Sheath−Core Differences Caused by Rapid Thermoxidation during

1140

Ind. Eng. Chem. Res. 1998, 37, 1140-1153

Sheath-Core Differences Caused by Rapid Thermoxidation during Melt Blowing of Fibers Stephen L. Kelley and Robert L. Shambaugh* Department of Chemical Engineering and Materials Science, University of Oklahoma, Norman, Oklahoma 73019

In melt-blown fibers there exists a radial variation in molecular weight. This difference is caused by the very rapid ( rs

(16)

rs2 - rc2 R2 - rs2 c + h R2 - rc2 R2 - rc2

for rc e rs

(17)

MWcore )

and

rs2 R2 - rs2 c + h R2 R2

MWsheath ) h

for rc g rs

(18)

Figure 8 shows how well the linear function fits the experimental data for case 1. A best fit of eq 8 to the data gave m ) -5440 g/mol‚µm and b ) 264 000 g/mol. The S (standard error) was 9200 g/mol for the fit of the predicted curve to the sheath data, while the S was 1300 g/mol for the fit of the predicted curve to the core data. The best fit values for m and b were determined by minimizing the sum of the S values for the sheath and core. A Newton-Raphson routine in Microsoft Excel Solver was used to minimize the sum of the squares of the error by changing the slope and intercept. Figure 8 also shows how well a step function fits the data. A best fit of eqs 12 and 13 gave h ) 53 000 g/mol and c ) 164 000 g/mol. For these values of h and c, rs ) 30.2 µm (this radius was directly calculated using the known average molecular weight of the fiber). The standard error S is 4200 g/mol for the fit of the predicted curve to the sheath data, while the S is 3800 g/mol for the fit of the predicted curve to the core data. Similar to what was previously done, the best fit values for h and c were determined by minimizing the sum of the S values for the sheath and core. As a visual examination of Figure 8 shows, both the linear and the step functions provide a good fit to the data. The average S value is 5250 g/mol for the linear function, while the average S is 4000 g/mol for the step function. For cases 2-6 of Table 1, the linear and step functions also provide good fits (these plots are not shown). If all S values for all six cases are examined, one finds that the average S for the step function fit is

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1145

Figure 9. Results of thermal degradation experiments run on a TGA. The molecular weights were determined via intrinsic viscosity measurements. The lines are empirical first-order and secondorder fits to the data (see eqs 19-22).

usually slightly lower (better) than the average S for the linear function fit. Both the linear function fit and the step function fit are empirical. The actual f(r) must be determined from careful consideration of the rapid diffusion of oxygen into the fiber and the reaction of oxygen with the polymer chains. This modeling procedure will be discussed below in the Modeling of Diffusion and Reaction section. Thermal Degradation Experiments. To separate oxidative degradation from thermal degradation, separate thermal degradation experiments were run in a TGA according to the procedure previously described. Figure 9 shows the results of these studies. The curves in Figure 9 are fits to the data based on first-order and second-order degradation. The first-order equation is

dMw/dt ) k1Mw

(19)

where k1 has the Arrhenius form

k1 ) A1e-E1/RT

(20)

The second-order equation is

dMw/dt ) k2(Mw)2

(21)

k2 ) A2e-E2/RT

(22)

where

The best-fit values of A1 and E1 are 1.03 × 106 s-1 and 24.9 kcal/mol, respectively; the best-fit values of A2 and E2 are 230 mol g-1 s-1 and 28.6 kcal/mol, respectively. The r2 for the Arrhenius first-order fit is 0.910,

Figure 10. Temperature and calculated molecular weight of the polymer as it travels from the extruder inlet (feed hopper) to the spinneret exit.

