I n d . E n g . C h e m . Res. 1989, 28, 711-719
711
Mathematical and Experimental Foundations of Linear Polycondensation Modeling. 3. Experimental Study of Catalyzed Polyesterification of Adipic Acid and Triethylene Glycol: Equilibrium and Kinetics Mario Rui N. Costa Centro de Engenharia Quimica da Universidade do Porto, Rua dos Bragas, 4099 Porto Codex, Portugal
Jacques Villermaux* Laboratoire des Sciences d u Glnie Chimique, C N R S - E N S I C , 1 rue Grandville, 54001 Nancy, France
Equilibrium chain-length distributions (CLD) and water content were measured as a function of water vapor pressure a t 130 “C for adipic.acid + triethylene glycol polyesterification, with 4toluenesulfonic acid as a catalyst. The CLD are close to the Schulz-Flory distribution, and the apparent equilibrium constant K for the reaction iswound 1.8. The vapor-liquid equilibrium of water cannot be described by Flory-Huggins theory, the observed interaction parameter x being a strong function of end group concentrations. A thermodynamic theory based on the principle of group contributions can explain this behavior. It also predicts a change of K with medium composition and its independence on chain length, but its full development needs data for other systems (predictions based on UNIFAC are not quantitatively correct). Kinetic data obtained in a small batch reactor sparged with nitrogen agree with the assumption of equal reactivity of end groups. The influence of the reverse reaction could be quantitatively assessed taking into account masstransfer resistance to water evaporation. The alcoholysis rate constant was measured by monitoring the relaxation of the CLD toward equilibrium after mixing triethylene glycol and polyester. Similar experiments by Flory are comparatively discussed.
Materials and Analytical Techniques Materials. Adipic acid (AA), triethylene glycol (TEG), and monohydrated p-toluenesulfonic acid (TSA) (puriss. or p.a. grade) were used as supplied by Fluka without further purification. The purities of AA and TSA were checked by measuring their acid content by titration with KOH (see below). The presence of other ethylene glycol oligomers in TEG was checked by gas chromatographyand found to be negligible. Nitrogen used for sparging the reaction medium was “R” grade supplied by Air Liquide, containing small levels of oxygen and water which had no influence in this study. End Group Analysis. The carboxyl end group content was determined by titration with a solution of KOH in ethylene glycol, in THF (tetrahydrofuran) medium, under nitrogen, with phenolphthalein as an indicator, after correction for acidity in the solvent. The hydroxyl end group content was indirectly determined from the knowledge of the carboxyl end group content, from the initial AA/TEG ratio, and from the amount of evaporated TEG, which was trapped in a scrubber containing an aqueous solution of 1% ethylene glycol as an internal standard for gas chromatography determination. Determination of Water. The water content in the liquid polyester was found by rapidly sampling 2-4 cm3 with a calibrated syringe and immediately injecting the sample into an automatic Metrohm 633 Karl Fischer analyzer. Size-Exclusion Chromatography. The polymer was analyzed by size-exclusion chromatography (SEC) using a Waters chromatograph, with a differential refractometer detector and a set of three p-Styragel columns (nominal pore sizes IO4, lo3,and lo3A) and one Ultrastyragel column (nominal pore size 100 A). The solvent was stabilized THF, with a volumetric flow rate of 1 cm3/min. Chlorobenzene was used as an internal standard. Figures 1and 2 show two examples of chromatograms, corresponding to samples of polyester at chemical equi-
librium, one prepared without solvent (Figure l),and the other made in a 8.59% solution in chlorobenzene (Figure 2). It is interesting to notice the separation of peaks for the lower linear and cyclic oligomers. The dilution by a solvent favors the formation of cyclic oligomers (Figure 2).
