Ind. Eng. Chem. Prod. Res. Dev. 1983,22, 86-89
86
Effect of Variable Storage Times on the Calculation of Diffusion Coefficients Characterizing Small Molecule Migration in Polymers John L. Ayres, James L. Osborne, Harold B. Hopfenberg,’ and Wllllam J. Koros Department of Chemical Engineering, North Carolina State Universw, Raleigh, North Carolina 27650
The error which results from calculation of a diffusion coefficient from desorption measurements which fail to account for the nonuniform concentration profile developed during prior sample storage is determined for the case of migration from a plane sheet into an infinite, well-stlrred bath. The incorrectly calculated diffusion coefficient, D’, neglecting prior storage time, is smaller than the actual diffusion coefficient, D. For small storage times, the ratio D‘JD is close to unity; however, D‘JD approaches an asymptotic value of 0.70 for very large storage times if the well-known “half-time’’ method is employed for evaluating the diffusion coefficients. I f the so-called “initial slope” method is used for determination of diffusion coefficients, the calculational error is large, with the ratio D’lD approaching zero for large storage time. A sigmoid response in plots of fractional loss vs. the square root of time is sometimes taken as evidence for the existence of non-Fickian transport processes. It is shown here that such sigmoid plots are also observed when elution occurs prior to the initiation of the formal desorption experiment, even when Fickian transport controls the desorption process.
Introduction It has recently become important to characterize accurately the migration of monomers or other small molecules from polymers because of increasing concern regarding the introduction of undesirable substances into food. These concerns are typically addressed in research and development programs, carried out in industrial laboratories, involving plaques, whole packages, or containers fabricated from polymeric materials. These studies involve, for example, acetaldehyde migration in poly(ethy1ene terephthalate) bottles, vinyl chloride migration in PVC pipe or bottles, acrylonitrile migration in acrylonitrile copolymers, and the elution of extraneous low molecular weight residuals from thermoset container liners. The experimental and analytical constraints discussed in this paper are relevant to all of these studies. The true value of the diffusion coefficient, which quantitatively characterizes migration for a specific polymer-penetrant system, must be determined in order to predict accurately the rate of migration of such small molecules from polymers. Errors introduced by unrecognized and variable storage times of samples prior to desorption analysis may indeed be significant. Alternatively, diffusion coefficients, correctly obtained, are often applied incorrectly to the analysis and prediction of the migration kinetics attending the elution of a low molecular weight residual from a container which has been subjected to prior storage or aging. This paper addresses not only the recognition of rigorous protocols for meaningful data analysis but also presents techniques useful for developing accurate predictions and analysis attending migration from containers which have experienced significant desorption of the small molecule during prior storage. Discussion General Analytical Criteria and Methodology. Small molecule migration from a polymer sheet into an infinite bath is described phenomenologically by Fick’s law, viz. N = -D-ac
ax where N is the rate of transfer per unit area, C is the local concentration of the diffusing substance, x is the space coordinate measured in the direction of diffusion, and D 0 196-432118311222-0086$01.50/0
is the effective diffusion coefficient (Crank, 1975). By considering a mass balance on an element of volume and assuming that Fickian diffusion occurs only in the x direction with a concentration independent diffusion coefficient, the following relationship, Fick’s second law, is developed (Crank, 1975)
where t is time. If the concentration of the diffusing substance is assumed to be initially uniform within the sheet, at time t = 0, and the surface concentrations are instantaneously brought to zero, the following equations can be derived from eq 2 using separation of variables and Laplace transforms, respectively.