Figure 11. Fiber radius determined by high-speed photography.

while the r2 for the second-order fit is 0.949. The second-order equation was used for calculations in our study. As suggested by Buntin et al. (1974), thermal degradation dominates for T > 343 °C, while oxidative degradation dominates for T < 343 °C. The temperatures considered in Figure 9 are representative of temperatures in the vicinity of the transition from oxidative to thermal dominance. For cases 1-6, Table 2 (column 2) shows the calculated molecular weight at the spinneret discharge. The residence times used to calculate column 2 values were based on the geometry of the extruder, spinpack, and die head. For case 1, Figure 10 shows the temperature and molecular weight profile in the extruder, spinpack, and die head. Figures 11 and 12 show the experimentally determined diameter (radius) and temperature profiles for the threadline (i.e., these are post-spinneret data). The techniques used to produce these profiles were described

Table 2. Molecular Weight Loss during Processing case

calculated MW at die exit based on second-order degradationa

highest core value in dissolution experimentsb

final MW based on only thermal degradation

final calculated fiber average MWc

final experimental fiber average MW

1 2 3 4 5 6

148 000 127 000 99 000 84 000 148 000 148 000

162 000 121 000 120 000 94 000 144 000 141 000

148 000 127 000 99 000 84 000 148 000 148 000

115 000 98 000 91 000 73 000 113 000 109 000

109 000 100 000 94 000 78 000 109 000 109 000

a Thermal degradation according to eq 21. b The core corresponding to the largest γ. c Using highest core value for the initial molecular weight.

1146 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

Placing eq 25 in eq 23 gives

( )

( )

∂CA ∂CA N 1 ∂ )D r + sk(CA)a ∂t r ∂r ∂r NAV

b

(26)

The model involves the simultaneous solution of eqs 24 and 26. The extent of bond scission is defined as

ξ ) (N0 - N)/N0

Figure 12. Fiber temperature determined by infrared measurements.

earlier in the Experimental Equipment and Procedures section. Based on these profiles, the final molecular weights of the collected fibers were calculated with the assumption that only thermal (no oxidative) degradation occurs; see column 4 in Table 2. Details on how these final molecular weights were calculated will be discussed in the Modeling of Diffusion and Reaction section. As the information in Table 2 suggests, very little spinline degradation can be attributed to thermal degradation. Oxidative degradation must have occurred, and this degradation will now be addressed. Modeling of Diffusion and Reaction. If the solubility of oxygen in the polymer is assumed to be low and if the gas-side resistance is negligible, then the appropriate equation for describing the diffusion and reaction of oxygen in a cylindrical geometry is (Welty et al., 1984)

( )

∂CA ∂CA 1 ∂ r + RA )D ∂t r ∂r ∂r

(23)

where CA is oxygen concentration, t is time, r is radial position, D is the diffusivity of oxygen in the polymer, and RA is the consumption of oxygen by chemical reaction. The RA in the above expression needs to be related to the loss of molecular weight in the fiber. Kelen (1983) discusses how the randomness of the bond scission process (in degradation) implies that the probability of cleavage is proportional to N. If the discussion of Kelen is expanded, the loss of bonds along a polymer chain can be described by

d(N/NAV)/dt ) k(CA)a (N/NAV)b

(24)

where N ) number of reactive bonds per volume, NAV ) Avogadro’s number, k ) rate constant ) k0 exp(-E/ RT), and a, b ) exponents. Based on the stoichiometry of the reaction of oxygen with the polymer bonds, the loss of oxygen can be expressed as

RA ) dCA/dt ) sk(CA)a (N/NAV)b

(25)

where s is the stoichiometric factor. Reich and Stivala (1969) give a reaction mechanism that suggests s ) 2, while the mechanism of Kelen (1983) implies that s ) 1.5. For most calculations in this work, s ) 2 was assumed.

