Diffusion and Saponification Inside Porous Cellulose Triacetate Fibers

Biomacromolecules 2005, 6, 152-160

152

Diffusion and Saponification Inside Porous Cellulose Triacetate Fibers Jennifer L. Braun† and John F. Kadla*,†,‡ College of Natural Resources, North Carolina State University, Raleigh, North Carolina 27695, and Biomaterials Chemistry Group, Faculty of Forestry, University of British Columbia, Vancouver, British Columbia, V6T 1Z4 Received June 18, 2004; Revised Manuscript Received September 7, 2004

Cellulose triacetate (CTA) fibers were partially hydrolyzed in 0.054 N solutions of NaOH/H2O and NaOMe/ MeOH. The surface concentration of acetyl groups was determined using ATR-FTIR. Total acetyl content was determined by the alkaline hydrolysis method. Fiber cross-sections were stained with Congo red in order to examine the interface between reacted and unreacted material; these data were used to estimate the rate constant k and effective diffusivity DB for each reagent during the early stages of reaction by means of a volume-based unreacted core model. For NaOH/H2O, k ) 0.37 L mol-1 min-1 and DB ) 6.2 × 10-7 cm2/sec; for NaOMe/MeOH, k ) 4.0 L mol-1 min-1 and DB ) 5.7 × 10-6 cm2/sec. The NaOMe/MeOH reaction has a larger rate constant due to solvent effects and the greater nucleophilicity of MeO- as compared to OH-; the reaction has a larger effective diffusivity because CTA swells more in MeOH than it does in water. Similarities between calculated concentration profiles for each case indicate that the relatively diffuse interface seen in fibers from the NaOMe/MeOH reaction results from factors not considered in the model; shrinkage of stained fiber cross-sections suggests that increased disruption of intermolecular forces may be the cause. 1. Introduction Cellulose is the most abundant natural polymer.1 Cellulose and its derivatives are widely utilized in textiles, packaging materials, films, membranes, and engineered thermoplastics. The utilization of cellulose is however limited due to poor solubility in most solvents and the fact that cellulose decomposes prior to melting. As a result, cellulose is derivatized, typically esterified to facilitate processing, e.g., molding or fiber spinning. Cellulosic products can then be desubstituted to varying degrees in order to recover some of the properties of native cellulose. Selective hydrolysis of surface acetyl groups can change surface reactivity without significantly altering the tensile properties of the fibers and films. Alternatively, acetyl groups can be removed throughout the material in order to produce “regenerated” cellulose. Understanding the desubstitution reactions is critical to the development of cellulosic materials with unique surface properties. Saponification (Figure 1) is a three-step process.2 Initially, the base attacks the ester; this step is slow and reversible. However, the next two steps are fast and irreversible, so the rate of the overall process is determined by the * To whom correspondence should be addressed. Address: 4034 Forest Science Centre, University of British Columbia, Vancouver, BC, Canada V6T 1Z4. Telephone: (604) 827-5254. Fax: (604) 822-9104. E-mail: [email protected]. † College of Natural Resources, North Carolina State University. ‡ Biomaterials Chemistry Group, Faculty of Forestry, University of British Columbia.

Figure 1. Saponification.2

first step. Consequently, the rate of consumption of acetyl groups may be represented by eq 1.3 ∂A ) -kAB ∂t

(1)

where A and B are, respectively, the concentrations of acetyl groups and base at a given location within the fiber and k is the rate constant. Equation 1 has been written as a partial differential equation because the distribution of A varies radially within the fiber as the reaction progresses. The rate of change of B within the fiber can be determined from a material balance for a cylindrical shell with radius r (see appendix).4 B in - B out - consumption of B ) rate of change of B (2) More specifically

[(

]

∂2B 1 ∂B ∂B + 2 - kAB ) DB ∂t r ∂r ∂r

)

where DB refers to the effective diffusivity.