(4) where Mt is the amount of diffusant lost by the sheet at time t , C is the thickness of the sheet, and M , is the equilibrium desorption attained theoretically at infinite time (Crank, 1975). For M J M , > 0.6, eq 3 is especially useful, since at long times terms beyond n = 0 become insignificant and eq 3 reduces to a more manageable expression presented as Mt 8 -r2Dt M , = 1- 772 exp( T )
Similarly, for values of M J M , < 0.6, eq 4 may be approximated, with very little error, by the equation
Equations 3 and 5 are, therefore, useful relationships for description of so-called ”long time” data ( M t / M , > 0.6), whereas eq 4 and 6 are much more useful for so-called “short time” data. Errors Resulting from Prior Storage Using the “Half-Time” Method for Calculating Diffusivities. A 0 1983 American Chemical Society
Ind. Eng. Chem. Prod. Res. Dev., Vol. 22,
No. 1, 1983 87
method for estimating diffusion coefficients involves further reduction of either eq 5 or 6 by substituting M , / M , = 0.5 into either equation. The following relationship results
D=- 0.049C2 tl/Z
(7)
where tlIz corresponds to the time at which M J M , = 0.50. Therefore, for a given sheet thickness, measurement of the half-time (tl,z) permits calculation of the diffusion coefficient, assuming that the previously mentioned initial condition and boundary conditions apply and Fickian transport occurs. If a polymer sheet is stored for a period of time, t,, some of the diffusant escapes from the sheet prior to the desorption experiment, thereby invalidating the simple preceding treatment (Berens and Hopfenberg, 1977) since the initial conditions will not conform to the uniform profile assumed in the derivation of eq 3 and 4. In the following treatment, it is assumed that the concentration of the diffusing substance is initially uniform within the sheet, at time t = 0, and the surface concentrations are instantaneously brought to zero and maintained there during a period of storage, t,, prior to formal desorption testing. Under typical storage conditions, the polymer might be stored in air at ambient temperature in a container of some sort with varying volume ratios of polymer to the surrounding air space. The assumed conditions, therefore, provide a conservative worst case estimate of the effects of prior storage on the results obtained in the subsequent formal desorption experiment. After storage of the polymer, the surface concentrations are assumed to be maintained at zero during the formal desorption experiment. Typical experimental procedures might consist of measuring the desorption of the diffusing species in a water bath at temperatures near ambient and under conditions approximating an infinite, well-stirred bath. The assumption of zero concentration of migrant at the polymer extremities is, therefore, a reasonable limiting case for storage and experimental conditions as long as the subtle differences which might be associated with elution into water compared to air do not contribute significantly. For nonpolar polymers, this assumption is especially reasonable. The apparent fractional loss during the experiment, M,‘/M,’, observed at some apparent time, t’, after starting the formal desorption experiment is given by
where M , is the amount of diffusant lost during storage prior to the formal experiment (from t = 0 to t = t,), M,’ is the apparent loss observed from the beginning of the formal desorption experiment up to an apparent desorption time of t’, and M,’ is the apparent equilibrium desorption at infinite time in the formal desorption experiment. The time, t , is the actual total time (storage plus apparent experimental) which has elapsed since the boundary concentrations were dropped to zero (Le., t = t , + t?. Consistent with the case described above, if the polymer sheet is stored in an infinite bath at the same temperature as that of the formal desorption experiment, the fractional loss during storage, M,/M,, may be determined from eq 3 or 4 by substituting M , for Mt and t, for t. For a given storage time, t,, one can predict the plot of M,‘/M,’ as a function of (t?’l2,which would be observed in an experiment, by substituting M,/M, and M,/M, (for various values of t ) into eq 8.
/ir
I
(days)$2
Figure 1. Apparent fractional loss as a function of the square root of the apparent desorption time ( t ’ = t ts). Note the tendency for the fractional loss curves to approach an asymptotic sigmoid-shaped form at long storage times.
-
1.0,
Figure 2. Apparent fractional loss as a function of the square root of the apparent dimensionless desorption run time, where 7’ = tDIC2.
As an example, if one assumes a film thickness of 40 mils (0.1016 cm) and a diffusion coefficient of 2.0 X cm2 [a typical value for migration of small molecules in glassy polymers (Berens and Daniels, 1976)], the plots in Figure 1 which describe the apparent fractional loss (MiIM,’) as a function of the square root of apparent time (t? for given storage times can be calculated using eq 8 in conjunction with eq 3 or eq 4. A similar plot of apparent fractional loss as a function of dimensionless apparent time, 7’ Dt’/C2,for given dimensionless storage times, T, Dt,/C2,is presented in Figure 2. Elution losses during storage prior to formal desorption characterization cause the plot of apparent loss vs. the square root of apparent time to take on a sigmoid shape. Also, the apparent time required to reach a given value of Mt’/M,’ increases as storage time, t,, increases. For large storage times where M,/M, is greater than 0.6, eq 5 may be rewritten as eq 9, substituting M , for Mt and t , for t , assuming that the boundary conditions and pro-
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Ind. Eng. Chem. Prod. Res. Dev., Vol. 22, No. 1, 1983
1.0L
I
I
I
,
0.8 -
t
-t
‘--I
4
D 2.0 x ~ 1 - 1 2cm*/sec 1- 0.102cm
02
0.0
102
lo4
103
105
106
10’
t s (minutes)
Figure 3. Ratio of the apparent to the actual diffusion coefficient, determined by the “half-time” method, as a function of storage time, t,, during which unacknowledged prior desorption occurs. Note the tendency of the ratio D’/D to approach a limiting value of 0.70 at long storage times.
xc2
D’= 16 where slope, equals the measured initial slope of M;/M,’ vs. (t?”’. If one draws tangents “by eye” to the initial portions of the curves presented in Figure 1, apparent diffusion coefficients for various storage times can be calculated from the slopes of these tangents using eq 13. The variation of D’/D vs. storage time, using the initial slope method described above, is presented in Figure 4. The values plotted are obviously somewhat poorly defined because of the method used to obtain the limiting slopes, but it is clear that the ratio D‘/D appears to approach unit for short storage times and decreases sharply as storage time increases. It can be shown that the initial slope of the plot of M,’/M,‘ vs. (t’)’/2 in fact approaches zero even for very small nonzero storage times. The slope, given by
tocols corresponding to storage are identical with those used in the development of eq 5 , therefore
7)
Ma = 1 - 8 ex.(
M,
-x2Dt,
i?