(27)

where ξ is the conversion of bond cleavage and N0 is the initial concentration of bonds. As discussed by Kelen (1983), ξ is actually a time-dependent probability, and ξ can be related to the number-average degree of polymerization with this equation:

Pn )

Pn0 1 + (Pn0 - 1)ξ

(28)

where Pn is the number-average degree of polymerization and Pn0 is the starting number-average degree of polymerization (at the start of the iteration step). The number-average molecular weight can be determined by simply multiplying Pn by the repeat unit molecular weight. The kinetics just presented involve Mn, the numberaverage molecular weight, while our experimental measurements involve Mw, the weight-average molecular weight. These two quantities will be related by assuming that Mw ) 2Mn, i.e., the molecular weight has the most probable distribution. This is a good assumption since, prior to the spinneret, the polymer is degraded in the extruder and spinpack; see Becker et al. (1996). (Equations 24 and 26 will be applied to the polymer after the spinneret.) Hence, the high molecular weight tail of the molecular weight distribution is preferentially attacked, and the distribution is narrowed. Equations 24 and 26 were solved numerically by iterating along the fiber threadline. In applying these equations to a fiber threadline, both the changing temperature and changing diameter of the threadline were considered. Changing temperature affected both the diffusivity D and the reaction rate constant k ) k0 exp(-E/RT). The dependence of D on temperature will be discussed below. The activation energy E was assumed to be 30.5 kcal/mol, the value suggested by Stivala et al. (1983) for the thermoxidative scission of polypropylene main chains. The value of k0 was found by a least-squares fit of the model prediction to the experimental molecular weights. The changing diameter of the fiber was accounted for by iterating down the fiber in steps. As shown in Figure 13, the diameter of the fiber was approximated by the average measured diameter over a segment length l. The iteration was performed with time as the independent variable. With the continuity equation the velocity of any segment of diameter di is

V ) 4m/Fπdi2

(29)

where m is the mass flow rate of polymer. A value of F ) 0.73 g/cm3 was assumed; this is an average F based on the measured fiber temperatures (see Figure 12) and

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1147

of eq 26, the solubility of oxygen in the polymer must be known. The diffusivity is defined as (Yasuda and Stannett, 1975)

D ) D0 exp(-ED/RT)

(31)

where D ) diffusivity, D0 ) diffusivity leading coefficient, and ED ) diffusivity activation energy. Similarly, the solubility is defined as (Yasuda and Stannett, 1975)

S ) S0 exp(-∆Hs/RT)

Figure 13. Approximation of the fiber threadline as a series of segments.

threadline densities measured by Bansal and Shambaugh (1996). The distance traveled by the fiber (which is the length l of the segment) in time ∆t is

li ) (4m/Fπdi2)(∆t)

(30)

As the iteration proceeded down the filament, the segment lengths got progressively longer. A ∆t of 1 µs was used in the calculations, and time iteration proceeded until a position 5 cm below the spinneret was reached. About 1000 iterations (and, thus, 1000 stepped segments) were needed to reach 5 cm. As results presented later will show, by 5 cm all degradation had occurred. Equation 30 was applied to each stepped segment as the iteration proceeded. Essentially, each stepped segment approximated a cylinder exposed to oxygen for 1 µs. Equations 24 and 26 were numerically integrated with a spatial step size of 0.001 R; i.e., 1000 steps were used. The input (starting) profile of oxygen concentration for each segment was dependent on the output (final) profile from the previous segment. To account for the change in fiber diameter, the final oxygen concentration from a segment was expressed as a function of dimensionless radius (CA vs 2r/di). This dimensionless profile then became the starting oxygen concentration for the solutions of eqs 24 and 26 for the next segment. The numerical solution was done in Fortran using IMSL routines (IMSL, 1994). The computer used was an IBM RS6000. Approximately 400 min was required for each calculation (i.e., the determination of a k0 value for a particular case). The numerical solution began by assuming a value for k0. This procedure produced a final N profile for the final fiber segment. Then, using eqs 27 and 28, the N profile was converted into f(r), a molecular weight profile. Next, with eqs 5-7, this molecular weight profile was converted into a molecular weight versus γ profile. The predicted core and sheath molecular weights were then compared with the experimental core and sheath molecular weights; a least-squares calculation was used for this comparison. An optimization scheme selected new values for k0 until the best k0 had been found for each operating case (see Table 1 for the operating cases). Oxygen Diffusivity in Polypropylene. In order to solve eq 26, the diffusivity of oxygen in polypropylene is needed. Also, as a boundary condition in the solution