10.1021/bm0496413 CCC: $30.25 © 2005 American Chemical Society Published on Web 11/13/2004

(3)

CTA Saponification Kinetics

Biomacromolecules, Vol. 6, No. 1, 2005 153

Therefore, through the controlled desubstitution of preformed cellulosic products cellulosic materials with specific properties can be produced. The current study examines the effect of different saponification reagents on the distribution of acetyl groups in a commercial cellulose triacetate fiber using ATR-FTIR and stained fiber cross-sections to develop kinetic models for the distribution of acetyl groups in the fiber at a given time. 2. Experimental Procedures 2.1. Sample Preparation. The starting material was a wetspun cellulose triacetate fiber (DS ) 2.95 ( 0.08)5 that contained 33 wt % ethyl citrate plasticizer and low degree of crystallinity (determined by DSC analysis). The plasticizer was extracted by immersion in methanol (15 mL/100 mg of fiber) for 2 days. The methanol-swollen fibers were dried in air for several days and then transferred to a vacuum desiccator where they were dried until they attained a constant mass (about 4 days). 2.2. Hydrolysis. 4.0 cm samples of dried fiber were weighed; fiber mass varied from 4 to 10 mg. The dried fibers were preswollen overnight in 20 mL portions of methanol. The swollen fibers were transferred to 20 mL portions of 0.05 N NaOMe/MeOH for alkaline hydrolysis, with continuous stirring. The reaction was interrupted by transferring the fibers back into 20 mL portions of methanol and allowing them to sit for 24 h in order to extract the reaction products and excess reagent. Afterward, the fibers were removed and then dried under vacuum for several days until they attained a constant mass. The preceding procedures were repeated for the alkaline hydrolysis in 0.05 N NaOH/H2O; in this case, water was used instead of MeOH for the preswelling and extraction steps. For each system multiple (10-20) replicates were performed. 2.3. Fiber Characterization. ATR-FTIR was performed on the partially hydrolyzed fibers using a Thermonicolet Nexus 670 FTIR with Smart Omnisampler accessory. The samples had been swollen in MeOH prior to measurement in order to make them pliable enough for good contact with the ATR crystal. Depth of scan was 0.66 µm. Total acetyl content of the partially desubstituted fibers was determined by the alkaline hydrolysis method.5 Samples were cut into very small pieces and incubated at 55 °C in 500 µL portions of water for 30 min. 500 µL of 0.5 N NaOH were added to each sample, and then incubation at 55 °C was continued for another 15 min. Next, the samples were removed from the heat and allowed to sit at room temperature for 4 days. After the room-temperature incubation, the sample solutions were titrated with 0.012 N HCl, using phenol red indicator. Fiber samples were embedded in methacrylate resin (Spurr Low-Viscosity Embedding Media) and cut into 1.5 µm thick cross-sections. The sections were mounted on glass microscope slides and stained for 30 min with a 0.1 wt % aqueous solution of Congo red, followed by a brief rinse with water to remove the excess stain. 2.4. Kinetic Model. Appendix A2 shows how eqs 1 and 3 can be used to calculate the distributions of A and B in the

Figure 2. ATR-FTIR scans of CTA fibers after incubation in 0.054 N NaOMe/MeOH.

Figure 3. ATR-FTIR scans of CTA fibers after incubation in 0.054 N NaOH/H2O.

fiber at any given time, provided DB, k and the boundary condition BS are known. These values were calculated indirectly from the total acetyl content of the fiber and measurements from ATR-FTIR and stained fiber crosssections, as illustrated in Appendix A3. 3. Results and Discussion 3.1. Experimental Results. Figures 2 and 3 show ATRFTIR scans for the NaOMe/MeOH and NaOH/H2O reactions. The peaks at 1700-1800, 1350-1400, 1200-1300, and 1000-1150 cm-1 correspond respectively to carbonyl stretching, methyl deformations, acetate C-C-O stretching and C-O stretching.6 Some contamination of the methyl and C-O regions is expected because the fiber samples had been swollen in MeOH to make them pliable enough for good contact with the ATR crystal. The alcohol and methyl peaks in the 2800-3600 cm-1 region were also affected, so that part of the spectrum has been omitted. The carbonyl peak was used to monitor the reaction because it is not affected by the presence of the solvent and does not overlap with peaks from the other functionalities present on the fiber. Figure 4 illustrates the reaction kinetics near the surface for both reagent systems. Concentration values calculated from ATR-FTIR data for the NaOMe/MeOH reactions indicate pseudo-first-order kinetics; in other words, BS, the surface concentration of base, is approximately constant with

