1663
Macromolecules 1985,18, 1663-1676
decrease the positive charge a t the carbocation active center and favor free ion formation. In contrast, the chloro substituent is electron withdrawing, which would increase the positive charge a t the growing carbocation center and favor the formation of solvated ion pairs. If these resonance effects occur to a significant extent, then the polymerization reaction of the p-chloro monomer could be much more sensitive to the environment of the carbocation active center than that of the p-fluoro monomer. The values of syndiotactic and isotactic sequence lengths, as calculated from the equations above, for the fluoro-substituted series of polymers obtained in mixtures of methylene chloride and hexane or toluene are collected in Table VI. It is again clearly seen that the isotactic configuration is very difficult to form, and the average isotactic sequence length is only one for all of the polymers; that is, no units longer than diads were formed and only very few of those. The syndiotactic sequence length shows a very slight solvent effect, becoming longer with increasing solvent polarity.
Experimental Section All polymerization reactions were carried out at -78 "C under an atmosphere of dry argon with reagents purified and dried by techniques as previously reported.' Proton NMR spectra for all the polymers obtained in mixtures of methylene chloride and hexane were obtained on 10% (w/v) solutions of the polymers in o-dichlorobenzene at 100 "C with p-dioxane as the internal standard. Spectra for all the polymers obtained in mixtures of methylene chloride and toluene were obtained on 10% (w/v) solutions of the polymers in toluene-d8 a t 100 "C with p-dioxane as the internal standard. Triad tacticities were determined from the relative areas of the three a-methyl proton resonances.
Acknowledgment. The high-field NMR experiments
were performed a t the NMR Facility for Biomolecular Research located at the F. Bitter National Magnet Laboratory, M.I.T., and a t the NMR Facility at the University of Massachusetts a t Amherst. We are grateful to the National Science Foundation for the support of this work under Grant No. DMR 80-22273.
References and Notes Lenz, R. W.; Faullimmel, J. G.; Jonte, J. M. In "Initiation of Polymerization", Bailey, F. E., Jr., Ed.; American Chemical Society: Washington, DC, 1983;p 103. Lenz. R. W.: Sutherland. J. E.: Westfelt. L. C. Makromol. Cheh. 1976,'177,653. ' Lenz, R. W.: Westfelt. L. C. J. Polvm. Sci.. Polvm. Chem. Ed. 1976,'14,2147. "CRC Handbook of Chemistry and Physics", 56th ed.; CRC Press: Boca Raton, FL, 1984. Ramey, K. C.; Statton, G. L.; Jankowski, W. C. J. Polym. Sci., Part B 1969,7,693. Lenz, R. W.; Regel, W.; Westfelt, L. C. Makromol. Chem. 1975, 176,781. Kunitake, T.; Aso, C. J. Polym. Sci., Part A-1 1970,8, 665. Szwarc, M. Makromol. Chem. 1965,89,44. Braun, D.; Heufer, G.; Johnson, U.; Kolbe, K. Ber. BunsenGes. Phys. Chem. 1964,68,959. Flory, P. J. "Principles of Polymer Chemistry"; Cornel1 University Press: Ithaca, NY, 1953;p 553. Higashimura, T.; Kishiro, 0. Polym. J. (Tokyo) 1977,9,87. Lenz, R. W.; Jonte, J. M.; Gula, D. J. Polym. Prepr. (Am. Chem. SOC.,Diu. Polym. Chem.) 1985,26 (1). Ohsumi, Y.; Higashimura, T.; Okamura, S. J. Polym. Sci., Part A-1 1966,4, 923. Sawamoto, M.; Masuda, T.; Higashimura, T. Makromol. Chem. 1976,177,2995. Imanishi, Y.; Higashimura, T.; Okamura, S. Kobunshi Kagaku 1960,17,236. Coleman, B.D.; Fox, T. G. J. Polym. Sci., Part A 1963,I , 3183. Bovey, F. A. "Polymer Conformation and Configuration"; Academic Press: New York, 1969;p 41. Brown, H. C.; Okamoto, Y. J. Am. Chem. SOC.1957,79,1913. 1
-
Morphology of Poly(ethy1ene terephthalate) Fibers As Studied by Multiple-Pulse lH NMR John R. Havens' and David L. VanderHart* Polymers Division, National Bureau of Standards, Gaithersburg, Maryland 20899. Received August 20,1984
ABSTRACT: Drawn poly(ethy1ene terephthalate) (PET) fibers annealed under various conditions are investigated by proton spin diffusion as detected through nuclear magnetic resonance. The primary objective is to study morphology on the 1-50-nm scale, the smaller dimensions of whicl have proved difficult to characterize for P E T by conventional techniques. The spin diffusion experiment is comprised of three periods: generation of a magnetization gradient among different domains, relaxation of the gradient by diffusion for a variable time, and separate detection of the magnetization corresponding to each domain. The use of a multiple-pulse sequence permits spin diffusion t o be confined to the second