(32)

where S ) solubility, S0 ) solubility leading coefficient, and ∆Hs ) solubility activation energy. Often, literature data are reported in the form of permeability P. Diffusivity, solubility, and permeability are related by the expression P ) DS. Nearly all diffusivity and solubility data for polymers have been taken at or near ambient (room) temperature. Recently, Marcandalli et al. (1991) determined diffusivities and solubilities of oxygen in polypropylene films at temperatures from -10 to +50 °C. In the format of eqs 31 and 32, the following values were determined for annealed films:

D0 ) 1.5 × 10-5 cm2 s-1 ED ) 11.9 kcal mol-1 S0 ) 5.1 × 10-3 cm3(STP) of oxygen/cm3 of polymer ∆Hs ) -2.8 kcal mol-1 Diffusivity and solubility data are difficult to determine when the diffusing material reacts with the polymer. For the case of oxygen diffusion into hot polypropylene, this is indeed what happens. An additional complication in our work is that the polymer is a molten liquid, not a solid. Typically, the diffusivity of gases is orders of magnitude higher in liquids than in solids. For example, Billingham (1993) lists the diffusivity of oxygen in high-density polyethylene as 1.6 × 10-7 cm2/s at 25 °C, while the diffusivity of oxygen in liquid hexane (a chemically similar material) is given as 3 × 10-5 cm2/s at 25 °C. More generally, Welty et al. (1984) give data for the diffusivity of gases in various solids and liquids. Typically, a gas has a diffusivity on the order of 10-5 cm2/s in a liquid such as water, while a gas has a diffusivity on the order of 10-7-10-11 cm2/s in solids such as glass or metal. To account for the larger diffusivity of oxygen in molten polypropylene, a range of diffusivity values was tested in the model. As described later, this testing procedure gave an estimated diffusivity that was about 4 orders of magnitude higher than that expected in solid polypropylene. Oxygen Balance. The oxygen that is absorbed into the fiber either reacts with the polymer or builds up in the fiber. Mathematically, the oxygen balance can be expressed as

O2,absorbed ) O2,remaining + O2,reacted

(33)

1148 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

Figure 14. Predicted oxygen profile in the fiber at t ) tf. Profiles are shown for four different assumed diffusivities. The activation energy ED ) 11.9 kcal/mol.

The absorption takes place along all segments of the fiber from the spinneret to the end of the fiber. Hence, the absorption term can be written as

O2,absorbed )

∑∫t)t

t)ti+∆t i

all i

|

∂CA

D

∂r

πdili dt

(34)

s

where the oxygen concentration gradient is evaluated at the surface. The oxygen remaining (unreacted) in the fiber is the oxygen contained in the final segment at tf, the time at the end of all calculations. This oxygen amount can be written as

O2,remaining )

r)R CA|t ∫r)0 f

end

× 2πrlf dr

(35)

The oxygen reacted can be determined from the stoichiometry of the bond-breaking reaction. The equation is

O2,reacted )

r)R 2 ∫r)0 f

[

]

N0 - N NAV

× 2πrlf dr

(36)

tend

A Fortran routine (Simpson’s rule) was used to determine the integrals in the above three equations. Model Results Effect of Diffusivity on the Oxygen Concentration Profile. For a preliminary calculation, assume that the consumption of oxygen due to degradation is small. Then, eq 23 can be solved with RA ) 0. The previously discussed Marcandalli solubility values were used as a boundary condition. Figure 14 shows the result of this calculation for case 1. For t ) tf (i.e., at the end of the threadline iteration procedure) the figure shows the expected oxygen concentration profile in a fiber whose radius is 44 µm. The different profiles show the effect of different assumed diffusivities; for the diffusivity calculations, the activation energy was assumed to be 11.9 kcal/mol (the Marcandalli value) and D0 was varied. Now, examination of Figure 8 shows that, for approximately γ > 0.5, no further change occurs in the core molecular weight. According to eq 5, γ ) 0.5 corresponds to a radius of 31 µm. Returning to Figure 14, the D0 ) 1.5 cm2/s assumption is too high,