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Braun and Kadla

Figure 4. Reaction kinetics at the fiber surface for (a) NaOH/H2O and (b) NaOMe/MeOH.

Figure 7. Variation of swelling with conversion. Table 1. Physical Properties Related to Fiber Swelling

V/VCTA

ω

F (g/cm3) H2O MeOH H2O MeOH CTA cellulose water methanol

1.3 1.5 1.000 0.791

1.15 1.90

1.30 1.52

ω)

time.3 In this case, eq 1 can be integrated to produce eq 4, which is linear. AS ) kMeONaBSt AS0

MeOH

0.031

0.018 0.027

m L + mP mP

(5)

where mL and mP refer, respectively, to the mass of liquid retained and the dry mass of the polymer fiber. The quantity ω is nearly constant throughout the NaOMe/MeOH reaction, but varies with conversion during the NaOH/H2O reaction due to the increasing hydrophilicity of the polymer. The quantity ω can be used to estimate V, the volume of the swollen fiber, if the volume of polymer VP and the volume of retained liquid VL are assumed to be additive.

Figure 6. Total conversion and conversion at the surface.

( )

1.00 0.68

As the reaction progresses, hydrogen bonds form between oxygen on the pyranose rings and the newly formed hydroxyl groups. These hydrogen bonds may form crystallites around acetyl groups, thus isolating them from the reagent.7 Figure 7 shows a plot of ω versus conversion. ω represents the ratio of swollen fiber mass to dry fiber mass; in other words

Figure 5. Conversion at the surface vs total conversion.

-ln

1.00 0.97

BS (mol/L) H2O

(4)

where kMeONaBS is the slope. - ln(AS/AS0) vs t is never linear for the NaOH/H2O system, most likely due to swelling of the newly forming cellulose as the reaction progresses; consequently, BS is not constant for this reaction. Figure 5 shows that the differences between the overall conversion and the conversion at the surface are much greater for NaOH/H2O than for NaOMe/MeOH; this indicates that the former has a much greater barrier to diffusion. The barrier has a very great impact on the time required for the reaction to reach its ultimate extent, as is illustrated by Figure 6. In this particular study, conversion never exceeded ∼0.95, possibly due to the inaccessibility of certain acetyl groups.

V ) V L + VP )

[

mL mP (ω - 1) 1 + ) mP + FL FP FL FP

]

(6)

where FL and FP refer, respectively, to the solvent and dry polymer densities. Table 1 lists relative values of V for regenerated (∼95% converted) cellulose and unreacted CTA in H2O and MeOH. V/VCTA ≈ 1 for regenerated cellulose in H2O; this indicates that the volume of solvent gained during the reaction is nearly equal to the volume of polymer lost by removal of the acetyl groups. V/VCTA < 1 for regenerated cellulose in MeOH, which indicates that swelling is not sufficient to offset the loss of polymer volume in this case. Figure 8 shows fiber cross-sections that have been stained with Congo red dye. Congo red has azo groups that form hydrogen bonds with hydroxyl groups, so the reacted material is stained red.8 The interface between reacted material (‘ash’) and unreacted material (‘core’) is sharp for the NaOH/H2O series and diffuse for the NaOMe/MeOH series. The latter



Figure 8. Stained cross-sections of CTA after incubation in NaOH/ H2O [(a) 3 h; (b) 24 h; (c) 48 h] and NaOMe/MeOH [(d) 10 min; (e) 30 min; (f) 80 min]. Scale bars indicate 100 µm. The NaOMe/MeOH series had a tendency to contract and become detached from the resin, especially at longer incubation times.