period, resulting in enhanced resolution among the domains. This procedure allows the magnetization decay observed during the detection period to be decomposed into three components, which are assigned to mobile noncrystalline, constrained noncrystalline, and crystalline domains. Rates of polarization redistribution among these three components are studied as a function of the diffusion time. Computer modeling is carried out in order to relate these measurements to the spatial arrangement and size of the three components. The results quantify the increase in crystallinity and in crystallite size upon annealing. Information pertaining t o the structure of the noncrystalline region, the importance of noncrystalline chain orientation, and the relative surface areas of the crystallites is also presented.
Introduction A. PET Morphology. Much of the usefulness of polymeric derives from the ability to their macroscopic properties by processing. T~ design the optimum treatmentfor a particular application requires a Present address: Raychem Corp., Menlo Park, CA 94025.
knowledge of chain structure on the 1-50-nm scale, a region which can strongly influence mechanical performance. In this work we characterize the local morphology of uniaxially oriented poly(ethy1ene terephthalate) (PET) fibers. Such fibers have important structure in the 3-15-nm regime. Previous efforts to characterize this region have been made through a variety of measurements: small-angle and wide-angle X-ray scattering,'-' heat of fusionF8d e n ~ i t y , ~ @ , ~
This article n o t subject t o U.S. Copyright. Published 1985 by the American Chemical Society
1664 Havens and VanderHart
Macromolecules, Vol. 18, No. 9, 1985
nation of PET fibers below T since the noncrystalline and infrared,l@16nuclear magnetic resonance (NMR),l6-l9and crystalline T i s are comparabfe. Moreover, if one attempts optical birefringence.16Js Interpretation of the results has to go above Tgfor an oriented PET fiber sample in order proved difficult, however, because many of these techniques give information which either is localized within a to provide better contrast between crystalline and noncrystalline Tis,the fibers change irreversibly. On the other repeat unit or involves an extrapolation to the local level hand, if one prepares magnetization gradients based on from an average macroscopic measurement. Another, more different intrinsic TI, behavior in different regions, then direct means of characterization is desirable. In this paper we assess the use of some recently developed NMR techa certain amount of confusion results from the fact that spin diffusion proceeds even in the presence of a strong, niques to investigate the morphology of PET samples with resonant radio-frequency field, albeit the corresponding different processing histories. Areas of particular interest diffusion constant is only half that characterizing spin include determination of percent crystallinity, validity of diffusion in the laboratory frame.29 For example, a T1, a two-phase (crystalline/amorphous) model, size and preparation period of 1ms would result in significant spin spatial arrangement of the domains, and structure of the diffusion over a distance of 1nm, thus blurring structural noncrystalline material. We will also explore the meafeatures at this level. Although this fuzziness can be acsurement of crystallite surface area through spin diffusion behavior. counted for to some extent in the mathematics of modeling the diffusion behavior, the ability to generate sharp graThis research is part of a joint effort with Firestone Tire dients experimentally reduces both the complexity of the and Rubber Co. to examine the morphology of drawn and mathematical analysis and the ambiguity of the morphoannealed P E T fibers. Further details concerning sample logical models. preparation, physical properties, and morphological Our approach has been to use a multiple-pulse radioanalysis by DSC and X-ray diffraction will be published frequency ~ e q u e n c during e ~ ~ ~ the ~ ~ preparation period. separately.20 This accomplishes two purposes. First, magnetization B. Spin Diffusion Experiment. For many heterogradients are created by different relaxation rates of the geneous solid samples NMR can be used to study domain various domains in the sample. Relaxation under multisizes by monitoring spin diffusion. Common synthetic ple-pulse schemes is related to the spectral density of polymers, with their abundance of hydrogen, are candimolecular motion in the mid-kilohertz regime.31-33In this dates for such