Figure 15. Fit of the model for first-order dependence on oxygen concentration (a ) 1) and first-order dependence on bond concentration (b ) 1). D0 ) 0.15 cm2/s, ED ) 11.9 kcal/mol, and E ) 30.5 kcal/mol.

since oxygen (and its related degradation effect) has penetrated too deeply into the fiber. Contrarily, the D0 ) 0.075 and D0 ) 0.015 cm2/s values are too low, since the penetration of oxygen is too shallow. It appears that the assumption D0 ) 0.15 cm2/s is of the correct order. This D0 value corresponds to D ) 1.7 × 10-7 cm2/s at 165 °C, the polymer melting point. As previously discussed, this diffusivity is roughly what is expected for the diffusion of a gas in a liquid. Use of the Model To Predict Molecular Weight. The full model (i.e., the solutions of eqs 26 and 24) can be used to predict the molecular weight distribution [f(r)] in the final fiber. To solve eq 24, values for a and b are needed. Kelen (1983) suggests b ) 1; this value was used in our solution. For the a value, Billingham (1993) suggests that both zero- and first-order dependence (a ) 0 and a ) 1) are possible; exponent values in the range 0 e a e 1 were tested in our solution. Figure 15 shows the results of solving the model with a ) b ) 1 for case 1 (see Table 1). The results are presented in the form of molecular weight as a function of mass fraction removed. The experimental data for case 1 are also shown on the graph for comparison. The rate constant k0 in eq 24 was determined by optimizing the fit of the model to the data via a least-squares technique. As can be observed, the model does not fit the data very well. Similar poor fits resulted when the other five cases were tested with a ) b ) 1. Figure 16 shows the effect of varying the a value between 0 and 1 for case 5. The zero-order assumption apparently fits the best. One additional assumption must be made when using a ) 0: a limiting concentration must be selected for the oxygen concentration. That is, even though the reaction is zero order with respect to oxygen concentration, some oxygen is necessary to make the reaction proceed. For Figure 16, a limiting concentration of CA ) 1 × 10-20mol/cm3 was used. For case 5, Figure 17 shows how different limiting concentrations can affect the solution. Similar results are shown in Figure 18 for case 6. Examination of Figures 17 and 18 and similar results for cases 1-4 showed that a limiting concentration of 1 × 10-20 mol/ cm3 fit well in all cases. As can be seen, the solution needs only an order of magnitude estimate of limiting concentration. Figures 19-22 show the predicted molecular weights for cases 1-4. The previously discussed best values for

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1149

Figure 16. Fit of the model for b ) 1 and 0 g a g 1. D0 ) 0.15 cm2/s, ED ) 11.9 kcal/mol, and E ) 30.5 kcal/mol. For a ) 0, an oxygen concentration limit of CA,limit ) 1 × 10-20 mol/cm3 was set for the reaction.

Figure 19. Fit of the model to case 1. D0 ) 0.15 cm2/s, ED ) 11.9 kcal/mol, CA,limit ) 1 × 10-20mol/cm3, and E ) 30.5 kcal/mol.

Figure 20. Fit of the model to case 2. Figure 17. Effect of oxygen concentration limit on the fit of the model to the data. D0 ) 0.15 cm2/s, ED ) 11.9 kcal/mol, and E ) 30.5 kcal/mol. The data are for case 5.

Figure 21. Fit of the model to case 3.