Introduction A comprehensive description of linear polycondensation processes requires a good knowledge of chemical equilibrium and kinetics, as well as of the “computer chemistry” allowing the prediction of the concentrations of the various reactive groups and species, including the CLD. These problems are dealt with in the preceding papers of this series (Costa and Villermaux, 1988, 1989). This information is not sufficient, as mass-transfer effects such as micromixing and volatile byproduct removal (in reversible polycondensations) also play an important role. Experimental research in this area is scarce, and this work was mainly intended to yield a clear view of the most crucial problems in this domain. The choice of an aliphatic polyesterification (adipic acid with triethylene glycol) seemed adequate for this bench-scale study, given the comparative simplicity of its chemistry and ease of analysis by liquid chromatography. The polyester resulting from this polycondensation was poly(oxyadipoyltris(oxyethylene)), abbreviated as PATOE in this work. The relationship between the degree of polymerization x (within a homologous series of molecules) and the elution volume V , for higher x has to be indirectly established. A linear relationship was obtained by plotting the logarithm of the degree of polymerization against V, for the first linear oligomers. V, values for the uneven degree of POlymerization species BB, = B(QVPV),-,QB and AA, = A(PVQV),-,PA (with A = C02H, B = OH, V = COO, P = (CH,),, and Q = (CH)20(CH2)20(CH2)2) can only be distinguished for AA, and BB,. They differ by no more than 0.1 cm3 for the trimers AA2 and BB2. The V , values for the hydroxyacids AB, = A(PVQV),-,(PVQB) also fall
0888-5885/89/2628-0711$01.50/0 0 1989 American Chemical Society
712 Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989
ii
AB.
., X
30 20
10
5
WITHDRAWA L O F SAMPLES
L v..
-
\ '
2
Figure 1. Normalized chromatogram of polyester at equilibrium without solvent with r = 1 and u = 0.734. Elution volumes are indicated in abscissas.
'
SCRUBBER
Figure 3. Experimental setup for batch reactor operation and equilibrium data acquisition.
Figure 2. Normalized chromatogram of polyester at equilibrium in a 8.59% by weight solution in chlorobenzene. This favors the formation of cyclic oligomers. (---) Tail of the peak for linear oligomers.
on the same line, so that the relationship between x and V , is valid for higher oligomers whatever the nature of the end groups. A linear dependence between log x and V , was found to hold for linear polystyrene in the same column set. This is why it was assumed that the same kind of relationship was valid for PATOE, namely, In x = V,, - PV, This expression only involves one free parameter, as it is constrained to hold in the oligomer range (say for x = 5). P was obtained by comparing the values of x , from integration of the observed chromatograms and those independently determined by end group analysis. This was consistent with the relationship observed for the oligomers. Besides the x ( V,) relationship, the evaluation of the CLD from the observed chromatogram also requires the knowledge of the response factors, i.e., the ratio of the area of individual chromatographic band for each species to its injected amount. In addition, a means for taking into account the peak overlap has to be found in order to evaluate the areas of the individual bands. The first problem was dealt with assuming a group-contribution relationship for the molar refractivity (product of refraction index and molecular mass-see Van Krevelen and Hoftijzer (1976)). This allows the unknown response factors of an oligomer to be interpolated between the observed response factors of the monomers and those of high molecular weight polymers. Finally a linear relationship between the response factors and l l x was obtained. In spite of the peak overlap, the individual contribution of each species was deduced from the overall area under the chromatogram, taking into account the fact that the peak shape is known and writing down a system of linear algebraic equations, summing up the contribution of each peak in a series of consecutive "windows". A different method, described by Costa and Villermaux (1978), was used in the higher molecular weight region
where no individual peaks are seen. Owing to the practical difficulty of distinguishing the molecular species according to their end groups, except for the very lowest members of their series, the results of the analysis are presented as the distribution of generic species S,having a degree of polymerization x , such as S, = AB, for even x and S, = AA, + BB, for odd x . A detailed presentation of the treatment of SEC data can be found elsewhere (Costa, 1983). Water-Polyester Equilibrium The polyesterification equilibrium in the bulk phase was studied by carrying out the reaction in a 100-cm3batch stirred tank reactor under known constant feeds of water vapor and nitrogen until a steady state was achieved. The experimental setup is shown in Figure 3. Equilibration experiments in chlorobenzene solution were performed in a 30-cm3 closed vessel heated during 30 h. The equilibrium CLD can be easily found by a probabilistic reasoning (Flory, 1953), if the equal reactivity principle is assumed to hold. Normalizing all concentrations by 2[P] (twice the mole concentration of the (CH,), units in adipic acid, APA = AA,), the CLD of the linear species is given by -2
BBn = AB, =
b2 2(r - u,)
___ Y"-'
ab(u - u,) Y"-' (1 - u J ( r - u,)
The concentration v, of the linking groups V (the later groups) belonging to cyclic molecules is obtained by adding the concentrations of cyclic oligomers D, = (PVQV),, themselves obtained via their formation constants Kc(n): u, = C2nD, n-1
2IPID, = K,(n)yn
(24
(2b)
Jacobson and Stockmayer (1950) have shown that K,(n) is proportional to n-2.5for a Gaussian chain, a condition
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 713 j
;
1
'
10
o
20
x
30
1
LO
Figure 4. Experimental CLD of linear species for chemical equilibrium in bulk and for batch polyesterification starting from the monomers, with equivalent amounts of the monomers ( r = 1). Solid u = 0.734; ( 0 )u lines: Schulz-Flory CLD. Equilibrium data: (0) = 0.808. Batch reactor data: (*) u = 0.331; (+) u = 0.510; (X) u = 0.651; (A) U = 0.794; (A) V = 0.898.