IH NMR measurements. The secular respect the magnetization behavior is similar to that in the Hamiltonian describing the many-body proton-proton 'H TIPexperiment. The rigid crystalline regions of the dipolar couplings includes a spin-exchange term. This sample relax slowly because the molecular motion is not term forms the basis for understanding magnetization centered a t mid-kilohertz frequencies; moreover, the amtransport in inhomogeneously relaxing spin systems. Spin plitude of motion is limited. The more mobile noncrysdiffusion is the name given to this transport process. The talline domains relax relatively quickly. (NMR measurespin diffusion experiments described herein are analogous ments of motional rates and amplitudes in PET have been to heat conduction experiments in the presence of a temp ~ b l i s h e d . ~ For ~ ! ~some ~ ) PET samples the time constants perature gradient where magnetization and temperature characterizing these exponential decays (Tlz,'s) differ by are the parallel quantities. 2 orders of magnitude, as will be shown below. The second The importance of spin diffusion in interpreting magpurpose in using multiple-pulse cycles during the prepanetic resonance relaxation was recognized in the 1 9 4 0 ~ 1 . ~ ~ ration (and detection) period is to slow spin diffusion. More recently, efforts to determine domain sizes in solid Coherent averaging t h e ~ r y ~ Odemonstrates >~l that the polymers have been made by following 'H spin diffusion multiple-pulse schemes ideally result in elimination of the behavior. Most of this work has been based upon the dipolar couplings among protons, which are necessary for Goldman-Shen experiment.2z P01yethylene:~S polyamide spin diffusion to occur. In practice, the 'H dipolar line blends,24and polyurethanesz5all have been investigated width of crystalline polymers can be reduced by 2 orders by the original Goldman-Shen sequence, which exploits of m a g n i t ~ d eusing ~ ~ , this ~ ~ method. Spin diffusion is then a difference in Tz's to generate the magnetization gradient effectively quenched, and the magnetization gradients between domains and which records the free induction generated by mobility differences among the domains decay (FID) during the detection period. For various spin maintain their sharpness until the diffusion period in the diffusion times polarization in each domain is then quanlaboratory frame begins. The suppression of spin diffusion tified by resolving the fast and slow contributions to the during both preparation and detection also carries with FID. Several modifications of this sequence also have been it the advantage that heterogeneity of molecular motion developed. In their study of PET films, Cheung et a1.26 in the noncrystalline regions may be identified and the used a difference in lH Tip's to generate the gradient and corresponding domain sizes determined, even though dodetected the signal during a dipolar-narrowed Carr-Purcell main sizes may be very small. sequence,27 which effectively quenches spin diffusion The pulse sequence we have chosen for the diffusion among the protons. Packer et a1.28have used a T,, prepmeasurements is shown in Figure 1. The multiple-pulse aration period with detection of the free induction decay scheme used in both preparation and detection periods is following an optional second selection period also based designated MREV-8%3and may be represented as follows: on 'H spin locking. In each of these experiments, the T - P ~ - ~ - P7-P-y-T-P-x-2 ~-~ 7-P-,-T-Py-2 T-P-y-7-Px-T magnetization decay during the detection period is decomposed into two or three components, the relative amwhere P, denotes a 90" pulse about the ith axis in the plitudes of which are then correlated with the variable spin rotating frame and where T and 2~ signify pulse separadiffusion time in the laboratory frame. The rate of distions. This sequence is diagrammed in Figure 2. Vega appearance of the original magnetization gradient can be and V a ~ g h a nhave ~ ~ studied the relaxation behavior for related to domain sizes by making a reasonable estimate this eight-pulse sequence in the toggling frame (the inof the diffusion coefficient. teraction frame of reference defined by the radio-frequency Each of the spin-diffusion experiments referred to above pulses). Spin-lattice relaxation along the (101) direction has disadvantages if applied to P E T fibers. The Goldin the toggling frame, normally one of the principal axes man-Shen experiment is not appropriate for the examiof relaxation, is characterized by the time constant Tlxz.