Figure 18. Effect of CA,limit on the fit of the model for case 6.

a, b, and limiting oxygen concentration were used to produce these plots (Figures 17 and 18 include the best fits for cases 5 and 6). As can be seen, the model results compare well with the experimental results. Table 3 contains the values of k0 that were determined for fits of the data from cases 1-6. The average k0 is 1.10 × 1010 s-1. As described earlier, a diffusivity constant of D0 ) 0.15 cm2/s was selected. With the kinetics (a ) 0 and b ) 1) and oxygen limit (CA,limit ) 1 × 10-20 mol/cm3) now

established, let us return to this selected value for D0. Figure 23 shows what would have happened if different values of D0 had been used. As can be seen, the fits of the curves are much worse for the higher and lower values of D0. For D0 ) 0.015, the average standard error S is 23 000 g/mol; for D0 ) 1.5, the average S is 38 000 g/mol. But for the previously selected best value of D0 ) 0.15, the average S is only 6300 g/mol. Other Model Predictions. Our experimental molecular weight measurements were made on collected fiber. Thus, our measurements show only the final degradation values. With the model, however, molecular weight predictions can be made for any position

1150 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

Figure 22. Fit of the model to case 4.

Figure 24. Average fiber molecular weight along the threadline for cases 1-6.

Figure 23. Effect of varying the oxygen diffusivity. For the calculations, a ) 0, b ) 1, ED ) 11.9 kcal/mol, CA,limit ) 1 × 10-20 mol/cm3, and E ) 30.5 kcal/mol. For D0 ) 0.015 cm2/s, k0 ) 9.85 × 1010 the core, sheath, and average standard error are respectively 18 000, 28 000, and 23 000 g/mol. For D0 ) 0.15 cm2/s, k0 ) 1.31 × 1010 the core, sheath, and average standard error are respectively 1600, 4700, and 6300 g/mol. For D0 ) 1.5 cm2/s, k0 ) 5.17 × 109 and the core, sheath, and average standard error are respectively 50 000, 27 000, and 38 000 g/mol. Table 3. Values of k0 for Cases 1-6 (D0 ) 0.15 cm2/s, ED ) 11.9 kcal/mol, and CA,limit ) 1 × 10-20 mol/cm3) standard error (g/mol) case

k0 ×

10-9

(s-1)

core

sheath

average

1 2 3 4 5 6

7.00 9.14 10.1 10.9 16.1 13.1

9000 4000 4000 7000 11000 2000

16000 3000 7000 10000 6000 5000

12000 3000 5000 8000 8000 3000

average

11.0

6000

8000

7000

along the threadline, i.e., as the process occurs. Figure 24 shows such predictions for all six experimental cases. The average molecular weight (the weight average in the radial direction) is plotted versus the exposure time. As can be seen, all significant degradation occurs in the first 30 ms for case 1. The degradation is even faster for the other five cases where the polymer or air temperature is higher than in case 1. Figure 25 shows the predicted sheath and core molecular weights along the threadline for case 1. Results for different levels of mass removed (γ) are given. Observe that the core molecular weight is less and less affected as γ increases.

Figure 25. Sheath and core molecular weights along the threadline for case 1.

Table 2 compares calculated versus experimentally measured molecular weight loss for the six cases. Column 2 shows the calculated molecular weight at the die exit based on eq 21 and our measured values of A2 ) 230 mol g-1 s-1 and E2 ) 28.6 kcal mol-1. Column 3 shows the highest core values from the dissolution experiments, i.e., the core values corresponding to the highest γ values. As can be seen, the values in column 3 are close to the values in column 2. This result is supported by Figure 25: as Figure 25 shows, the core molecular weight for large γ does not change between the spinneret and the collection screen. Since the standard errors for fits with column 3 values were slightly smaller than fits with column 2 values, all results in this paper are based on column 3 values. (This is no surprise since experimental measurements are generally superior to calculated values.) Column 4 in Table 2 shows the calculated final molecular weight based on thermal degradation alone. In other words, eq 21 was used to calculate the degradation along the threadline (no oxidative degradation was considered). As can be observed, the thermal degradation along the threadline is so small that column 4 is essentially identical to column 2 (more specifically, the columns are identical to three significant figures). Column 5 shows the calculated final molecular weight based on our model with a ) 0, b ) 1, CA,limit ) 1 × 10-20mol/cm3, and the best-fit value of k0 for each case (see Table 3). Column 6 shows the experimentally measured fiber molecular weight (from intrinsic viscos-

