Sorption and Diffusion of Ethylene Oxide in Semidry Potato Starch

Oct 15, 2003 - The Fickian diffusion coefficient D of ethylene oxide in rubbery potato starch appears to have a maximum, Dmax = 5.38 × 10-13 exp(0.22...
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Ind. Eng. Chem. Res. 2003, 42, 6068-6079

Sorption and Diffusion of Ethylene Oxide in Semidry Potato Starch Granules Norbert J. M. Kuipers,*,† Harry F. Vervelde,‡ Eize J. Stamhuis,‡ and Antonie A. C. M. Beenackers‡,§ Department of Chemical Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands, and Department of Chemical Engineering, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands

The sorption and diffusion of gaseous ethylene oxide in semidry potato starch granules was studied by measuring the time-dependent mass-uptake rate of ethylene oxide by starch in an isothermal pressure-controlled semibatch reactor as a function of the ethylene oxide solubility (cEO,s < 8.7 kmol‚m-3), moisture content of the granules (5.9 e W e 22.5 wt % dry basis), and temperature (293 e T e 368 K). The data obtained, together with results of the chemical kinetics, are needed for the design of a reactor for the gas-solid hydroxyethylation of potato starch. The distribution coefficient of ethylene oxide, mEO, defined as the ratio of the concentrations of ethylene oxide in the starch and in the gas phase at equilibrium, could be fitted by the expression mEO ) 1.69 × 10-4 exp(-0.65aEO + 6.1/W + 31.6 × 103/RT), with aEO ) pEO/poEO as the ethylene oxide activity. Depending on W and T, the diffusion of ethylene oxide in potato starch is either Fickian, anomalous, or relaxation-controlled (case II diffusion). The latter is observed at T ) 313 K for 5.9 e W e 9.9 wt % d.b. and cEO,s > c/EO, where c/EO is the threshold concentration for case II diffusion. Here, the case II front velocity u can be described by a power-law equation that takes into account the preswelling of the granules by water, namely, u ) Kw + K(cEO,s - c/EO)n with Kw ) 2.14 × 10-10(W - 5.8)0.58, c/EO ) 6.5 - 0.53W, K ) 8.7 × 10-10 + 8.6 × 1011/W24.6, and n ) 7.0-0.33W. The Fickian diffusion coefficient D of ethylene oxide in rubbery potato starch appears to have a maximum, Dmax ) 5.38 × 10-13 exp(0.22W - 15.9 × 103/RT), as a function of cEO,s at cmax EO,s ) 28.9 - 0.071T - 0.33W. For rubbery starch, the diffusion coefficient max max is fitted by D/Dmax ) 0.19 exp(1.58cEO,s/cmax ) 1.64 EO,s) for cEO,s/cEO,s < 1 and by D/D max max exp(-0.34cEO,s/cEO,s) for cEO,s/cEO,s g 1. These relations are valid above the glass transition temperature Tg, i.e., for W g 14.2 wt % d.b. and T g 313 K or for W < 14.2 wt % d.b. and T g 313 K, provided cEO,s g c/EO. The diffusion coefficient for potato starch in the glassy state is described by D ) 2.12 × 10-8 exp(0.49W + 0.51cEO,s - 57.2 × 103/RT) for 293 e T e 313 K, 5.9 e W e 9.9 wt % d.b., and cEO,s < c/EO. For temperatures very well below Tg, so-called twostage sorption was observed because of a slow increase in the solubility of ethylene oxide in potato starch with time. Based on sorption, diffusion and data concerning the chemical kinetics of the uncatalyzed and catalyzed hydroxyethylation of semidry potato starch particles, a schematic summary of the physical behavior of ethylene oxide in semi-dry glassy potato starch is proposed as function of aEO. 1. Introduction Hydroxyethyl starch is a commercially important product in, for example, the paper and textile industries. The classical method of manufacturing it is by a batch reaction of starch with ethylene oxide in aqueous slurries catalyzed by hydroxide. This process has several disadvantages:1 (1) Part of the ethylene oxide in the aqueous phase is hydrolyzed to ethylene glycol, resulting in a low yield. (2) To prevent gelatinization of the starch granules, the temperature must be kept below 323 K, and the sodium hydroxide concentration must remain relatively low. * To whom correspondence should be addressed. Tel.: +3153-4894289. Fax: +31-53-4894821. E-mail: n.j.m.kuipers@ ct.utwente.nl. † University of Twente. ‡ University of Groningen. § Deceased.

To overcome these disadvantages, we developed a novel gas-solid semidry process. A reactor design according to this new principle requires sorption and diffusion data on ethylene oxide in semidry potato starch granules. Such data are not available in the open literature and are presented here. Data on the chemical kinetics of the uncatalyzed and catalyzed gas-solid hydroxyethylation of potato starch are published elsewhere.2 2. Theory 2.1. Diffusion of a Penetrant in a Polymer. Diffusion of a penetrant in a glassy polymer is known to show “anomalous” or “non-Fickian” behavior, especially when the penetrant causes extensive swelling. In a rubbery polymer, on the other hand, diffusion is often close to Fickian. The physical reason for this difference in behavior is the time-dependency of the properties of a glassy polymer, which is due to the finite rate of

10.1021/ie020944s CCC: $25.00 © 2003 American Chemical Society Published on Web 10/15/2003

Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6069

The following features are readily apparent: (1) Case II diffusion is confined to relatively high penetrant activities and temperatures in the vicinity of or below the Tg of the system. In its most extreme form, crazing can occur. (2) The region of case II diffusion is separated from the Fickian diffusion regions by a region where both mechanisms play a role, resulting in anomalous diffusion. 2.1.1. Case I or Fickian Diffusion (Regions Ia, Ib, and V). If the uptake of penetrant by a spherical polymer grain can be described by Fick’s law, then the relative mass uptake, Mt/M∞, as a function of time t is given by5 Figure 1. Schematic diagram of the glassy core and rubbery shell for a swelling-promoting penetrant diffusing into a grain of a glassy polymer at a rate of u ) -dr0/dt.