Macromolecules, Vol. 18, No. 9, 1985
Morphology of PET Fibers 1665
45
45,
?RE?ARATIOh
DETECTIOV
SPI b. Dl F E'JSI02
fVRE\-8'
.I;
1'4RFV-3)
LOC4NG
x
FIELD
X
Figure 1. Pulse sequence for measuring spin diffusion rates in PET samples. Multiple pulse is performed during both prepa-
ration and detection periods. The coordinate system represents the toggling frame of reference, in which a resonance offset generates an effective spin-lockingfield along the (101) direction. -
-
z
i
7.
=-
i
F
' Figure 2. Representation of the MREV-8 multiple-pulse sequence. All pulses are 90'. For average Hamiltonian theory to where llHdll apply, the cycle time t, must be much less than is the magnitude of the dipolar Hamiltonian.
lvdll-l,
If the relaxation measurement is performed with spinlocking and if the cycle time is sufficiently short so that average Hamiltonian theory applies, Tlxrobeys the following equation^:^^,^^ l/Tlxz = 2Ahf2r2/9rC, l / T l x z= 2 A M 2 r C / 3 ,
for for
7,
7,
>> 7 0. The second relation refers to flux continuity a t the interfaces. This condition stipulates that the interfaces cannot act as independent sources or sinks of magnetization. Initial conditions must also be required for a numerical solution to the diffusion equation. Knowledge of the magnetization level in each domain a t an early time in the diffusion experiment is desirable. We estimate these levels by considering the decomposition of a Tlx. decay accumulated after a short (300 ps) period of spin diffusion. The magnetization integrals which are obtained from the decomposition must then be converted into magnetization levels by considering the relative domain sizes from Table 11. Because the gradients at t = 300 ps are assumed to be perfectly sharp, calculated magnetization levels a t short diffusion times ( I 1ms) may be incorrect. For this reason t = 3 ms is the first diffusion time for which agreement between calculated and experimental results is used to judge the validity of the model. The relaxation term in the diffusion equation is characterized by a rate R, which is different for each of the domains in the model. R represents the rate of recovery of magnetization due to laboratory-frame spin-lattice relaxation. For the PET samples under study all proton Tl's are a t least 2 orders of magnitude longer than the time required to equilibrate spin temperatures in the three domains (cf. Table 111). Therefore, all R values are small and have little effect on the solutions of the diffusion equation. For heterogeneous polymers with substantially shorter Ti's, the determination of accurate R values for the different domains is an important part of modeling the morphology by spin diffusion. 2. Solutions. With the foregoing information, one can calculate a numerical solution to the diffusion equation, with magnetization profiles given for various diffusion times. Numerical solutions were obtained through DISPL, a software package written by Leaf et al. at Argonne National labor at or^.^^ Representative results for sample N
EXPERIME\-A
Ce-CbLA't
Figure 8. Comparison of calculated and observed diffusion for sample N. Calculationsare based on the one-dimensionalmodels of Figure 6. Magnetization levels are plotted on the ordinate, with the dashed lines indicating the equilibrium levels.