that holds approximately for long flexible chains. This condition would allow (2a) and (2b) to be rewritten as 2[Pluc = 2K(1)4(~;1.5) (3) in which
121 '
'
'
'
'
'
'
'
'
'
'
\
'
100
50
'
'
' 150
'
'
'
'
Figure 5. Experimental CLD of linear species for chemical equilibrium both in the bulk and in chlorobenzene solution and for batch polyesterification starting from the monomers. (-) Experimental CLD (the points corresponding to each individual species were joined by a continuous line). (A) Equilibrium CLD for bulk polyester (r = 1);u = 0.933. (B) Equilibrium CLD in a chlorobenzene solution with 8.59% by weight polymer; r = 0.995; u = 0.964; u, = 0.38. (C, D) Batch reactor starting from the monomers; r = 0.995; u = 0.959 ( C ) and u = 0.976 (D). (-.-) Theoretical CLD with u, = 0. (---) Theoretical CLD for sample A (predicted uc = 0.032). (---) Theoretical CLD for sample B (predicted u, = 0.404).
m
(4) is a function studied by Truesdell (1945). It can be evaluated either by a series in terms of the { function or by a numerical quadrature. Finally a mass action law should relate the end group concentration to the water concentration w: K = (u - uc)w - (u - u,)w (5) ab (1 - u)(r - u ) All these predictions of the classical treatment agree quite well with the CLD of the linear species experimentally observed for the model system in the present study, both in bulk and in chlorobenzene solution (see Figures 4 and 5). The experimental cyclization equilibrium constants Kc(n) were correlated by Kc(n) = 0.080n-2.33mol kg-l 1I n I 5 (6) This seems valid, at least for n = 1and m = 2, both in the bulk and in chlorobenzene solution. This kind of behavior is found for similar chemical systems (see Semlyen (1976)).
Water-Polyester Vapor-Liquid Equilibrium As a first approximation, water-polyester vapor-liquid equilibrium was interpreted according to the Flory-Huggins theory (Huggins, 1942; Flory, 1942a,b, 1944). In the context of this theory, it will be assumed that water is a molecule composed of a single segment and that the polyester is a blend of an infinite number of molecules S , with various numbers of segments X ' J 'becoming indefinitely large as the molecular size increases. In the original theory, the number of segments in a molecule was taken to be proportional to the molar volume. Instead, the van der Waals volume will be used. It is computed by a group contribution method according to Bondi (1968). The segment count is not affected very much by this choice (see Sayegh and Vera (1980)).