Mt M∞

)1-

6 π

∞

∑ 2 n)1

1

e-n π Fo 2 2

(1)

2

n

where Fo is the Fourier number, given by Fo ) Dt/R2; D is the Fickian diffusion coefficient; and R is the radius of the grain. 2.1.2. Case II Diffusion (Region II). For case II diffusion, eq 1 must be replaced by6

4 3 3 r0 Mt 3π(R - r0 ) ) )1M∞ 4 3 R πR 3

()

Figure 2. Transport mechanisms in the various regions of the temperature-penetrant activity plane.4 a, b, c, and d are lines of constant activation energy, which decreases from a to d. The effective glass transition temperature is also given.

adjustment of the polymer chains to the presence of the penetrant. Relaxation times associated with changes in the polymer structure decrease with increasing mobility of the polymer segments, e.g., with increasing temperature or sorbate concentration. The temperature at which a polymer changes from glassy to rubbery is called the glass transition temperature, Tg. Its value depends on the sorbate concentration. Alfrey et al. distinguished three classes of diffusion in polymers, based on the relative rates of relaxation of the polymer chains and diffusion, i.e., the so-called Deborah number:3 (1) Case I or Fickian diffusion occurs when the rate of diffusion is much less than the relaxation rate. (2) Case II or relaxation-controlled diffusion occurs when the rate of diffusion is rapid relative to the relaxation rate. (3) Anomalous diffusion occurs when the diffusion and relaxation rates are comparable. Case I systems are controlled by the Fickian diffusion coefficient D. In case II systems, the controlling parameter is the constant velocity u of an advancing front that separates an outer shell saturated with penetrant from an inner core that is essentially unpenetrated and, therefore, remains glassy (see Figure 1). For many penetrant/polymer systems, all three transport mechanisms can be observed by changing the temperature and/or the penetrant concentration (see Figure 2 for a chart of anomalous transport phenomena4).

3

(

)1- 1-

)

ut R

3

(2)

where r0 () R - ut) is the time-dependent radius of the glassy core and u is the linear velocity of the moving boundary between the rubbery shell and the glassy core (see Figure 1). The rubbery shell is assumed to be saturated with penetrant, whereas the glassy core is unpenetrated. The uptake of penetrant is complete at t ) R/u. According to Astarita and Sarti,7 u can be described by a power-law (assuming ethylene oxide to be the penetrant)

u≡-

dr0 ) K(cEO,s - c/EO)n dt

for cEO,s > c/EO (3)

where K and n are empirical constants. Thus, the driving force for swelling is the difference between the sorbate concentration cEO,s at the penetrant front (r ) r0) and some threshold concentration for swelling c/EO. For case II diffusion, the first concentration equals the solubility of ethylene oxide. 2.1.3. Anomalous Diffusion (Region III). Figure 2 shows that, between the region of case II diffusion (region II) and the regions of Fickian diffusion (regions Ia, Ib, and V) exists a transition regime of anomalous diffusion (region III). The latter type of diffusion occurs if both the diffusion rate of the penetrant and the relaxation rate of the polymer chains determine the overall mass uptake. Then, the penetrant concentration at the swelling front is less than its solubility. Mt/M∞ depends on the value of the dimensionless group χ ) D/uR, which has an order of magnitude of 1 in this regime, as has been discussed by Kuipers and Beenackers.8 Because of this intermediate region between case I and case II diffusion, eq 3 for u has to be adapted. Thus, we define a new threshold concentration, c#EO, that marks the boundary between Fickian diffusion region Ib and anomalous diffusion region III, whereas c/EO

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marks the boundary between anomalous regime III and region II of case II diffusion

u ) Kw + K(cEO,s - c/EO)n u*0 u)0

for

c#EO
0.35, the solubilities predicted by the BET model are too high. 4.1.1. Influence of Moisture Content W. The shape of the sorption isotherm depends on the moisture content of the starch. Figure 5 shows distribution coefficients and sorption isotherms for W ) 7.7, 9.9, and 18.3 wt % d.b. at T ) 313 K. Initially, the two drier starches are glassy, whereas the starch with W ) 18.3 wt % d.b. is rubbery.10 Consequently, the Tg’s of the two drier starches will probably be reduced below 313 K by the uptake of the plasticizer ethylene oxide. The sorption isotherm for W ) 18.3 wt % d.b. is of type III, and the other two are of type II.