Figure 9. Best fit of calculated and experimental values of pA. Table 2. Calculation Results reaction

method

NaOH/H2O cross-section NaOMe/MeOH ATR-FTIR NaOMe/MeOH cross-section

k (L mol-1 min-1) 0.37 2.4 4.0

φ′v

DB (cm2/sec)

7.16 6.2 × 10-7 7.1 × 10-6 7.88 5.7 × 10-6

also had a marked tendency to shrink and pull away from the embedding resin. This may indicate that more disruption of intermolecular forces occurs during the NaOMe/MeOH reaction than during the NaOH/H2O reaction. 3.2. Calculation Results. Figure 9 shows the best fit of calculated and experimental values of pA, the total acetyl fraction, for both reagent systems. The calculations are outlined in Appendix A3. Values of k, φ′V, and DB are listed in Table 2. Both calculation methods used for the NaOMe/ MeOH case produced comparable results. The small dis-

Figure 10. NaOH/H2O concentration profiles of (a) acetyl groups and (b) base calculated from stained fiber cross-sections.

crepancies most likely result from the slight swelling that occurred during the reaction. Swelling may have contributed to error in the ATR-FTIR method, although Figure 4 indicates that this was not a problem in the early part of the NaOMe/MeOH reaction. For the other method, the interface between converted and unreacted material was measured from dry fiber cross-sections, so the relative dimensions of the ash and core may be different from those of the swollen state, since the converted material tended to swell more than the unreacted material did. Again, this was not a major problem for the NaOMe/MeOH reaction, but it may have caused spuriously low values of k and DB for the NaOH/ H2O reaction. The sharper decrease of the NaOMe/MeOH points in Figure 9 is as much a consequence of diffusion as of reaction. The rate constant and effective diffusivity for the NaOMe/ MeOH system are both about 10 times greater than the corresponding values for the NaOH/H2O system. The DB values differ because CTA swells more in methanol than it does in water. Note that the data for the NaOH/H2O reaction were collected in the early stage of the reaction, when only the layers near the surface were affected to a significant degree. DB values from late stages of the NaOH/H2O reaction are likely to be much higher due to the increasing content of cellulose, which is hydrophilic and, consequently, swells in water. The difference between the rate constants is due to (1) the greater ionic strength of MeO- and (2) solvent effects. The dielectric constants for H2O and MeOH are 78 and 32.7, respectively; because MeOH is less polar, charge dispersion is more favorable.9,10 Charge dispersion helps to stabilize the ion formed in the first step of the reaction mechanism (Figure 1); this increases the reaction rate by shifting the equilibrium to the right. The rate constant for

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Braun and Kadla

Figure 12. Halved fiber cross-section illustrating the location and dimensions of the shell used to derive equation (A2).

the NaOMe/MeOH reaction are thought to originate from factors not considered in the model, most likely increased disruption of intermolecular forces, as indicated by shrinkage of stained fiber cross-sections.

Figure 11. NaOMe/MeOH concentration profiles of (a) acetyl groups and (b) base calculated from stained fiber cross-sections.