I
200
x
Kl
+
K'
+l -
1Y
xx
0
-
xXX
" " " " ' ~ ~ " ~ ~ " ' ' ~ ~ ' " " " '
714
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989
According to the Flory-Huggins theory, the free energy of mixing of a polymer-solvent system is the sum of two contributions: an entropic or combinatorial term, which stems from the different ways the segments of polymer and solvent can be arranged in a lattice of coordination number 2; and a residual term, which includes the heat of mixing. the logarithm of the activity of a generic component j is thus the sum of these two contributions: log A, = log A,CoMB+ log AIRES
(9)
Defining the standard state for j as the pure component with its molecules aligned in a specific ordered state, Flory (1944) has obtained, for a molecule with symmetry number ul
f
log A,CoMB = 1 - x / , / x ’ , ,
+ log f J + log ( u , / x ; )
l)[log (2 - 1) - 11 (10)
= 1, uAA, - uBB, = 2 , and uDn = 2n. (x;
In our case, uAB, = (T, There are some corrective terms to this expression (see Sayegh and Vera (1980)),which are of minor importance when dealing with chainlike molecules. They will therefore be neglected. Notice that a molecule with normalized concentration GI has a mole fraction G,/[n, + (a + b)r + w], and therefore
+ (a + b ) / 2 + w ] / x ;
f, = x’,G,/[n,
(11)
The residual activities for the Flory-Huggins theory become log AwRES= ~ (- fl, ) 2 log AIRES = x.z.‘,fW2 The interaction parameter through
x
=
~
P
x
(j
(124 #
w)
(12b)
was originally defined
/ (ZR, S T)
(13)
in which hEps is the exchange energy of interaction between polymer and solvent segments, but it was later verified that it includes other entropic contributions. Equation 12a should, in fact, be considered as the definition of x (see Flory (1970)). The experimentally observed activities of water in equilibrium with PATOE, taken as the ratio of the water vapor pressure to its saturation value of 2.67 atm at 130 “ C ,were used to compute x from (8), (9), (lo), and (12a):
x f,
+ 1 / x \ - log f,) = 10/[5.019 + 3.551(r - 1) + 1.412~+ w] = (1 - fJ2(log A, - 1
(14a) (14b)
The interaction parameter x was found not to be a constant. It could be correlated, for r = 1 (see Figure 7 ) as a function of end group concentration according to the empirical relationship x = 1.410 - 1 . 9 4 5 ~+ ( 3 . 7 8 5 ~- ~2.414~+ 0.422)’12 (15)
It is probable that an independent change of x with w becomes important for higher values of w ,but this influence cannot be distinguished through the available data because of a correlation between a and w. This change of x with end group concentration is caused by the greater water affinity of carboxyl and hydroxyl end groups as compared to that of the repeating unit of PATOE. Other effects, such as the so-called state equation contribution (Flory, 1970; Patterson, 1969) due to the difference of expansivities of solvents and polymers, could also play a role. A preliminary calculation of this effect, based on literature values of expansion coefficients of water (Perry
0
a5
a
1
Figure 7. Interaction parameter x as a function of end group concentration at r = 1 for water-polyester. Solid line: regressed relationship (15). Dashed line: UNIFAC predictions. (+) Data for Continuous-stirred tank reactor sample at chemical equilibrium. (0) data.
and Chilton, 1973) and similar polyesters (Flory, 1940), showed that this effect is rather small for this kind of system. A convenient framework for describing the thermodynamic properties of the complex mixtures of molecular species in the course of a polycondensation is provided by group contribution methods, the most well-known being UNIFAC (Fredenslund et al., 1975). They are based on an additivity law assuming that the residual free energy of species j is the sum of the free energies of its groups, which is translated, in terms of activity coefficients, by log yj = log yjCOMB = log 77“+
c vk’(1Og
NG
rkREs - log rk,REFRES)
(16)
k=l
The expression for evaluating the activity coefficients according to the Flory-Huggins theory agrees also with (16), as it assumes that log yTV = 0 and that each polymeric species is made of x equal segments ( =vzj, k = 1 denoting the group “w”). An attempt at predicting the water activity in PATOE using UNIFAC parameters according to Tiegs et al. (1987) was not quantitatively successful (see Figure 7 ) , although the overall change of the residual water activity x with end group concentration is similar to the one shown by the experimental data. It may happen that the interactions with the nearby ether and ester groups are not simple additions of their individual interactions. Only the study of vapor-liquid equilibria with other similar systems can give a more definite answer.