Under these conditions, after rearranging and combining eq 5, 8, and 9 (with t ’ = t - t,), it is easily shown that for large storage times the apparent fractional desorption, Mt’/M,’, at a specific t’, is given by
-Mt’ M” Therefore, for large storage time, M;/M,‘ is virtually independent of the precise value oft,. This limiting behavior is presented graphically in the plot furthest to the right in Figure 1. Using the plots in Figure 1, an apparent diffusion coefficient, D’, may bP calculated (using an expression analogous to eq 7) by incorrectly assuming that the initial condition used to derive eq 3 and 4 is applicable to the desorption experiment 112
where t is the apparent time required for an apparent fractional loss equal to 0.5. The apparent diffusion coefficient decreases with increasing storage time since the apparent half-time increases with storage until it reaches an upper limit for large storage times. A plot of the ratio of the apparent diffusion coefficient to the actual diffusion coefficient, D‘/D, vs. t, is given in Figure 3. For large storage times, one may, using eq 10 with M,’/M,‘ = 0.5 and t’= t I l j 2 , obtain an expression for the true diffusion coefficient in terms of ttll2and, in conjunction with eq 11, yield D’/D = 0.049a2/ln 2 N 0.70 (12) Clearly, D Y D is approximately unity for low storage times and decreases as storage time increases to an asymptotic lower limit of 0.70 for large storage times. Errors Resulting from Prior Storage Using the “Initial Slope Method” for Calculating Diffusivities. Another method which may be used experimentally to determine a diffusion coefficient involves determination of the initial slope of MJM, as a function of t1/2. If MlIM,’ is plotted as a function of (t?’’’, an experimenter might determine the initial slope and then calculate an apparent “early time” diffusion coefficient as
can be evaluated in terms of eq 8 using either eq 5 or eq 6. It can be shown that the derivative d(M,’/M,’)/dt’is finite in the limit as t ’approaches zero in all cases for which t , > 0; therefore, the limiting slope of a plot of M;/M,’ vs. (t’)’I2 (as t’approaches zero) is equal to zero for all nonzero values of t,. Casual placement of tangents to the respective curves when analyzing actual experimental data would not reveal this subtlety. To demonstrate the above points, consider the case MJM, < 0.6, so that eq 6 accurately represents Mt/M, and M,/M,. One can write eq 8 in the form
-M,’ M,’
- K(t)l/’ - K(t,)’lz
(15)
+ t,.
For this case
1 - K(t,)’i2
where K = ~ ( D / T P ) ’ and / ~ t = t’ (Mt/M, < 0.6)
These equations demonstrate the validity of the conclusions stated above concerning the initial slope method. Specifically,the initial slope is actually zero, but increases rapidly with t’ and approaches K/[1 - K(t,)1/2]. This upward curvature occurs more quickly for small storage times than for large storage times, and for sufficiently small values of storage time will not be apparent as a practical problem. One can see, however, that the determination of diffusion coefficients by the initial slope method is more prone to errors than the half-time method if unacknowledged elution has occurred during storage. Invalid qualitative conclusions concerning the transport mechanism may even result if unacknowledged desorption occurs prior to the formal desorption experiment. It is common to identify so-called “anomalous diffusion” systems by their sigmoid response when plotted on square root time coordinates (Rogers, 1965). In such cases, where experimental artifacts are absent, it is believed that the mobility of penetrants in the polymer matrix is a function of time (due to relaxation processes occurring) as well as a function of local penetrant concentration. When sigmoid responses of fractional desorption vs. t1/2are observed, one must, therefore, be certain that the behavior is not due to a simple artifact such as described here before concluding that an independent phenomenon is occurring which confounds simplistic expectations.
Ind. Eng.
1
\
0.6
u 0.0
to
1
102 t,
103
104
105
(minutes)
Figure 4. Ratio of the apparent to the actual diffusion coefficient, determined by the "initial slope" method, as a function of storage time, t,, during which unacknowledged prior desorption occurs. Note the tendency for the ratio D'/D to approach zero at long storage times. The values of D' were estimated from the initial slopes arbitrarily determined by eye. As shown in eq 14, the initial slopes of curves such as those in Figure 1 actually approach zero for nonzero storage time desorption losses; however, this limit is not easily appreciated in considering plots of fractional loss data at short and intermediate times.
