results obtained encompass the whole variety of kinetic processes in any NC-solvent system. Evidently, the construction of a full kinetic model of mass transfer in these systems appears to be a sufficiently complex problem. In order to obtain an idea of its most general qualitative features in NC systems, we shall at first characterize the mass transfer by means of a phenomenological criterion--the quantity 3' in the dependence M t ~ t ~, where M t is the amount of the substance which had diffused into (or out of) the sample during the time t. It is well known that the kinetics of mass transfer in polymers is often considered from the point of view of two limiting mechanisms--the

2427

Mass transfer in the system NC--ethyl acetate 10

t, min

I

J0 i

-1

\

-Z

8

-;3'

!

t x 10 2 min

I

,Y I

-0.5

-a. 7

X .......

J ~2

K

I

I

-I.0

e-" ...a

10

12

-1.5

- 0,0'

I

I

I

1

3

5

t x 10 3 rain

FIG. 2. Kinetic curves of stepwise desorption of ethyl acetate from NC film (100 ktm). Intervals of change in

P/Ps are 0.92-0.87 (1); 0.87-0.79 (2); 0.79-0.71 (3); 0.71-0.65 (4); 0.65-0.55 (5); 0.55-0.47 (6); 0.47-0.40 (7); 0.40-0.30 (8); 0.30-0.21 (9); 0.21-0.10 (10); 0.10-0.05 (11); 0.05-0 {12).

2428

YE. D. PoPOVA and A. YE. CHALYKH IF l

(a) !

t

I 0

!

l

0,6

I

l

~I

I

~-~__.,

I

I

I

~)

7 7.2

I

i

I

I I

-----4

0,~

p~ ~

N

-.-4

I

~.---I

I

I

o,z

o.~

I

p/p~

7

Ftc. 3. Values of 3~in variousp/ps intervals for NC films 100 (a) and 25 p.m (b). Full lines---sorption, dashed lines----desorption.

Fickian case, and the so-called Case II [9]. The first corresponds to normal diffusion, and its phenomenological criterion is the value 3' = 0.5. Normal or Fickian diffusion is observed in materials the structure of which does not change with time, or else changes very rapidly relative to the rate of diffusion. In both cases there exists in the system only one characteristic relaxation time, corresponding to the establishment of the diffusion equilibrium. All cases with 3'4:0.5 are usually regarded as anomalous diffusion [10]. Here the situation with 3' = 1 is defined as Case II kinetics, controlled by the relaxation of the material structure [11]. The case with ~/>1, also called Super-Case II, is observed when the rate of diffusion exceeds the rate of structure relaxation, leading to gradual increase of the equilibrium concentration of the diffusant in the material [12]. The general tendency of the change in 3' in dependence on P/Ps is the same for both films (Fig. 3): 3' decreases with increasing p/ps, i.e. the contribution of structural relaxation to mass transfer decreases. However at p/p~>~0.7, 3" again begins to grow; it is incorrect to connect this with an increased contribution of relaxation processes caused by segmental mobility of NC chains, as the polymer now exists already in the highly elastic state. During stepwise sorption, all 3,> 0.5, and their numerical values differ for the films studied. Integral sorption at p/p~ = 0---~0.85 is characterized by