the NaOH/H2O reaction is lower than values reported for the saponification of methyl acetate (∼9 L mol-1 min-1) and ethyl acetate (∼6 L mol-1 min-1).11 Steric hindrance is likely the major reason for this, since most of the reaction volume is occupied by polymer. Figures 10 and 11 show concentration profiles from calculations of A and B for hypothetical fibers with mass ) 7.00 mg. The values of k and DB from Table 2 were used in the calculations. Both figures show that the reaction is pretty much limited to regions near the surface during the early stages; this is consistent with the data from the stained fiber cross-sections shown in Figure 8. The similarities between the profiles calculated for the NaOH/H2O and NaOMe/ MeOH reactions suggest that the relatively diffuse interface seen in fiber cross-sections from the latter results from factors not considered by the model, for example, the increased disruption of intermolecular forces indicated by Figure 8. 4. Conclusions ATR-FTIR, stained fiber cross-sections and total acetyl content indicate that saponification of CTA fibers occurs more rapidly in NaOMe/MeOH than in NaOH/H2O. Calculations from a volume-based unreacted core model indicate that the rate constant and effective diffusivity for the NaOMe/ MeOH system are both about 10 times greater than the corresponding values for the NaOH/H2O system. The former has a larger rate constant due to solvent effects and the relatively higher nucleophilicity of MeO- as compared to OH-; its effective diffusivity is higher in the early stages of the reaction because CTA swells more in MeOH than it does in water. Calculated concentration profiles for each case are similar, so the relatively diffuse interface seen in fibers from

Acknowledgment. The authors greatly acknowledge the Charles A. and Anne Morrow Lindbergh Foundation for financial support of this work. We would also like to thank Prof. Glenn Lipscomb (Department of Chemical Engineering, University of Toledo), Prof. George Roberts (Department of Chemical Engineering, North Carolina State University) and Prof. Richard Gilbert (Department of Wood and Paper Science) for their very helpful advice. Appendices A1. Material Balance for Component B. The rate of change of B within the fiber can be determined from a material balance for a cylindrical shell with radius r (Figure 12).4 B in - B out - consumption of B ) rate of change of B (A1) More specifically 2π(r + ∆r)LNBr|r+∆r - 2πrLNBr|r - 2πr∆rLkAB ) ∂B (A2) 2πr∆rL ∂t

( )

where NBr refers to the molar flux of B along a radius at a distance r from the center of the fiber (the fibers are relatively long, so flux through the ends may be neglected). Division by 2π∆rL and letting ∆r f 0 produces eq A3. lim

∆rf0

[

]

(

)

rNBr|r+∆r - rNBr|r ∂ ∂B ) (rNBr) ) r kAB + (A3) ∆r ∂r ∂t

No attempt will be made to describe diffusion inside the tortuous void passages of the fiber; instead, an “effective diffusivity” DB is defined.4 NBr ≡ DB

(∂B∂r )

(A4)



Substitution of eq A4 into eq A3 produces eq A5.

[(

]

∂B ∂2B 1 ∂B + 2 - kAB ) DB ∂t r ∂r ∂r

)

(A5)

A2. Numerical Solution of Material Balances for Components A and B. Equations 1 and A5 can be solved numerically using an implicit method. In this case Bj+1,i - Bji ) ∆t Bj+1,i+1 - 2Bj+1,i + Bj+1,i-1 1 Bj+1,i+1 - Bj+1,i-1 + DB ri 2∆r (∆r)2

[(

]

)

kAjiBji (A6) where i ) 0, 1, 2, ..., m + 1 and j ) 0, 1, 2, ..., n + 1 represent increments along the r and t axes, respectively.12,13 Equation A6 can be rearranged to produce eq A7. βi-Bj+1,i-1 + γBj+1,i + βi+Bj+,i+1 ) RjiBji βi- )

∆r 1 -1) -1 2ri 2i

(A9)

∆r 1 -1)- -1 2ri 2i

(A10)

2

Rji )

(A8)

(∆r)2 DB∆t

γ)2+ βi+ ) -

(A7)

(∆r) 1 - kAji DB ∆t

[

]

Table 3. Required Input for Solution of Eqs A14 and A15

(A11)

At the surface of the fiber, r ) R, i ) m + 1 and Bj+1,m+1 ) BS

(A12)

where BS refers to the concentration of base in the surface layer. At the center of the fiber, r ) 0, i ) 0 and

[

Bj+1,0 ≈ Bj+1,1

][ ] [ ]