are shown in Figure 7. Here the magnetization level is given in arbitrary units as a function of position. The calculation is based upon the first (one dimensional)model of Figure 6. The magnetization levels a t 300 ps are represented by the solid horizontal line segments in Figure 7. After longer diffusion times the gradients are attenuated, until finally a uniform magnetization density exists throughout the sample. (The intersection of the diffusion profiles a t a single point is consistent with the behavior of an extended source of infinite extent.53) For each model under consideration, the calculated magnetization integral and average magnetization level for each domain can be compared to those from experimental observation. Agreement over the entire range of diffusion times is expected. For convenience, the single parameter which was varied to improve the fit was the diffusion coefficient D , rather than a scaling factor for the domain dimensions. From this best-fit value of D (indicated by D*),the scaling factor for the domain dimensions was determined. The relationship between D and x dictates that an increase in D by a factor f is equivalent to scaling x by f - l 1 2 a t constant D. Values of the diffusion coefficients were varied in steps of 0.5 X cm2/s, and those which fit the experimental data best were selected for the calculations of domain size. In order to convert this scaling factor to real dimension, the value of D appropriate to P E T was estimated. By taking the diffusion coefficient of polyethylene (PE) to be 6.2 X cm2/s58and by assuming that the diffusion constant should scale as the concentration of protons to the 1 / 3 power, a reasonable estimate of 5.0 X cm2/s is obtained for the diffusion coefficient of PET. For the two-dimensional model D was varied independently in both dimensions. With DISPL it is also possible to specify different D's for the various domains. While this flexibility is not required for PET, it would be necessary for any polymer which exhibits a strong contrast in T2behavior. Figure 8 shows the best fit obtained between observed and calculated magnetization levels for sample N. The first model of Figure 6 was used to establish conditions on the diffusion equation. The agreement in Figure 8 suggests that the first model is a good representation of the annealed film morphology. However, the plot on the right is for the average of the short TI,, components; the agreement for each of the individual constituents is not nearly so good, as seen in Figures 10 and 11. This indicates that the noncrystalline domain structure is not well represented by the first model. In Figures 9-11 the best fits for all
Macromolecules, Vol. 18, No. 9,1985
1670 Havens and VanderHart
TablgV PET Domain Sizes for One-Dimensional Models
s
a
'O:
d o m a i n size, nm xtal inter amorDh
EXPERIMEYTAL
0
-
XTAL- I Y T E Q - A M X T A L - b t 4 . INTEQ
...
2 D I U
samale
D*
L. nm
C N D257-0 D200-22 D200-0
16.0 8.0 7.5 15.0 13.0
11.2 15.8 16.4 11.6 12.4
2.9 7.4 7.1 3.9 4.2
2.3 2.3 2.6 2.0 2.3
3.7 3.8 4.1 3.8 3.6
% xtal
26 47 43 34 33
Table VI PET Domain Sizes (nm) for Two-Dimensional Model xtal inter amorph 'io 1 0
20
4c
30 TINE
DIFFUSION
60
50
lmSe:i
Figure 9. Plot of percentage of magnetization contributing to the long Tl, as a function of diffusion time. Experimental data are for sample D257-0. The curves represent the best fits for calculated results for the three models.
-
XTAL INTER
__
XTAL-AM
20
DIFFUSION
AW
50
% xtal
4.4 6.2 6.4 4.0
8.3 8.4 9.4 7.6
3.6 5.0 5.2 4.2
26 47 43 34
long period r 11.2 15.8 16.4 11.6
long period z 8.0 11.2 11.6 8.2
Table VI1
Percent Crystallinities lit. valuesb WAXS" DSCe p c WAXS 54 65 65
13
28d
48
54
47
43
39' 39f 41d
Table VI11 Long Periods (nm)
lit.
. .