Mass transfer in the system NC-ethyl acetate

2429

7 -- 0.77. During desorption, the interval of 7 variation is smaller (7 = 0.58-0.42), while for integral desorption y = 0.42. Let us note that the case with y > 1, observed by us in a thin film at low p/p,, has also been described for other NC systems [13]. It differs from a similar case observed for example in the system PS-hexane [14] by the circumstance that in the latter system y > 1 appears at the final stage of sorption, when the diffusant fronts moving from opposite sides of the film meet at its centre. This leads to a drop of the internal stress in the central part of the film and acceleration of the establishment of sorption equilibrium as compared to the initial stage, when the mass transfer proceeded according to a different law within the same P/Ps interval. Let us stress that in the case of NC systems, each p/p~ interval is characterized by a single value of y, indicating a single and unchanging mechanism for the establishment of a sorption equilibrium. At first sight it might seem that normal diffusion occurs during stepwise desorption from a thick film in the following P/Ps intervals: 0.30-0.21; 0.21-0.10; 0.10-0.05, where y ~ 0 . 5 . However, the last condition, though necessary, is not sufficient for determining normal diffusion in films. From the solution of the diffusion equation for the given case it follows that the kinetic curves of desorption at t = 0 should extrapolate to the value ln(8/-~ 2) ~ - 0 . 2 1 . Figure 2 shows that at any P/Ps value they unambiguously extrapolate to higher values; with the thin film, some curves extrapolate also to lower values. This feature of the NC-ethyl acetate system is connected with the circumstance that at large desorption times, for all p/ps intervals, the linear segments are absent; these should exist and determine the value of the coefficient of mutual diffusion D(c) at solvent concentrations corresponding to any given experimental point on the desorption isotherm. If, in spite of all the mentioned discrepancies between the experimental data and the model of Fickian diffusion from a semi-infinite plate, the derivatives to the curves in Fig. 2 are nevertheless formally used to estimate the effective diffusion coefficient by the formula [2] l 2 d l n ( 1 - M r / M ~ ) c....

D(Co) = ~2

dt

'

(1)

then the overall concentration dependence D(c) will be of the shape shown in Fig. 4. The vertical dashed lines correspond to the upper and lower concentration limits in the successive p/p~ intervals. The sharp drop of D(c) in each interval reflects the fact that dln (1 - M/M~ )/dt---~0 at t--* ~. As the concentrations where D(c) exhibits a discontinuity simply coincide with the limits of the P/Ps intervals in the given case, they do not represent any special points. Any other concentration value in the given system can be made to a discontinuity point of D(c) with a corresponding selection of the desorption interval. These effects probably indicate a strong time dependence of the effective diffusion coefficient D(c). In our opinion, the explanation of the experimental behaviour presented requires some refinement and development of the previous notions. In references [1, 3, 4] it has been shown that the solutions of NC cannot be regarded as a homogeneous system. From reference [15] it follows that the structure of each actual system at anyp/ps can be characterized by the fraction of bound and free solvent. Let us imagine that the studied NC-ethyl acetate system is divided into two subsystems: (NC-ethyl acetate) b°und and (NC-ethyl acetate) free. The former consists of polar polymer centres together with solvent molecules bound on them, the other represents a system of the Flory type. Evidently, the overall mass transfer is determined both by the amount of solvent in both subsystems, and by the transport properties and topological organization of the subsystems themselves. Let us assume that the transport of solvent within (NC-ethyl acetate) b°und is determined by the kinetics of escape from the traps, while in (NC-ethyi acetate) fro° it proceeds by

2430

YE. D. POPOVA and A. YE. CHALYKH

log D

I

/

-7

{ { { I -8

{

/

-9

l,

I

II ZO

/

I

I}

{

l { 1

{ { t I I I

I { I t t I JBCE4, mass%

-tO

-11

/

F~o. 4. Formal calculation of the concentration dependence of D by means of formula (1) for the kinetic curves shown in Fig. 2.

normal Fickian diffusion, with solvent exchange between the subsystems. Then the problem is reduced to finding a model that could describe mass transfer processes in such a complex unordered medium. Let us take into account the circumstance that the topological organization and dimensions of the elements of both subsystems evidently are closely connected with the analogous structural characteristics of the original NC films, i.e. of a hierarchical fractal object built of stereoregular sequences of NC monomeric units. As a hierarchy of structures signifies the formation in the system of clusters with a broad distribution of dimensions, the corresponding structural elements should exhibit a broad distribution of diffusional relaxation times. As a result of this, generally the decay of correlation functions in similar media is well described by a universal phenomenological dependence--the Kohlrausch-William-Watts equation [16]

~(t )~exp[-(t/~')~],

(2)

where the parameters fl and ~-depend on the material, the studied property, and they may depend

Mass transfer in the system NC-ethyl acetate

2431

on T and p. The value of /3 obeys the condition 0 < / 3 < 1, and is sometimes called the non-exponentiality factor [17]. ~0(t) denotes that part of the total change in the property ~ which has not yet relaxed at time t. In our case ~o(t) corresponds to Mt*--the total amount of ethyl acetate that has not desorbed at time t, i.e. MI* = M= - MI. Thus formula (2) can be rewritten as