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1151 Table 5. Effects of Reaction Stoichiometry on Model Predictions (Case 1) standard error

oxygen (mol)

stoichiometric core sheath average absorbed reacted remaining factor (g/mol) (g/mol) (g/mol) (×1012) (×1016) (×1012) 1 2 10

Figure 26. Development of the distribution function f(r) along the threadline.

9000 9000 9000

16 000 12 000 16 000 12 000 16 000 12 000

2.88 2.88 2.88

3.99 7.97 39.9

2.75 2.75 2.75

and corresponding positions. Since so little oxygen is used in the degradation reaction, the profiles in this figure are nearly identical with the profiles for when RA ) 0. Table 5 shows the effect of assuming a different value for s, the stoichiometric coefficient. More oxygen is consumed as s increases. However, if s is varied from 1 to 10 in the model equations, the best-fit value for k0 remains essentially unchanged and the standard errors of fit are also essentially unchanged. These results are due to (a) the zeroth-order dependence of degradation on oxygen concentration and (b) the small fraction of oxygen consumed in the degradation reaction. Conclusions

Figure 27. Oxygen concentration profile along the threadline. Table 4. Results of Oxygen Balance on the Fiber (for These Calculations, a ) 0, b ) 1, D0 ) 0.15 cm2/s, ED ) 11.9 kcal/mol, and CA,limit ) 10-20 mol/cm3) case

O2,absorbed (mol × 1012)

O2,reacted (mol × 1016)

O2,remaining (mol × 1012)

1 2 3 4 5 6

2.88 1.82 2.57 1.81 2.27 1.74

7.97 2.79 3.25 3.26 3.99 3.98

2.75 1.73 2.49 1.74 2.16 1.65

ity measurements). As can be seen, the numbers in column 5 are close to the values in column 6. This is expected, since previous figures have shown how well the model fits the data. Column 4 values are significantly lower than the column 6 (or column 5) values. Hence, thermal degradation alone cannot account for the change in the molecular weight. In Figure 26, the molecular weight distribution function f(r) is plotted against the reduced radius for case 6. Curves for six different timessand their corresponding positionssare given in the figure. By 25 ms, the dimensionless profile has almost reached its final shape (at t ) 50 ms). Table 4 shows the results of an oxygen balance on the fiber for cases 1-6. The absorbed, reacted, and remaining oxygen were calculated with eqs 34-36. As can be seen, only a fraction of the oxygen is used in the degradation reaction. Figure 27 shows the buildup of oxygen concentration in the fiber as a function of reduced radius. Profiles are shown for various times

When producing fibers via the melt-blowing process, significant degradation of the fiber sheath can occur. In contrast, very little core degradation occurs during the melt-blowing process. Higher polymer and/or air temperatures cause higher sheath degradation. The speed at which this degradation occurs is quite rapid. Most degradation occurs within the first 25 ms of the process at positions within 1 cm of the spinneret. A complex mathematical model was developed for the degradation process. This model includes both oxygen diffusion and reaction in the polymer fiber. Oxidative degradation dominates over thermal degradation for locations beyond the spinneret tip. The oxidative degradation appears to be zeroth order in oxygen concentration and first order in the concentration of reactive bonds. With these assumptions, the mathematical model was used to determine a best-fit value of k0 ) 11.0 × 109 s-1. Acknowledgment We thank the following organizations for their support: The National Science Foundation (Grant DDM9313694), the State of Oklahoma (Project AR4-109/ 4844), 3M Company, and Conoco/DuPont. We also thank Vishal Bansal for his measurements of the on-line fiber diameter and fiber temperature. Nomenclature a ) order of dependence on oxygen concentration in eq 24 A1 ) leading coefficient in the Arrhenius expression of firstorder degradation, s-1 A2 ) leading coefficient in the Arrhenius expression of second-order degradation, mol g-1 s-1 b ) order of dependence on bond concentration in eq 24 c ) parameter in step function, g/mol CA ) oxygen concentration, mol/cm3 CA,limit ) oxygen concentration necessary for polymer degradation to occur during zeroth-order (a ) 0 in eq 24) degradation, mol/cm3 di ) diameter of a segment of the fiber (used in iterative procedure), µm D ) diffusivity of oxygen in the polymer, cm2/s