For W ) 7.7 wt % d.b., mEO becomes almost independent of the ethylene oxide activity for aEO > 0.4 (i.e., sorption according to Henry’s law). This transition coincides with a hole saturation constant of c′H ) 2.3 kmol‚m-3 d.b. The results for W ) 18.2 wt % d.b. show a transition only at about aEO ) 0.73. Here, mEO starts to increase with increasing aEO, again probably as a result of clustering. 4.1.2. Influence of Temperature T. Figure 6 shows cEO,s and mEO for W ) 9.9 wt % d.b. at three temperatures. The sorption isotherms at the two lower temperatures are sigmoid type II isotherms. Above Tg (about 321 K for W ) 9.9 wt % d.b.), such a sigmoid type II sorption isotherm is expected to change into a type III isotherm. Unfortunately, this could not be confirmed, because we could not measure above pEO ) 2.5 bar. However, where measured, the sorption isotherm at T ) 353 K appears to be linear. 4.1.3. Distribution Coefficient as a Function of W and T. Figure 7 shows a parity plot of the experimentally determined distribution ratios and their fit according to

mEO )

(

1.69 x 10-4 exp -0.65aEO +

)

6.1 31.6 × 103 + (11) W RT

with 5.9 e W e 22.5 wt % d.b. and 293 e T e 368 K. This equation holds only in the region where mEO decreases with increasing aEO, because only the points in this region were used to obtain the above equation. These are the open symbols in Figure 7. As expected, mEO decreases both with increasing W and with increasing T. The former is caused by the competition between water and ethylene oxide for the available sorption sites,

Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6073

Figure 7. Parity plot of the experimental mEO and the fit according to eq 11.

and the latter is caused by the negative heat of sorption. The decrease of mEO with increasing T follows the van’t Hoff equation, but the heat of sorption of 31.7 kJ‚mol-1 appears to be higher than the heat of vaporization of ethylene oxide at T ) 303 K, which is 24.5 kJ‚mol-1 according to Kirk-Othmer.23 The excess sorption energy is due to the negative heat of sorption liberated by the sorption of ethylene oxide in the holes in the glassy potato starch. 4.1.4. Isotherm Analysis with Dual-Mode and BET Models. Table 1 gives the results of the sorption isotherm analysis with the dual-mode model. It shows that the hole saturation constant c′H decreases both with increasing W and with increasing T. This is expected, because c′H approaches zero at and above Tg. Also, with increasing W, the holes in the glassy potato starch are filled with more water, leaving less space for ethylene oxide and, hence, decreasing c′H. The Henry’s law constant H for dissolution in the polymer matrix increases with increasing W, but decreases with increasing T. Table 1 also shows that c′H ) 0 for W ) 18.3 wt % d.b. (T ) 313 K) and for T ) 353 K (W ) 9.9 wt % d.b.). The latter means that Tg < 353 K for W ) 9.9 wt % d.b. Extrapolation of the data at T ) 303 and 313 K yields c′H ) 0 at T ) Tg ) 334 K for W ) 9.9 wt % d.b. This value of Tg does not deviate significantly from Tg ) 321 K as reported by Van den Berg.8 Extrapolation of the data at W ) 7.7 and 9.9 wt % d.b. yields c′H ) 0 for W ) 11 wt % d.b. at T ) 313 K. This value of W approximately equals that for the boundary between class I and class II water.8 Table 2 presents the results of the sorption isotherm analysis with the BET model. It shows that the monolayer concentration c′ and the BET adsorption constant cB increase and decrease, respectively, both with increasing W and with increasing T. The increase of c′ can be attributed to the swelling of the starch granules with increasing W and T; i.e., more sites are freed for sorption with increasing W and T. A decrease in cB means that the ratio between the adsorption rate and desorption rate decreases, resulting in a negative heat of sorption.

Figure 8. Two-stage relative mass uptake of ethylene oxide as a function of time t for potato starch with W ) 9.9 wt % d.b., at T ) 293 K and aEO ) 0.86. Curve 1: Fickian diffusion with D ) 4.17 × 10-14 m2‚s-1. Curve 2: eq 12 with R ) 2.40 and β ) 2.62 × 10-4 s-1.

Unfortunately, not enough data are available to determine the dependence of W and T on the dual-mode and BET parameters. 4.1.5. Two-Stage Sorption. Figure 8 shows the relative mass uptake of ethylene oxide by (glassy) potato starch as a function of time t for W ) 9.9 wt % d.b. at a relatively low temperature of 293 K. After about 3500 s, an equilibrium is reached. However, the final true equilibrium is not reached until about 30 000 s. This two-stage sorption was not observed above T ) 300 K. Figure 8 also shows the fits of both stages of the sorption curve. The first stage of sorption is fitted using Fick’s law with D ) 4.17 × 10-14 m2‚s-1 (curve 1), assuming the surface concentration to be the concentration at quasi-equilibrium. An adapted version of eq 6 is able to fit the second stage of sorption

Mt - Mi ) 1 - Re-β(t-t1) M∞ - Mi

(12)

The factor R appears in this equation due to the slightly sigmoidal character of the experimental sorption curve,