Mt*/M~o = exp [ - ( d r ) °]

(3)

An evaluation of experimental desorption data by means of this equation has shown that the whole collection of kinetic curves in Fig. 2 is resolved into two families of parallel straight lines (Fig. 5). Starting from the interval P/Ps -- 0.92-0.87 in the thick film,/3 = 0.80_+ 0.01. Relating the whole range of pips (five intervals) in which/3 = 0.80, to the division of the isotherm for this film into ranges of free and bound solvent ]15], we may assume that/3 = 0.80 corresponds to the escape of only free ethyl acetate, as the amount of bound ethyl acetate remains constant in the interval 0.55 <~p/ps<~0.92 during desorption. Starting from P/Ps = 0 . 5 5 ~ 0.47,/3 = 0.58 + 0.01. This means that the contribution of the remaining free solvent to the mass transfer is negligibly small as compared to desorption of the solvent bound on the polar centres. This is not surprising, as the amount of free ethyl acetate in the considered interval already amounts to not more than - 0 . 1 of the bound solvent. Let us note that the possibility to determine only two values of parameter/3 during stepwise

bn (--I,n

(m~/m=O

q v

! o

Z I ,b// /

L/

V

F[o. 5.

PS 33=12-F

Caption overleaf.

q

2432

YE. D. PoPOVA and A. YE. CIaALYKH

~n(-t.(M~IMoo

IJ

/8

3 ×

I

-I

//

J

/

V

1

o

/r~

/, #

bn%

4- +

FIG. 5. Representation of the kinetic curves of stepwise desorption shown in Fig. 2, in the coordinates of equation (3). Numbers correspond to the same intervals of change in P/Ps as in Fig. 2. For all straight lines, /3 = 0.80 (1-5) and 0.58 (6--12).

reduction ofp/ps from 0.92 to 0 is in agreement with the possibility to separate two subsystems in the studied system. It also means that the solvent escapes from them practically independently, in the studied case--successively: at p/ps> 0.5, the kinetics of desorption is determined by the escape of free solvent, at p/p~ < 0.5, by the escape of bound solvent. Evidently, an independent escape of free and bound ethyl acetate can only occur at a certain topological organization of the subsystems within the given NC sample, namely: (NC-ethyl acetate) free and (NC-ethyl acetate) b°und should represent infinite clusters. The transition from/3 = 0.80 to/3 = 0.58 probably is connected with the end of the existence of an infinite cluster of the first subsystem (true solution) owing to the decreasing amount of ethyl acetate during desorption, and its decomposition into separate non-bound clusters. A similar analysis for the thin film shows that the kinetic curves of desorption also split into two groups of parallel straight lines. At high p/p~, as for the thick film, /3 = 0.80+0.01, in the intermediate interval of p/p~ = 0.61-0.49, /3 = 0.60+0.01, while on further lowering p/p~, two

Mass transfer in the system NC-ethyl acetate

2433

5

0

0

.(;oOO- 7 _ /

~_o

l

I

J~"

I 7

-!

9'o

/

x.

uj,~-

/

Lnt

S °

"7

FIG. 6. Representation of the kinetic curves of integral desorption from NC film (100/xm) in the coordinates of equation (3) forp/p~ = 0.25 (1); 0.38 (2); 0.61 (3); 0.89 (4); 0.93 (5); and 0.96 (6). For the initial sections (up to the break),/3 = 0.50 (1-3) and 0.58 (4--6).