1152 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 D0 ) diffusivity leading coefficient, cm2/s E ) activation energy for polymer degradation, kcal/mol ED ) diffusivity activation energy, kcal/mol E1 ) activation energy for first-order degradation, kcal/ mol E2 ) activation energy for second-order degradation, kcal/ mol f(r) ) radial distribution of the fiber molecular weight (in terms of Mw), g/mol h ) parameter in step function, g/mol k ) rate constant for polymer degradation (units depend on a and b) k1 ) first-order thermal degradation rate constant, s-1 k2 ) second-order thermal degradation rate constant, mol g-1 s-1 k0 ) leading coefficient in the Arrhenius expression for polymer degradation (units depend on a and b) l ) segment of the fiber length (used in iterative procedure), µm L ) length of a collected fiber, m m ) polymer flowrate, g/s Mw ) weight-average molecular weight, g/mol Mn ) number-average molecular weight, g/mol Mavg ) molecular weight averaged over the entire fiber (given as Mw), g/mol Msheath ) sheath molecular weight (given as Mw), g/mol Mcore ) core molecular weight (given as Mw), g/mol N ) number of reactive bonds per volume, cm-3 NAV ) Avogadro’s number (6.02 × 1023) N0 ) value of N at the start of an iteration step, cm-3 O2,absorbed ) oxygen absorbed along the entire length of the threadline, mol O2, remaining ) unreacted oxygen that remains in the threadline at time tf, mol O2, reacted ) oxygen that has reacted with the polymer by time tf, mol Pn ) number-average degree of polymerization Pn0 ) number-average degree of polymerization at the start of an iteration step Qair ) air flowrate at die exit, L/min at 21 °C and 1 atm pressure r ) radial position in the fiber, µm rc ) radius of the remaining (after dissolution) core fiber, µm rs ) radius at which step change in f(r) occurs, µm R ) fiber radius, µm; also, gas constant, kcal mol-1 K-1 RA ) consumption of oxygen by chemical reaction, mol cm-3 s-1 s ) stoichiometric coefficient: the number of oxygen molecules required to break one polymer bond S ) solubility, cm3(STP) of oxygen/cm3 of polymer S0 ) solubility leading coefficient, cm3(STP) of oxygen/cm3 of polymer t ) time, s tf ) time at the end of all calculations; this time corresponds to when the polymer in the fiber reaches z ) 5cm, s T ) temperature, K Tpolymer ) spinpack temperature, °C Tair ) temperature of air at the die exit, °C z ) position along the threadline; z ) 0 at the die exit, cm Greek Letters γ ) mass fraction of polymer dissolved from the polymer surface ∆Hs ) solubility activation energy, kcal/mol ∆r ) thickness of a fiber shell, m ∆t ) time increment used in iteration, µs ξ ) the extent of bond scission defined in eq 27 [η] ) intrinsic viscosity

F ) fiber density, g/cm3 Favg ) average fiber density, g/cm3 Fi ) density of a radial shell of thickness ∆r, g/cm3

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Received for review August 7, 1997 Revised manuscript received November 10, 1997 Accepted November 12, 1997 IE970549F