1 -1 Bj+1,0 0 β1 γ β1+ Rj1Bj1 Bj+1,1 Rj2Bj2 Bj+1,2 β2- γ β2+ - - ) - - - - Rj,mBj,m βm- γ βm+ Bj+1,m BS B j+1,m+1 1 (A14)

Equation A14 can be solved very efficiently using Gaussian elimination. However, Aj+1,i must be calculated after every increase in j. In finite difference form, eq 1 becomes

which can be solved for Aj+1,i.

symbol

identity

source

t L V BS A0

measured directly measured directly dry and swollen fiber masses polymer and solvent densities fiber mass and swollen volume

k

total reaction time length of swollen fiber volume of swollen fiber surface concentration of B initial concentration of A at ri initial concentration of B at ri rate constant

DB φ′v ∆r ∆t

effective diffusivity Thiele modulus radius increment time increment

B0

(A13)

Equations A7, A12, and A13 can be expressed in the matrix format.13

Aj+1,i - Aji ) -kAj+1,iBj+1,i ∆t

Figure 13. Program flow chart.

(A15)

none present prior to reaction guess or ATR-FTIR and swelling data guess (see text) stained fiber cross-sections guess (see text) guess (see text)

A3. Calculation of Acetyl Group Distributions. Figure 13 shows a flowchart for the solution of eqs A14 and A15. Two slightly different methods were employed, depending upon the availability of input values for k and DB. Table 3 lists the required input. Changes in overall swollen fiber dimensions during the reaction were neglected in order to simplify the calculations. Table 4 lists results of mesh refinement studies. For the simulations, values of ∆r and ∆t were selected such that convergence occurred during a reasonable number of iterations. For NaOH/H2O, these values were ∆r ) 0.5 µm and ∆t ) 0.1 min; for NaOMe/MeOH, they were ∆r ) 0.5 µm and ∆t ) 0.01 min. As Table 4 shows, a slight improvement was attained by using smaller step sizes. A3.1. Calculation of the Rate Constant from ATRFTIR Data. As was noted previously, swelling was minimal

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Braun and Kadla

Table 4. Values of pAdata Calculated for Mesh Refinement Studies of the Fiber Cross-Section Method (a) NaOH/H2Oa ∆t (min) ∆r (µm)

0.5

0.1

0.05

10 5 1 0.5

0.603 0.582 0.594 0.605

0.602 0.581 0.588 0.596

0.602 0.581 0.587 0.591

(b) NaOMe/MeOHb ∆t (min) ∆r (µm)

0.1

0.05

0.01

10 5 1 0.5

0.566 0.584 0.596 0.628

0.564 0.582 0.582 0.603

0.563 0.579 0.569 0.577

Initial fiber mass ) 6.50 mg. Reaction time ) 720 min. pAdata ) 0.598. b Initial fiber mass ) 5.57 mg. Reaction time ) 40 min. p Adata ) 0.577. a

for the NaOMe/MeOH reactions. Consequently, Figure 4 provides a valid description of reaction kinetics at the fiber surface for this case. The rate constant kMeONa may be determined from the slope of the line kMeONaBS. The surface concentration of base BS can be determined from BB, the bulk concentration of base. BS )

BBVLS BBVL ≈ VS V

(A16)

where VLS and VS refer, respectively, to the volume of liquid in the surface layer and the total volume of the surface layer. BS is approximately constant, so eq 6 can be substituted into eq A16 to produce an expression for BS in terms of initial values of mass and density. BS ) 1+

BB FL

[

FP(ω - 1)

]

Vi ) πri L - π(ri - ∆r) L ) π(2i - 1)(∆r) L 2

2

m+1

AjiVi

∑ i)0 A V

(A19)

0

where j ) t/∆t and A0 is the initial concentration of A, which is uniform throughout the fiber. The experimental value for the total acetyl content was designated pAdata. The error for pAcalc was defined by eq A20.13 q

E)

(pAcalch - pAdatah)2 ∑ h)1

{ [ ( )]}

DB′ d dB′ r ) kAS0(B′ - Bo) r dr dr B′ ) BS at r ) R dB′ ) 0 at r ) 0 dr

(A21) (A22) (A23) (A24)

where Bo and B′ refer, respectively, to equilibrium and reaction zone concentrations. AS0 is the initial concentration of acetyl groups at the surface. The fiber was immersed in a continuously stirred solution of the base, so diffusion through the bulk solution could be neglected. Equations A21-A24 can be solved in terms of the Thiele modulus φ′V and dimensionless values of core radius ξ and time θV.