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which deviates from that of pure two-stage sorption. The fit, using eq 12 with R ) 2.40 and β ) 2.62 × 10-4 s-1, is reasonably good (see Figure 8, curve 2). It yields a value of 1.86 for the dimensionless parameter βR2/D, which means that the total sorption rate is determined by both diffusion and the increase in surface concentration with time. According to the literature, this results in a two-stage sigmoidal sorption curve, as, indeed, can be observed in Figure 8.13 The value of the diffusion coefficient as determined from the first stage of sorption, D ) 4.17 × 10-14 m2‚s-1, is high relative to the single-stage Fickian diffusion coefficients as will be discussed in section 4.2. This is due to the unusual concentration-activity relationship in two-stage sorption.24 The initial Fickian stage occurs when there is a large increment in the vapor pressure pEO (aEO) and a relative small increment in the concentration cEO,s (volume fraction φEO), resulting in high values of the thermodynamic correction factor Γ ) 1 + (∂ ln γEO)/(∂ ln φEO). This causes the effective diffusion coefficient to be 7-9 times larger than the diffusion coefficient for ideal behavior.24 If this correction is applied here, we see that the diffusion coefficient for diffusion in a glassy polymer is still 4-5 times lower than the diffusion coefficient as determined from the first stage of sorption. This difference is not yet understood. 4.1.6. Nonideality Effects of Clustering. The activity coefficient γEO of ethylene oxide in potato starch can be estimated from the sorption isotherm25 because pEO ) γEOφEOpoEO (see Figures 4-6). Thus

γEO )

pEO φEOpoEO

)

aEO aEO ) φEO kcEO,s

(13)

where k is a proportionality constant between the volume fraction, φEO, of ethylene oxide in the starch and the molar concentration cEO,s. An ideal solution with γEO f 1 should be obtained in the case in which aEO f 1 and φEO f 1. Thus, k ) aEO/ cEO,s at aEO f 1. Therefore

γEO )

( ) ( ) aEO cEO,s

aEO cEO,s

(14)

aEOf1

which can be determined from the sorption isotherm. Once the relation between γEO and cEO,s is known from the sorption isotherm, the average number of ethylene oxide molecules in a cluster can be calculated using eq 9.26 Figure 9 shows the results for various values of W and T. Again, three regions can be distinguished for W ) 9.9 wt % d.b. at T ) 313 K. In region IIa, mEO and therefore γEO are constant, but the interaction between ethylene oxide and the starch chains becomes stronger with increasing cEO,s, because the average cluster size, nEO, decreases with increasing cEO,s. This is due to increasing dissolution in the starch matrix relative to hole filling. In region IIb, the average cluster size is almost constant and minimal, corresponding to a maximum affinity between ethylene oxide and the starch chains. This is because all of the holes are filled in this region, and therefore, additional ethylene oxide can only dissolve in the polymer matrix, resulting in substantial swelling

Figure 9. Average number of ethylene oxide molecules in a cluster, nEO, as a function of cEO,s for various W and T. The division in regions is given for W ) 9.9 wt % d.b. and T ) 313 K.

due to plasticization. A further increase of cEO,s results in a rapid increase of nEO (region III), first building up clusters of ethylene oxide molecules, probably through multilayer sorption, but for cEO,s > 7 kmol‚m-3, clustering disappears again. Because of a lack of data points, however, the inaccuracy in ∂γEO/∂cEO,s is large for high cEO,s, resulting in uncertainty in the value of nEO. Thus, clustering might occur even for cEO,s > 7 kmol‚m-3. This growing affinity of starch for ethylene oxide is accompanied by a steep rise of mEO with increasing cEO,s. Possibly, hydrogen bonds between the starch chains in both the amorphous and the crystalline phases are broken down (gelatinization). If so, this would create new sorption sites, though at the expense of granular structure. The maximum cluster size appears where mEO is minimal. This maximum size increases and occurs at decreasing cEO,s with increasing W. This trend is probably due to water, which prevents contact between ethylene oxide and the starch chains at higher W and therefore promotes clustering of ethylene oxide molecules. At higher temperatures, clustering starts at lower values of cEO,s, as is also known for water clustering in the starch/water system.22 4.2. Diffusion Mechanisms of Ethylene Oxide in Semidry Potato Starch Granules. Figure 10a and b shows the observed u and/or D of ethylene oxide in potato starch as a function of cEO,s at T ) 313 K for W ) 9.9 and 7.7 wt % d.b. In both cases, the starch is initially glassy.9 The different diffusion mechanisms shown in Figure 2, are also visible in these two plots; i.e., with increasing cEO,s, the diffusion mechanism changes from Fickian in glassy starch to anomalous in glassy starch to case II to anomalous in rubbery starch. The values of χ for curves 1-3 in Figure 11 appear to satisfy the criterion 0.001 , χ < 10, which indicates that anomalous diffusion indeed occurs here.7 However, it should be noted that χ is relatively low for curve 3 (cEO,s ) 8.69 kmol‚m-3 d.b.), indicating that anomalous diffusion becomes more Fickian with increasing cEO,s. This is in agreement with Figure 2. 4.2.1. Anomalous Diffusion. According to Figure 2, the transition from Fickian to case II diffusion is characterized by anomalous diffusion. The corresponding sorption curves in this region were fitted with the model of Kuipers and Beenackers,7 which uses eq 3 to describe u. However, if anomalous diffusion were too close to case II or Fickian diffusion to note any improve-

Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6075 Table 3. Swelling Kinetics According to the Modified Model of Kuipers and Beenackers7 at T ) 313 K

a

W (wt %)

aEO rangea

Kw × 10-10 (m‚s-1)

K × 10-10 (m3n+1‚kmol-n‚s-1)

c/EO (kmol‚m-3 d.b.)

n

7.7 9.9

0.52-0.68 (5) 0.28-0.62 (8)

3.1 ( 0.2 4.9 ( 0.4

9.8 ( 0.7 8.8 ( 0.7

2.40 ( 0.06 1.32 ( 0.05

4.5 ( 0.4 3.7 ( 0.4

Value in parentheses gives the number of data points incorporated in the applicable aEO range.