.~ .
sample C D257-0 D200-22 D200-0
L NMR
ax L SAXS"
11.2 16.4 11.6 12.4
16.1 17.9 14.1 15.5
transverse L SAXS" 8.7 13.6 17.0 11.1
valuesb ax L SAXS 17.5' 18.4d 13.2O 12.2d
OReference 20. bThese values correspond to samples whose treatment was similar to those used in this study. Reference 12.
20
i/
2
Reference 5. eReference 6.
EXPERlMENTAL 0'
I
fl
z
ImYcI
TIME
. . ....
p
r
€0
301
&
z
Reference 20. bThese values correspond to samples whose
INTER
Figure 10. Plot of percentage of magnetization contributing to the intermediate Tkz as a function of diffusion time. Experimental data are for sample D257-0. The curves represent results calculated from the same parameters as those used in Figure 9.
B
r 8.3 8.4 9.4 7.6
treatment was similar to those used in this study. cReference 5. dReference 55. eReference 8. 'Reference 18.
40
30
r 2.9 N 7.4 D257-0 7.0 D200-22 4.0
sample NMR DSC" p" C 26 46 42 N 47 D257-0 43 D200-22 34 42 47 D200-0 33 46 50
2 DIM
10
sample C
,'?
-
XTAL-1NTER - A M
__
X T A L - A M .INTER
10.;
2 DIM
1 '
10
20
30 DIFFUSIOV
50
40 TIME
€0
Im5ecl
Figure 11. Plot of percentage of magnetization contributing to the short T,,, as a function of diffusion time. Experimental data are for sample D257-0. The curves represent results calculated from the same parameters as those used in Figure 9. three components are shown for each of the models. Here magnetization integrals are presented for D257-0. Similar behavior also is exhibited by the other samples. Figure 9 indicates that all these models provide equally good fits to the dependence of the amount of rigid component on diffusion time. The diffusion coefficients were optimized for this component and then held fixed. Figures 10 and 11depict the behavior of the noncrystalline regions of the sample and allow differentiation among the models in terms of quality of fit. The worst agreement between observed and calculated recovery is obtained for the first model. The two-dimensional model provides a slightly
better fit than the second, especially in the early diffusion time regime. Once the optimum value of the diffusion coefficient has been found for each of the models, the domain sizes can be calculated directly. This information is presented in Table V for the two one-dimensional models. Despite the observation from Figures 10 and 11that the second model fits the experimental values considerably better than does the first, the optimum agreement for both models occurs for the same diffusion coefficient. The calculated domain sizes then are the same for the two linear models, although the spatial arrangements are different. The quantity labeled L in Table V is the structural repeat distance in the models and corresponds to twice the distance enclosed by the dashed boundary lines in Figure 6. This number is the analogue of the long period determined by small-angle X-ray scattering. In Table VI results from the two-dimensional model are shown. The letter r represents the horizontal direction in Figure 6; z represents the vertical. Diffusion coefficients were optimized independently for each of the dimensions. We are now able to compare the crystallinity and long-period results from NMR with those from X-ray,
Morphology of PET Fibers 1671
Macromolecules, Vol. 18, No. 9, 1985
I
B
!
.
C
XTAL
B
INTER
A
0200 - 22
C
I C
A
A
B
B
A
D257 - 0
c
L .....
~~~~
!
A ......
c
,.... .. ........
P
10 nm
Figure 12. Representation of the morphology of two PET fibers. Results are calculated from a two-dimensional solution to the diffusion equation. Both vertical and horizontal directions are drawn to the indicated 10-nm scale. Region C has infinite dimension in the vertical direction.