>
Dependence of Esterification Equilibrium Constant on Medium Composition Although quantitative predictions for this chemical system based on UNIFAC are not correct, it is worth discussing how group contribution methods can explain changes of the apparent chemical equilibrium constant K with medium composition, which are quite obvious for water-Nylon 6 (Giori and Hayes, 1970) and water-Nylon 6,6 (Ogata, 1960) equilibria. From the additivity law (16), the thermodynamic constant K can be related to the apparent constant K and correcting factors K K FV, and K RES: KO = KKCOMBKFVKRB (17a) Owing to the lack of vapor-liquid equilibrium studies for oligomer mixtures, it is not yet possible to state whether
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 715 there is an appreciable chain length dependence of K FV. If it does exist, it should be confined to reactions with the first oligomers, as K FV only depends on the values of reduced volumes of each component (mole volumes/van der Waals volumes) which are certainly quite close in a homologous series except for the very first oligomers. In this case, the prediction of CLD becomes more difficult: the mass balances of end groups are coupled to the mass balances of the first oligomers and must be solved together. A similar situation prevails when the ringforming reactions are included in the kinetic scheme (Costa and Villermaux, 1988). There are no insurmountable difficulties in computing the CLD in either case, and use of discrete-transform based methods actually helps in eliminating the closure problem appearing in the system of mass balance equations of the oligomers. and K RES, are conAs far as the other terms, K cerned, they are truly chain length independent. KCoMB can be easily computed according to (10). It is related to the change in the number of segments Ax ‘by the reaction A B V + W (-0.422 for an esterification):
+
-
KCoMB= exp(-Ax’/x’,)
(17b)
Unless the volume fraction of byproduct is not small, this term is close to 1. Of much more importance is the residual term, which can be written as K
~
=S rvrw/(rArB)
(174
The group activity coefficients can be computed by known thermodynamic models (such as UNIFAC) as a function of the medium composition (expressed by the fraction of contacts allocated to W, A, B, and V and to the chemical groups in P and Q ) . It depends thus on w, a, b, and the temperature. For r = 1 and w = 0, UNIFAC predicts KRESto vary almost linearly between 0.32 and 0.52 when a goes from 0 to 0.3. Experimental data in Figure 6 do suggest a slight increase of K for a 0. Application of UNIFAC in order to predict KRWapparently leads to a true equilibrium constant KO independent of composition. However, this is probably due to some compensation effect, since the activity coefficient of water was not correctly predicted by UNIFAC as shown in Figure 7 . Taking this chemical system as a representative example, it can be concluded that changes of apparent equilibrium and rate constants with the concentrations of end groups and of byproduct can be rather large. The effect on the rates of reaction between end groups and on equilibrium concentrations of end groups cannot thus be neglected; it is very unfortunate that no reliable predictions from a fundamental point of view can be made at present. On the contrary, the shape of CLD is affected only because of changes of ratios of kinetic constants like kBV/kAB,and this should seldom have great practical consequences, except possibly in analyzing the the results of some experiments devised to put into evidence the effect of exchange reactions.
to explain the observed time evolution of end or linking group concentrations a, b, and u and the water concentration w, given that they react according to
-A
-
+ B- + C A km/K -V-
+W +C
(C is acid catalyst)
such that the rate constant kABCof this reaction does not depend on the nature and size of the molecules carrying the A, B, and V groups. The chemistry of this reaction is such that first order with respect to each reactant is expected (Fradet and MarBchal, 1982). However, owing to the uncertainty in the activity coefficients of the reacting groups and of the activated complex, some dependence of kABC on the medium composition may arise, a feature which has long been pointed out by Flory (1939). At the temperature of the experiments (130 “C), there is also a small contribution of the esterification catalyzed by carboxylic end groups. This was assumed to be second order with respect to the carboxyls and first order with respect to hydroxyls, with a rate constant kABA= 2.4 X 10” kg2 mor2 s-l from esterification runs without TSA at 0 I u 5 0.12. The exact dependence of kABAon the medium composition is not critical, owing to its weakness. Because of the possible existence of this medium influence on kABC, much attention was given in this research to the kinetics at the beginning of the reaction. Some care was necessary to define the instant t = 0 well (the moment at which the catalyst solution was injected) and to control the temperature history as isothermally as possible. The occurrence of temperature oscillations in the first 15 min could not be avoided, but they were compensated in the data analysis by defining an effective time t instead of the clock time t ’by
-