Figure 11. Anomalous sorption curves at T ) 313 K and W ) 9.9 wt % d.b. fitted with the unmodified and modified model of Kuipers and Beenackers.7 Curves 1-3: fits with modified model of Kuipers and Beenackers7 assuming u to be constant in the region of anomalous diffusion (cEO,s > 2.7 kmol‚m-3). The parameters are shown in the figure. Curve 4: fit with unmodified model of Kuipers and Beenackers,7 assuming eq 3 to be valid also in the region of anomalous diffusion: cEO,s ) 8.69 kmol‚m-3, D ) 3.8 × 10-14 m2‚s-1, Kw ) 5.6 × 10-10 m‚s-1, K ) 5.86 × 10-13 m13‚kmol-4‚s-1, n ) 4, and c/EO ) 1.32 kmol‚m-3, so u ) 2.3 × 10-9 m‚s-1 and χ ) 0.96.

Figure 10. (a) Rate of ethylene oxide transport u and/or (b) D in potato starch for W ) 9.9 and 7.7 wt % d.b. as a function of cEO,s at T ) 313 K. Closed symbols correspond to anomalous behavior; open symbols to either pure case II or Fickian sorption. The various regions are indicated for W ) 9.9 wt % d.b. Drawn curve in part a: case II diffusion fitted with eq 15 (see Table 3). Dashed curve in parts a and b: anomalous diffusion with a concentrationindependent u.

ment in the fit results by applying this model, then the best value of either u or D was calculated. Anomalous diffusion is expected for 0.001 , χ ) D/uR < 10.7 In Figure 10a and b, u appears to be independent of cEO,s in region III. Apparently, the swelling (relaxation rate) has reached a maximum value. This is not in agreement with the model of Astarita and Sarti,6 and as a consequence, the model of Kuipers and Beenackers7 has to be adapted to allow for a constant u in this region. This was done for cEO,s ) 8.69 kmol‚m-3 d.b. Thus, the value of D was calculated assuming u ) 3.5 × 10-9 m‚s-1, which is the observed value for cEO,s ) 2.65, 3.47, and 8.69 kmol‚m-3 d.b. The related model predictions on the mass uptake of ethylene oxide are compared to the experimental results in Figure 11. 4.2.2. Case II Diffusion. Figure 12 shows the data in the region of case II diffusion for W ) 7.7 and 9.9 wt % d.b. at T ) 313 K. Those for W ) 9.9 wt % d.b. are fitted with the swelling kinetics of Astarita and Sarti (eq 3), with c/EO ) 1.3 kmol‚m-3 d.b., K ) 1.53 × 10-9 m5.74‚kmol-1.58‚s-1, and n ) 1.58 (curve 1 in Figure 12). This fit is not very good, probably because of plasticization, which causes preswelling and, consequently, causes the case II front velocity to deviate from zero for cEO,s )

Figure 12. Region of case II diffusion of ethylene oxide in potato starch for W ) 7.7 and 9.9 wt % d.b. at T ) 313 K. Curve 1: eq 3 with K ) 1.53 × 10-9 m3n+1‚kmol-n‚s-1, ca,r)r0 ) cEO,s, c/a ) c/EO ) 1.3 kmol‚m-3, and n ) 1.58. Table 3 lists the parameters for curves 2.

c/EO. We therefore propose a modified version of eq 3 that takes into account this preswelling

u ) Kw + K(cEO,s - c/EO)n u ) Kw

for cEO,s > c/EO

for cEO,s ) c/EO

(15)

with Kw depending on W. The fits for u, curves 2 and 3 in Figure 12, are in good agreement with the experimental data (see Table 3 for the fit results). 4.2.2.1. Influence of W. Figure 12 shows that, at the start of the case II diffusion region, there is a cEO,s range for which the swelling rate is almost concentrationindependent. It also shows that this penetrant concentration range starts at lower cEO,s with increasing W.

6076 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003

Figure 13. Parity plot of case II diffusion of ethylene oxide in potato starch according to the modified Astarita and Sarti eq 15 using eq 17.

The latter point means that c/EO decreases with increasing W. In addition to the Kw values presented in Table 3, also a value for W ) 5.9 wt % d.b. at T ) 313 K could be estimated: Kw ) 5.17 × 10-11 m‚s-1. Using this estimated value, the dependence of Kw on W could be determined. It appears to satisfy an Astarita and Sarti type of equation for 5.9 e W e 9.9 wt % d.b. at T ) 313 K

Kw ) 2.14 × 10-10(W - 5.8)0.58

Figure 14. Diffusivity of ethylene oxide as a function of the ethylene oxide solubility, cEO,s, in potato starch at various W and T. Open and closed symbols correspond to true Fickian diffusion and to anomalous behavior, respectively.

for cEO,s > c/EO (16)

Thus, no case II diffusion is expected for W e 5.8 wt % d.b. at T ) 313 K. Indeed, no case II diffusion region was found for W ) 4.4 wt % d.b. K, c/EO, and n in eq 15 appear to depend on W according to

n ) 7.0 - 0.33W c/EO

Figure 15. D/Dmax as a function of cEO,s/cmax EO,s for various T and W with T g 313 K and W g 9.9 wt % d.b. The curves show the fits of eqs 18 and 19.