[ ]

I0(φ′Vξ) AS ) 1 - θV AS0 I0(φ′V)

(A25)

B′ - Bo I0(φ′Vξ) ) I0(φ′V) BS - Bo

(A26)

ξ) φ′V ) R

r R

(A27)

x

kAS0 DB′

θV ) k(BS - Bo)t

(A28) (A29)

(A18)

The shell volumes were used to calculate pAcalc, the total acetyl content of the fiber at time t. pAcalc )

∂AS ) -kAS0(B′ - Bo) ∂t

(A17)

Shell volumes were calculated for each increment of i using eq A18. 2

about 4 mg to 10 mg, so the simulations were repeated for each fiber for a given value of the effective diffusivity DB. DB was varied until a minimum value of E was attained. A3.2. Calculation of the Thiele Modulus Using Data from Stained Fiber Cross-Sections. The effective diffusivity DB was calculated from the Thiele modulus using a volume-based unreacted core model, wherein a central core of unreacted material is surrounded by a layer of completely converted material (‘ash’).3,14 The model divides the reaction into two stages. The first stage precedes the formation of the ash layer. In this case, the mass balances can be stated as follows

(A20)

where q refers to the number of data points collected for a given reagent system. The initial dry fiber masses varied from

In the second stage of the reaction, the reaction zone is surrounded by a layer of ash. At this stage, the mass balances for the base in each zone can be stated as follows: d dB′ ξ ) (φ′V)2(B′ - Bo)ξ dξ dξ B′ ) Bm at ξ ) ξm

[ ( )]

dB′ ) 0 at ξ ) 0 dξ

(A30) (A31) (A32)

in the reaction zone and d dB ξ )0 dξ dξ

(A33)

B ) BS at ξ ) 1

(A34)

B ) Bm at ξ ) ξm

(A35)

[ ( )]



in the ash layer. The subscript m refers to the boundary between the two zones. Equations A30-A35 can be solved to obtain the relative concentration of base at any given position within the fiber.

( )[ ] [ ]

Bm - Bo I0(φ′Vξ) B′ - Bo ) BS - B o BS - Bo I0(φ′Vξm)

(A36)

B m - Bo ln ξ B - Bo ) -1 +1 o o ln ξm BS - B BS - B

(A37)

At the boundary between the two zones DB′

dB )D ( ) (dB′ dξ ) dξ

(A38)

B

B′ ) B at ξ ) ξm

(A39)

Equations A36-A39 can be solved in terms of ξm and Bm. B m - Bo BS - B

o

)[

{ (

]}

-1

DB′ ln ξm I1(φ′Vξm) DB I0(φ′Vξm)

) 1-

(A40)

The surface concentration of acetyl groups in the stage 2 reaction zone can be determined by integration of eq A21. AS ) AS|t)tc -

∫t tkAS0(B′ - Bo) dt

(A41)

Figure 14. Least squares curve fit for eq A46.