Batch Polyesterification Starting from the Monomers: Chemical Kinetics and Mass-Transfer Effects In the absence of ring-formation reactions, the CLD for a batch polyesterification starting from the monomers is the same as the equilibrium distribution, as can be proved using probabilistic arguments or by solving the mass balances for individual species. This still approximately holds when only a small amount of ring molecules is formed, and the experimental results for PATOE confirm this (see Figures 4 and 5). The main concern will be then
in which the reference temperature is equal to To = 403 K and the activation energy is E = 11 kcal mol-’ (Flory, 1940). The model for describing the evolution of end group concentration is based on the following assumptions: (i) Both liquid and gas phases are perfectly mixed. (ii) Mass accumulation in the gas phase is negligible. (iii) All resistance to water mass transfer is located in the liquid phase. (iv) The mass-transfer coefficient of water is a function only of liquid-phase properties (such as viscosity and diffusion coefficient) and of geometrical and hydrodynamic parameters (size and shape of the stirrer, rate of stirring, gas velocity, etc.). This latter assumption is not valid if the reaction rate is fast with respect to mass transfer. Just as it happens in the case of absorption with chemical reaction, the concentration profile of the transferred solute W close to the interface may be affected because of its participation in a chemical reaction. A further simplification arises if only an overall masstransfer resistance is considered. The molar flow rate of water being transferred from the liquid to gas phase is then written as F w = 2[P]V(w - w*)/tw (19) where the mass-transfer time constant tw is the reciprocal of the product of interfacial area by unit volume of liquid phase by a mass-transfer coefficient defined from a nonconventional expression for the driving force in terms of normalized concentrations. The time evolution of the concentrations of ester groups v and of water w can be
716 Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 Table I. Reaction Conditions for the Experimental Sums Used for Measuring the Polyesterification Rate Constant superficial final velocity of stoichiomet- inert gas, m space-time run C ric ratio r s-1 of gas 71, s 1
0.995 0.995 0.992 0.996 1.104
0.00102 0.00173 0.00196 0.00375 0.00231
2 3 4 5
0.031 0.061 0.080 0.080 0.080
306 144 92 88 82
1 1-v
Figure 9. Normalized water concentration w versus reaction time t (minutes) for polyesterification in a batch reactor starting from the monomers. (+, 0, A, X , A) Experimental values for runs 1-5 (see Table I). Solid lines: predicted relationships taking into account mass-transfer resistance. Dashed lines: predicted relationships assuming instantaneous mass transfer ( t w = 0). 20
1c
1 10
20
30
40
50
63
70
50
9C
‘00
1‘0
‘20
Figure 8. Normalized ester concentration u versus reaction time t (minutes) for polyesterification in a batch reactor starting from the monomers. (+, 0, A, X , A) Experimental values for runs 1-5 (see Table I). Solid lines: predicted relationships.
determined by solving the following system of ordinary differential equations:
dw dt
du dt
W-W* tw
There is, however, some misfit for the data in run 4 for v > 0.9 and to a smaller extent for run 2. This may question the validity of (21) for u > 0.92 (a range for which no accurate values of w could be obtained) but requires also that an important change in kABCwith u is postulated. In the absence of further support by data on other chemical systems, it is difficult to accept the implied change on IZABC, with a maximum a t u = 0.8-0.9. In these experiments, the influence of the hydrolysis reaction is not a major factor, but it is not negligible either. The order of magnitude of tw at large (a few minutes) was confirmed by other experiments in which a blend of monomers was added to high molecular weight polyester.
Kinetics of Transesterification Reactions Polyesterifications are unavoidly accompanied by the alcoholysis reaction between hydroxyl and ester groups, which may cause interchange of molecule fragments (transesterification)
@Ob) B
An algebraic equation allows w* and (w - w*)/t, to be computed from u, w, and tw. This equation is derived from a mass balance in the gas phase. Further details of this computation can be found in the supplementary material. The experimental relationships u(t) and w(t) for the runs described in Table I were used to regress the parameters relating k m and tw to the composition of the medium and to the gas superficial velocity. No definitive conclusions can be drawn on the influence of other factors on the rate of mass transfer, as the stirring speed was not varied in a wide range (500 f 100 rpm). Different relationships for the influence of the medium composition on kABC and tw were tried, based on the minimization of a weighted sum of errors on u and w,either absolute at long times or relative a t the beginning of the reaction. These are reported as supplementary material. The simplest results were that kABCis a constant equal to 0.0234 kg2 mol-2 s-l and that tw is independent of the gas superficial velocity but depends on the number-average degree of polymerization according to
No significant improvement was observed by using more sophisticated assumptions. As seen in Figures 8 and 9, fair agreement with experiment is obtained from these very simple assumptions.