) 6.5 - 0.53W

K ) 8.7 × 10-10 +

8.6 × 1011 W 24.6

(17)

Figure 13 shows a parity plot of the modified model of Astarita and Sarti, using eqs 15 and 17. This model appears to be satisfactory. 4.2.2.2. Influence of T. Figure 12 shows the observed u values for case II diffusion. Assuming a temperature dependence of u according to an Arrhenius law, then the activation energy for front propagation, Eu, increases from 43 kJ‚mol-1 at cEO,s ) 1.6 kmol‚m-3 to 110 kJ‚mol-1 at cEO,s ) 2.45 kmol‚m-3. These orders of magnitude agree with those found by Gall and Kramer.27 4.2.3. Fickian Diffusion. At the boundary between regions IIa and IIb in Figure 10, potato starch starts to change from the glassy state to the rubbery state through the uptake of additional ethylene oxide. However, in region III, up to aEO ) 0.9, no Fickian diffusion is observed. Fickian diffusion is observed in region I only, i.e., in the glassy starch. This, however, does not mean that there is no Fickian diffusion region for aEO > 0.9. Figure 14 shows the observed value of D as a function of cEO,s for W ) 18.3 and 14.2 wt % d.b. at T ) 313 K (in both cases, the potato starch is rubbery).

In agreement with Figure 2, a concentration-independent diffusion coefficient is observed for W ) 14.2 wt % d.b. when cEO,s < 1.5 kmol‚m-3. No concentrationindependent D is observed for W ) 18.3 wt % d.b. The observed initial increase in D is accompanied by a decrease in cluster size (see Figure 9), indicating that the ethylene oxide/potato starch interactions are favored, i.e., localized sorption due to strong interactions with the polymer chains, resulting in plasticization. With a further increase of cEO,s, the interactions between the ethylene oxide molecules become more predominant, causing clusters to form and, consequently, D to decrease (see Figure 9). 4.2.3.1. Influence of T. There seems to exist a maximum value of D as a function of cEO,s for W ) 9.9 wt % d.b. at T ) 353 K (see Figure 14). This maximum, Dmax (in m2‚s-1), depends on W and T according to

(

Dmax ) 5.4 × 10-13 exp 0.22W -

)

15.9 × 103 RT

(18)

The ethylene oxide concentration for which Dmax occurs, cmax EO,s, depends linearly on both W and T, i.e., the higher the degree of swelling, the lower the value of cmax EO,s

Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6077

(

)

cEO,s D/Dmax ) 1.64 exp -0.34 max cEO,s

if

cEO,s/cmax EO,s g 1 (21)

When potato starch was still in the glassy state after the system has attained physical equilibrium, D was found to depend on cEO,s, W, and T according to the following equation (see Figure 16)

D)

(

2.12 × 10-8 exp 0.48W + 0.51cEO,s -

Figure 16. Parity plot of the experimental D and the fit according to eq 22 for Fickian diffusion in glassy starch. T e 313 K and W e 9.9 wt % d.b.

cmax EO,s ) 28.9 - 0.071T - 0.33W

(19)

Equations 18 and 19 are valid above Tg only, i.e., for W g 11 wt % d.b. and 313 e T e 353 K or for W < 11 wt % d.b., cEO,s g c/EO, and 313 e T e 353 K. These equations can be used to make a dimensionless plot of D/Dmax as a function of cEO,s/cmax EO,s (see Figure 15). Note that D/Dmax > 1 because of inaccuracies in the fit of Dmax. The fit equations are

(

)

cEO,s D/Dmax ) 0.19 exp 1.58 max cEO,s

if

cEO,s/cmax EO,s < 1 (20)

)

57.2 × 103 RT (22)

for 5.9 e W e 9.9 wt % d.b., 293 e T e 313 K, and cEO,s < c/EO. 5. Conclusions/Description of Physical Interactions On the basis of the sorption and diffusion results as described above, as well as the chemical kinetics of the uncatalyzed and catalyzed hydroxyethylation of semidry potato particles starch published elsewhere,2 a schematic summary of the physical behavior of ethylene oxide in semidry glassy potato starch is given in Figure 17. The indicated boundaries, especially between region I and II, are not as sharp as suggested by the diagram, because most transitions proceed fairly continuously over the range of aEO. Four regions can be distinguished with increasing aEO. Initially, the starch takes up ethylene oxide molecules that are strongly sorbed and immobilized in existing holes and for a smaller part also between chains (solution according to Henry’s law) by Fickian diffusion in the glassy starch. Minor swelling occurs in the starch

Figure 17. Sorption, diffusion, and reaction characteristics of ethylene oxide in semidry starch as a function of aEO for W ) 9.9 wt % d.b. at T ) 312 K.