Division of eq A44 by eq A45 produces a normalized expression for the reaction time. θV θVo

) aξm2I0(φ′Vξm)[ln ξm - 1] + bI0(φ′Vξm) + c DB′(φ′V)2

c

a)

where tc is the time required for completion of stage 1. Substitution of eqs A25, A29, A36, and A40 into eq A41 produces an expression for AS in terms of ξ and θV.

b)-

[ ] ∫[ ]{ (

AS ) AS0 - AS0

AS0

θV

1

I0(φ′Vξ)

-

1-

I0(φ′Vξm)

[

]{(

)[

]}

dθV (A42)

)[

] }

DB′ ln ξm I1(φ′Vξm) -1 DB I0(φ′Vξm)

(A43)

At the end of the first stage, ξm ) 1 and θV ) 1. This condition may be substituted into the integrated form of eq A43 to produce an expression for θV in terms of ξm. θV )

DB′(φ′V)2

{ξm2I0(φ′Vξm)[ln ξm - 1] +

DBI0(φ′V)

I0(φ′V)} -

I0(φ′Vξm) I0(φ′V)

+ 2 (A44)

At ξm ) 0, θV ) θVο, the time required for completion of both stages. In terms of eq A44 θVo )

[DB′(φ′V) + DB]I0(φ′V)

DB′(φ′V)2 +1 DB

DB′(φ′V)2 + 2DB DB′(φ′V)2 + DB

(A47)

(A48)

(A49)

-1

DB′ ln ξm I1(φ′Vξm) DB I0(φ′Vξm)

At the boundary, ξ ) ξm and AS ) 0. Consequently, dθV/dξm may be determined from the derivative of eq A42. I1(φ′Vξm) dθV ) dξm ξmI0(φ′V)

DB 2

c)

I0(φ′V)

I0(φ′Vξ)

[DB′(φ′V)2 + DB]I0(φ′V)

(A46)

(A45)

The constants a, b and c can be determined from a least squares curve fit of eq A46 using experimental values of ξm from stained fiber cross-sections, as is shown in Figure 14. This method requires an initial guess for φ′V. Once a and b have been calculated, eqs A47 and A48 can be solved to obtain a new value for φ′V. I0(φ′V) )

1 a-b

(A50)

The preceding procedure is repeated until φ′V(guessed) ) φ′V(calculated). Once φ′V has been determined, DB′ may be calculated from eq A28 using a given value of the rate constant k. The rate constant k is varied until a minimum value of E is attained from eq A20. References and Notes (1) Gilbert, R. D.; Kadla, J. F. In Biopolymers from Renewable Resources; Kaplan, D. L., Ed.; Springer: 1998; pp 47-95. (2) Fessenden, R. J.; Fessenden, J. S. Organic Chemistry, 2nd ed.; Willard Grant Press: Boston, 1982; pp 630-631. (3) Levenspiel, O. Chemical Reaction Engineering, 2nd ed.; John Wiley & Sons: New York, 1972; pp 44-50, 357-375. (4) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960; pp 521-522, 542-544, 559. (5) ASTM Procedure D871.

160


(6) Yang, Y. In Polymer Data Handbook; Mark, J. E., Ed.; Oxford University Press: New York, 1999; p 49. (7) Kra¨ssig, H. A. Cellulose: Structure, Accessibility and ReactiVity; Gordon & Breach Science Publishers: Philadelphia, 1993; pp 8-10, 167. (8) Shore, J. In Cellulosics Dyeing; Shore, J., Ed.; The Alden Press: Oxford, 1995; pp 156, 163. (9) Carey, F. A.; Sundberg, R. J. AdVanced Organic Chemistry, Part A: Structure and Mechanisms, 3rd ed.; Plenum Press: New York, 1990; pp 232-234.

Braun and Kadla (10) Isaacs, N. Physical Organic Chemistry, 2nd ed.; John Wiley & Sons: New York, 1995; p 207. (11) Laidler, K. J. Trans. Faraday Soc. 1958, 54, 1026-1033. (12) Saul′yev, V. K. Integration of Equations of Parabolic Type by the Method of Nets; MacMillan: New York, 1964; pp 73-83. (13) Hornbeck, R. W. Numerical Methods; Prentice Hall: Edgewood Cliffs, NJ, 1975; pp 91-98, 122-125, 269-276, 285. (14) Ishida, M.; Wen, C. Y. AIChE J. 1968, 14, 311-317.

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