+ ,v / *
+
-
k0VC. k 6 V A
C,A
* V/*
* B V C . kEYA
+
B
\
+
C.A-
Other reactions of the same kind, the acidolysis and esterlester exchange, have a negligible importance in the temperature range and with the catalyst used in this study. These reactions leave unchanged the end group concentrations and the equilibrium CLD. In order to put them into evidence, Flory (1942) made a mixture of high average molecular weight polyester either with the monomeric glycol BB1 or with other alcohols or with lower average molecular weight polyesters. He then followed the time evolution of the low shear viscosity of the - mixture, which was shown to be well correlated with x., A full quantitative interpretation of these experiments has not been possible until now, due to the lack of suitable and the CLD. After the develmeans for computing opment of the methods based on discrete transforms described in part 2 of this series (Costa and Villermaux, 1989), an immediate application was the reanalysis of Flory’s kinetic data. Other results on this kind of reaction are scarce (Kotliar, 1981). For Flory’s experiments described in Table 11, the kinetic constants ~ B V Cand ~ B V A were -found by least-squares minimization of relative errors on x,. A good fit of the observed &(t) curves (Figure 10) was obtained. The standard deviations of relative error on is comprised between 1% and 2 % . This small remaining error is mostly due to uncertainties in the com-
.,
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 717 Table 11. Evaluation of Alcoholysis Rate Constant for Poly(oxyadipoyl(oxydecamethy1ene)) at 109 "C According to ExDeriments by Flory (1942a,b)
std dev wt fraction
of polyester run
C
A B
0.0008
C
0
1 0.424 0.4 0.197
0
a1
rl
a2
r2
0.0175 0.0175
1.047 1 1
0 0.1607 0.1607
1.170 1 1
0
Table 111. Evaluation of kBVC for PATOE at 130 "C wt fraction of added run C BBI a1 D 0.001 74 0.053 0.043
E
0.001 92
0.051
0.036
regressed constant name value, kg2 mol-2 s-l kBVC 2.8 x 104 2.4 X 10"
~BVA
of re1 on fw
0.011 0.017 0.024
PI
mean obsd w
mo1-l s-l
fw
std dev of abs error in BB1
0.997 0.997
0.002 0.007
0.0051 0.0051
0.036 0.036
0.0027 0.0027
std dev of re1 error on
regressed kBVC, kg2
l""""'I""""'l""""'l""
50k
B B,
Rw 50
I
0
- 0.05 - 0.04 - 0.03
-
30h 200
'
0.0 1 + - 0 430
-. .
'
0,02
,
500
a
,
i
-
~-r
0
-0
, 1000
t(P ")
6
Figure 10. versus reaction time t (minutes) for batch transesterification of two different samples of poly(oxyadipoyl(oxydecamethylene)) at 109 " C according to Flory (1942a,b). ( 0 ,0,m, A) Experimental values for runs A, B, and C (respectively) as described in Table 11. Solid lines: predicted relationships with fitted kBvc or kBVA
position of the initial polyesters. For runs B and C, kABA = 8.5 X kg2 mol-'s-l, as computed from experimental a ( t ) data. For run C, kBVA was not regressed, the reported standard deviation of the relative error of X, corresponds to k B V A = 2.4 x 10-6kg2 moF2 s-l as in run B. Similar experiments were carried out for the alcoholysis of PATOE with TEG at 130 OC. The results are reported in Table 111. In this case, the whole CLD, via the SEC analysis, was known and kBVC could be regressed by minimizing the sum of squared relative errors on plus 100 times the sum of squared absolute errors on BB1. The computed value for ~ B V C(kBVC = 0.0051 kg2 molT2s-l) is intermediate between (5% the optimum values minimizing the errors on lower) and the errors on BB1 (15% higher). As shown is Figure 11, the fit of experimental data is good and the observed deviation seems to be caused by random analytical errors. Similar experiments with low molecular weight equimolar PATOE instead of TEG did not yield good results because the water formed by the reaction between end groups was only slowly removed and caused an additional mechanism for relaxation of CLD toward equilibrium through the hydrolysis of longer chains.