6078 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003

whose structure remains rigid, causing the number of (re)active sites to remain constant (region I). The holes are saturated for c′H ) 0.92 (kmol‚m-3 d.b.). This corresponds to a molar hole saturation concentration [in kmol‚(kmol of AGU)-1, where AGU stands for anhydroglucose unit, i.e., the monomer unit of the starch molecule) of

MAGU 162 c′′H ) c′H ) 0.92 ) 0.089 F 1500

(23)

for starch with a moisture content W of 9.9 wt % d.b. A moisture content W of 9.9 wt % d.b corresponds to a molar moisture content [in kmol‚(kmol of AGU)-1] of

MAGU 162 ) 0.891 ) 9.9 W′′ ) W 100MH2O 100 × 18

starch/ethylene oxide solution. With increasing W and/ or increasing T, the starch becomes more swollen, which means that region I becomes smaller relative to regions II and III. Acknowledgment These investigations were financially supported by AVEBE, b.a. Veendam, the Dutch Carbohydrate Research Foundation (IOP-K, ‘s Gravenhage), The Netherlands Foundation for Chemical Research (SON), and the Technology Foundation (STW, Utrecht, The Netherlands). We thank Mr. J. van de Meer and Mrs. R. Ziengs of AVEBE Research Laboratory, Foxhol, The Netherlands, for advice and for performing part of the analytical work.

(24)

The total molar concentration of both plasticizers, ethylene oxide and water, therefore equals 0.98 kmol‚ (kmol of AGU)-1. Thus, stoichiometrically, these sorption sites can be identified as the anhydroglucose units of the starch. Without the presence of ethylene oxide, it takes 11 wt % water to obtain such a stoichiometrical sorption,10 i.e., to obtain W′′ ) 0.99 kmol‚(kmol of AGU)-1. Once the holes are saturated, additional class II ethylene oxide plasticizes the initially rigid starch structure, thus increasing the main mobility of both starch and ethylene oxide molecules. Most of the ethylene oxide sorbs in the vicinity of region I molecules in new holes or at newly created sites, both of which are created by the swelling and gradual weakening of the starch structure. Upon swelling, the number of interactions between starch chains decreases, thus favoring ethylene oxide-starch interactions. This is also shown by the increasing relative reaction rate REO/cEO,s with increasing penetrant concentration. The plasticization is increased further by the breakage of interstarch hydrogen bonds by ethylene oxide and its replacement by weak starch-ethylene oxide bonds, which leads to substantially increased molecular mobility and a reduction of the glass transition temperature. This explains why the swelling rate u increases with increasing aEO (region IIb). In region Ia, the internal pressure buildup by ethylene oxide is apparently too low for chain breakage to take place. In region III, ethylene oxide consists of almost freely moving penetrant molecules. Initially, a termination of the plasticization of the amorphous polymer structure is expected, but beyond this value, a starch solution is created. At about this level, all further ethylene oxide is taken up by the amorphous parts, which are now likely to behave as swelling gels. Ethylene oxide in region III behaves very similarly to free ethylene oxide, and consequently, its interaction with starch must be weak. Therefore, not all molecules in contact with the starch are available to react. This explains why the relative reaction rate REO/cEO,s decreases with increasing aEO. The ethylene oxide molecules appear to be mechanically entrapped in the void spaces of the now fully weakened amorphous parts of the swollen starch network. The isotherm for ethylene oxide sorption rises steeply in this region. This can be explained by clustering of ethylene oxide molecules to chains of ethylene oxide that extend from the H-bonding sites on the starch and grow in size up to the compatibility limit of the

Nomenclature Latin Symbols aEO ) activity of ethylene oxide: aEO ) pEO/poEO b ) hole affinity constant, bar-1 c ) total concentration of sorbate, kmol‚m-3 c′ ) penetrant concentration adsorbed when all sites contain one molecule (monolayer), kmol‚m-3 cB ) adsorption constant used in the BET equation c/EO ) threshold concentration marking the transition from anomalous to case II diffusion, kmol‚m-3 cEO ) threshold concentration marking the transition from Fickian to anomalous diffusion, kmol‚m-3 cEO,r)r0 ) concentration of ethylene oxide at the position of the swelling front, kmol‚m-3 cEO,g ) concentration of ethylene oxide in the gas phase, kmol‚m-3 cEO,s ) (equilibrium) solubility concentration of ethylene oxide in the starch, kmol‚m-3 max cEO,s ) concentration of ethylene oxide in the (rubbery) starch at maximum D, kmol‚m-3 c′H ) Langmuir hole saturation constant, kmol‚m-3 c′′H ) Langmuir hole constant, kmol‚(kmol of AGU)-1 ci ) initial equilibrium surface concentration for two-stage sorption, kmol‚m-3 c∞ ) final equilibrium surface concentration for two-stage sorption, kmol‚m-3 D ) diffusion coefficient of ethylene oxide in granular potato starch, m2‚s-1 Dmax ) maximum diffusion coefficient of ethylene oxide in granular (rubbery) starch, m2‚s-1 Eu ) activation energy for propagation of the swelling front, kJ‚mol-1 Fo ) dimensionless Fourier number () Dt/R2) H ) Henry’s law constant of ethylene oxide in glassy starch, kmol‚s2‚kg-1‚m-2 k ) proportionality constant, k ) φEO/cEO,s, m3‚mol-1 K ) swelling constant in the power-law rate equation of a swelling front, m3n+1‚kmol-n‚s-1 K# ) swelling constant in the power-law rate equation of a swelling front, m‚s-1 Kw ) preswelling constant in the power-law rate equation of a swelling front, m‚s-1 MAGU ) molecular mass of one monomer (anhydroglucose) unit of starch () 162), kg‚kmol-1 M1 ) amount of ethylene oxide sorbed by starch at (quasiequilibrium) time t1, kg mEO ) distribution coefficient of ethylene oxide in starch, mEO ) cEO,s/cEO,g at equilibrium Mt ) amount of ethylene oxide sorbed by starch at time t, kg M1 ) amount of ethylene oxide sorbed by starch at quasi equilibrium time t1, kg

Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6079 M∞ ) amount of ethylene oxide sorbed by starch at equilibrium, kg n ) exponent in the power-law rate equation for a swelling front, n g 0 n ) positive integer in eq 1 nEO ) average number of ethylene oxide molecules in a cluster p ) partial pressure of penetrant, bar pEO ) partial pressure of ethylene oxide, bar poEO ) saturation pressure of ethylene oxide, bar R ) average radius of the starch granules, m R ) gas constant, m2‚s-2‚K-1 r ) position within the starch granule, m r0 ) position of the swelling front, m t ) time, s t1 ) time at quasi-equilibrium for two stage sorption, s T ) absolute temperature, K Tg ) glass transition temperature of starch, K u ) velocity of the swelling front, u ) -dr0/dt W ) moisture content of the starch, wt % d.b. W ′′ ) moisture content of starch, kmol of water‚(kmol of AGU)-1 Greek Symbols R ) two-stage sorption parameter β ) parameter characterizing relaxation of surface concentration, s-1 γi ) activity coefficient of penetrant i in the solid γEO ) activity coefficient of ethylene oxide in the potato starch χ ) dimensionless ratio of the diffusion rate to the velocity of the swelling front, χ ) D/uR φi ) volume fraction of penetrant i absorbed φEO ) volume fraction of ethylene oxide in the potato starch F ) density of the starch () 1500), kg‚m-3 Γ ) thermodynamic factor [) 1 + (∂ ln γEO)/(∂ ln φEO)]

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(8) Kuipers, N. J. M. Gas-solid hydroxyethylation of potato starch, Ph.D. Thesis, University of Groningen, Groningen, The Netherlands, 1995. (9) van den Berg, C. Vapour sorption equilibria and other water-starch interactions: A physico-chemical approach. Ph.D. Thesis, Agricultural University of Wageningen, Wageningen, The Netherlands, 1981. (10) Brunauer, S. The Physical Adsorption of Gases and Vapours; Oxford University Press: London, 1945. (11) Barrie, J. A.; Machin, D. J. Diffusion and association of water in some poly(alkyl methacrylates) 1. Equilibrium sorption and steady-state permeation. Trans. Faraday Soc. 1971, 67 (1), 82, 244-256. (12) Bagley, E.; Long, F. A. Two-stage sorption and desorption of organic vapors in cellulose acetate. J. Am. Chem. Soc. 1955, 77, 2172-2178. (13) Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon: Oxford, U.K., 1975. (14) Long, F. A.; Richman, D. Concentration gradients for diffusion of vapors in glassy polymers and their relation to time dependent diffusion phemomena. J. Am. Chem. Soc. 1960, 82, 513-519. (15) Sada, E.; Kumazawa, H.; Yukashiji, H.; Bamba, Y.; Sakata, K.; Wang, S. T. Sorption and diffusion of gases in glassy polymers. Ind. Eng. Chem. Res. 1987, 26, 433-438. (16) Brunauer, S.; Emmett, P. H.; Teller, E. J. Adsorption of gases in multimolecular layers. J. Am. Chem. Soc. 1938, 60, 309319. (17) Zimm, B. H.; Lundberg, J. L. Sorption of vapours by high polymers. J. Phys. Chem. 1956, 60, 425. (18) Vieth, W. R. Diffusion in and through Polymers: Principles and Applications; Oxford University Press: New York, 1991. (19) Khamrakulov, N. J.; Myagkova, N. V.; Budtov, V. P. Water sorption and diffusion in cellulose and cellulose acetates. Polym. Sci. Ser. 1993, B36 (5), 699-701. (20) Starkweather, H. W. Clustering of water in polymers. J. Polym. Sci. 1963, B1 (3), 133-138. (21) Carillo, P. J.; Gilbert, S. G.; Daum, H. Starch/solute interaction in water sorption as effected by pretreatment. J. Food Sci. 1988, 1199-1203. (22) Scandola, M.; Ceccorulli, G.; Pizzoli, M. Water clusters in elastin. Int. J. Biol. Macromol. 1981, 3, 147-149. (23) Kirk-Othmer Encyclopedia of Chemical Technology; J. Wiley and Sons: New York, 1985. (24) Park, G. S. Transport PrinciplessIntramembrane Phenomena; Advanced Study Institute on Synthetic Membranes, North Atlantic Treaty Organization: Alcabideche, Portugal, Jun 26, 1983. (25) Moore, W. J. Physical Chemistry, 5th ed., Longman Group Limited: London, 1985. (26) Starkweather, H. W. Clustering of solvents absorbed in polymers. In Structure-Solubility Relationships in Polymers; Harris, F. W., Seymour, R. B., Eds.;Academic Press: New York, 1977; pp 21-31. (27) Gall, T. P.; Kramer, E. J. Diffusion of deuterated toluene in polystyrene, Polymer 1991, 32, 265-271.

Received for review November 25, 2002 Revised manuscript received August 5, 2003 Accepted August 25, 2003 IE020944S