straight lines meeting at a break can be drawn through the experimental points. The first section (at small desorption times) has a slope of 0.51+0.01, with smaller /3 values after the break. The appearance of the break can be connected with relatively larger pressure drops in each interval as compared to desorption from the thick film, and this enables us to define the structure of the system within each p/p** interval. The deviation at long times in our opinion is connected with the circumstance that the real process should be described by a more complicated dependence than equation (3). This is clearly manifested when the data on integral desorption from the thick film are presented in the given coordinates for various P/Ps (Fig. 6, curves 4-6). In this case the kinetic curves also consist of two linear segments. The slope of the first one coincides with the slope typical for the escape of bound solvent--/3 = 0.58. Evidently in this case both clusters operate simultaneously, but not independently--a strong exchange of solvent should occur between them. In consequence of this, the limiting step of mass transfer thus becomes the escape of bound solvent. The second linear segment with/3 = 0.16--0.29 indicates the formation of new conditions for the escape of solvent, and this requires a special study. As a rule, two segments are exhibited only at a sufficiently large drop of p/p.~. Integral desorption

2434

YE. D. PoPOVA and A. YE. CHALYKH

from the thick film at various pips has shown (Fig. 6) that at least up to P/Ps = 0.61,/3 = 0.50 in the first section up to the break; at p/ps>0.8,/3 -- 0.58. The presented results enable us to discuss the features of the kinetic processes in the NC-solvent system from a general point of view. Moreover, they have a number of applications and consequences. Thus by means of the proposed model it is possible to evaluate the individual contributions of the free and bound sorbates to the experimental sorption isotherm. It is also possible to draw qualitative conclusions concerning the nature of polar centres and to construct a general scheme of equilibria for sorbents with polar centres, with a free solvent at increasing p/ps. In this way the developed approach can be extended to other systems characterized by a complex hierarchic structure.

Translated by D.

DOSKOt~ILOV.~

REFERENCES 1. Ye. D. POPOVA, A. Ye. CHALYKH and A. N. POPOV, Vysokomol. soyed. A32: 1675, 1990 (translated in Polymer Sci. U.S.S.R. 32: 8, 1596, 1990). 2. A. Ya. MALKIN and A. Ye. CHALYKH, Diffuziya i vyazkost' polimerov. Metody izmereniya (Diffusion and viscosity of polymers. Experimental methods). P. 304, Moscow, 1979. 3. A. Ye. CHALYKH, Ye. D. POPOVA, A. N. POPOV and D. M. KHEIKER, Vysokomol. soyed. A29: 2609, 1987 (translated in Polymer Sci. U.S.S.R. 29: 12, 2871, 1987). 4. A. Ye. CHALYKH, Ye. D. POPOVA and A. N. POPOV, Vysokomol. soyed. B29: 841, 1987 (not translated in Polymer Sci. U.S.S.R.). 5. A. Ye. CHALYKH and Ye. D. POPOVA, Vysokomol. soyed. A28: 727, 1986 (translated in Polymer Sci. U.S.S.R. 28: 4,808, 1986). 6. D. S. COHEN and A. B. WHITE, J. Polymer Sci. Polymer Phys. 27: 1731, 1989. 7. A. KISHIMOTO, H. FUJITA, H. ODANI, M. KURATA and M. TAMURA, J. Phys. Chem. 64: 594, 1960. 8. H. ODANI, S. KIDA, M. KURATA and M. TAMURA, Bull. Chem. Soc. Japan 34: 571, 1961. 9. T. ALFREY, E. F. GURNEE and W. G. LLOID, J. Polymer Sci. C No. 12, p. 249, 1966. 10. K. G. URDAHL and N. A. PEPPAS, J. Appl. Polymer Sci. 33: 2669, 1987. 11. J. H. PETROULOS and M. SANOPOULOU, J. Polym. Sci. Polym. Phys. Ed. 26: 1087, 1988. 12. J. D. COSGROVE, T. G. HURDLEY and T. J. LEWIS, Polymer 23: 144, 1983. 13. T. J. LEWIS, Polymer 19: 285, 1978. 14. C. H. M. JACQUES, H. B. HOPFENBERG and V. STANNETT, Polymer Sci. Technol. 6: 73, 1974. 15. Ye. D. POPOVA and A. Ye. CHALYKH, Vysokomol. soyed. A32" 343, 1990 (translated in Polymer Sci. U.S.S.R. 32: 282, 1990). 16. M. I. KLINGER, Phys. Rept. 165: 275, 1988. 17. J. M. HODGE and A. R. BERENS, Macromolecules 15: 762, 1982.