Extension to Variable Storage Conditions. Much of the discussion so far has refered specifically to the case where storage conditions were identical with the experimental conditions. The form of the results is actually independent of storage temperature for a given constant experimental temperature. As long as zero concentration of diffusant is maintained at the boundaries and symmetry is maintained during both storage and testing, the form of the results will be identical with those discussed for the case where storage conditions are identical with experimental conditions. Diffusion from a slab is dependent on one parameter, i.e. the characteristic group D t / C 2 (see eq 3 and 4). The storage period imposes a nonuniform concentration profile a t the beginning of the desorption experiment. The equation for the time-dependent concentration profile, C ( x , t ) (Crank, 1975), is given in eq 17, where Co is the
C -_ CO
4
-
;ZoiX exp[-D(2n + 1 ) r 2 t / C 2 ]cos (17)
initial concentration of diffusant in the sheet. Equation 17 contains the time dependence within the characteristic group Dt/C2; therefore, for any storage temperature or series of storage temperatures, the same series of concentration profiles will exist in the slab for the same values of D t / C 2 . Different values of storage temperature will require varying storage times to achieve a given common profile. The Arrhenius temperature dependence of the diffusion coefficient will determine the value of D which, in turn, will set the value o f t through the dimensionless group D t / t 2 .
Conclusions Failure to account for the nonuniform concentration profile, which develops in a sample during storage prior to a desorption experiment, introduces error into the calculation of a diffusion coefficient. The value of the ratio of the apparent diffusion coefficient to the actual diffusion coefficient, D ' / D l decreases as the storage time increases and asymptotically approaches a value of 0.70 if the half-time formula is used to evaluate the apparent diffu-
Chem. Prod. Res. Dev., Vol. 22, No. 1, 1983 89
sion coefficient. If the initial slope method is used to evaluate the apparent diffusion coefficient,D 'ID decreases with increasing storage time and approaches an even more unrealistic lower limit of zero for large storage times. The error in determining the diffusion coefficient is, therefore, likely to be much higher with the initial slope method than with the half-time method. The estimation of the diffusion coefficient by the initial slope method is compromised further by the recognition that the limit of the initial slope is actually zero for any finite storage time although evaluation of the initial slope "by eye" typically suggests erroneous finite and systematically decreasing slopes with increasing storage time. Qualitatively incorrect interpretations can arise as well when unacknowledged desorption occurs prior to the formal desorption experiment since the resulting sigmoid kinetic response resembles so-called anomalous diffusion behavior produced when slow polymer relaxation controlled processes strongly influence the transport of small molecules in the polymer. Acknowledgment The authors gratefully acknowledge the financial support of this work provided by the Plastics Institute of America in the form of a fellowship to Mr. James L. Osborne. We appreciate the useful discussions with Dr. Alan R. Berens which stimulated our interest in this problem. Nomenclature D = actual diffusion coefficient cmz/s or cm2/min D' = apparent diffusion coefficient calculated neglecting elution which may occur during storage prior to experiment, cmz/s or cm2/min t = actual diffusion time starting initially with uniform concentration of diffusant in polymer, s or min t, = storage time of polymer prior to desorption experiment, s or min t' = apparent time measured by experimenter neglecting storage time, t' = t - t,, s or min t1/2= actual half-time,time required for the actual fractional loss of diffusant, M,/M,, to become 1 / 2 , s or min t 'l/z = apparent half-time required for the apparent fractional loss of diffusant, M,'/M,', to become l/z, s or min Mt = amount of diffusant that diffuses from polymer in actual diffusion time t M , = amount of diffusant that diffuses from polymer during storage M , = amount of diffusant that diffuses from polymer in infinite time M,' = amount of diffusant that diffuses from polymer during apparent time, t', M,' = Mt - M , M,' = amount of diffusant that diffuses from polymer in infinite time after storage, M,' = M , - M , C = diffusant concentration Co = initial diffusant concentration C = thickness of plane sheet of polymer, cm N = rate of diffusant transfer per unit area x = space coordinate measured normal to polymer section, cm T Dt/C2,dimensionless total time T, Dt,/C2,dimensionless storage time T' = D/C2(t- t,) Literature Cited Berens, A. R.; Daniels, C. A. Polym. Eng. Sci. 1076, 16. 552. Berens, A. R.; Hopfenberg, H. B. "Recent Developments in Separation Science"; Li, N. N., Ed.; CRC Press: Cleveland, 1977: Vol. 111, p 300. Crank, J. "The Mathematics of Diffusion", 2nd ed.;Clarendon Press: Oxford, 1975; Chapters 1 and 4. Rogers, C. E. "Solubility and Diffusivity in Organic Solids", in "Physics and Chemistry of the Organic Solid State": Fox, D.; Labes, M.: Weissberger. A,, Ed.; Interscience Publishers: New York, 1965; VoI. 11.
Received for review March 29, 1982 Accepted September 20, 1982