DSC,and density measurements. These data are shown in Tables VI1 and VIII. Values obtained from the literature for similarly treated samples are also presented. The NMR crystallinities are regarded as the fractions of the long T,,, component recorded in decays from equilibrium under multiple pulse. The NMR long periods apply to both the one- and two-dimensional modeling results for the repeat distance between consecutive crystalline lamellae. Finally, we present in Figure 12 schematic drawings of the morphology for two of the P E T fibers, calculated from the spin diffusion results. The models are drawn to scale and represent the best fit to the experimental data for the simplified block structures which we have considered. On the basis of experiments which we will not detail here, the crystallinities as determined by NMR are considered to be accurate to within f 5 % . A reasonable estimate of the uncertainty in domain sizes is h0.6 nm. D. Relative Crystallite Surface Areas. The interfaces between different domains in a heterogeneous polymer can influence much of its macroscopic behavior, yet the detailed nature of the interface, as well as the area it covers, is typically difficult to characterize. For the PET fibers considered here, we would like an estimate of the crystalline/noncrystalline surface area in a given volume of sample. On a relative basis such information can be extracted from the initial magnetization recovery as a function of spin diffusion time. Thus, the ratios of crystallite surface areas for different samples can be correlated with annealing treatment. Since we are interested primarily in relative surface areas of the crystallites, a twophase, crystalline/noncrystalline model suffices. The two short Tlxzfractions are summed to comprise the noncrystalline component. This simplification allows exact analytical solutions to be used in dealing with the spin diffusion behavior. We first summarize our approach to the problem and follow this with a more rigorous justification of the procedure. Finally, we present the experimental observations. 1. Summary of Approach. After preparation of a magnetization gradient under multiple pulse, the higher level of polarization present in the crystallites causes
magnetization to diffuse into noncrystalline regions. The amount of magnetization which flows via spin diffusion during a time t into the noncrystalline domains is designated AMNC(t),where AMNC(t)= MNC(t) - MNc(0). For short diffusion times AMNc(t)is directly proportional to the size of the initial magnetization gradient (at time t = 0), to the surface area though which diffusion occurs, and to some function of time which characterizes the diffusion rate. We can formulate this relationship as
AMNC(t) [@(O) Q:
- fiNc(0)]Af(t)
where @(O) represents the crystallite magnetization level (total magnetization in the crystallites divided by the fraction crystallinity) at time 0, A is the total surface area of the crystallites in a given volume, andf(t) represents the time dependence of noncrystalline magnetization growth. As shown in the following section, f ( t ) is proportional to t1/2for sufficiently short diffusion times, but the specific functional form of f ( t )is not critical in determining relative surface areas. The important factor is that we consider only those diffusion times short enough so that the same f ( t ) applies equally well for all samples-in other words, so that differences in domain sizes do not cause variations in the rates of propagation of the magnetization fronts. Thus, the diffusion times used in the analysis must be shorter than the time required for two diffusing magnetization fronts to meet in the middle of either crystalline or noncrystalline domains. If this condition is met, the crystallite surface areas of two samplts can be compared, since both AMNC(t)and [W(O) - MNc(0)]are experimentally measurable. Slopes of plots of MNC(t) vs. t 1 / 2can be ratioed (after appropriate corrections) to yield the desired surface area information. 2. Justification. The crystalline/noncrystalline finite-phase system corresponds to the one-dimensional lamellar models presented in Figure 6, if the two noncrystalline regions are assumed to be indistinguishable. Crank53has solved the diffusion equation analytically for such a system; the following conditions describe the situation. The magnetization is present initially only in the crystalline region with a concentration C,. This region extends from x = 0 to x = h. The magnetization diffuses a t times t > 0 into the noncrystalline region of width 1 h, which has an initial concentration of 0. Reflecting boundary conditions are applied at x = 0 and x = 1 such that dC/dx = 0, where C = C(x,t) is the concentration of magnetization. The profile of concentration after diffusion has occurred for time t is given as
h
+ 2 n l - x + erf h - 2 n l + x 2(Dt)1/2 2(Dt)'l2
where erf z is the error function, defined by
The experimental observable of interest is AMNc,which in this case is equal to SdC dx. From the above expression for C , it is clear that AMNcis directly proportional to C,, the initial size of the magnetization gradient. It is also readily apparent from the nature of the model that hMNc scales directly with the amount of surface area through which diffusion occurs. Unfortunately, the time dependence of aMNc is less easily evaluated, since it involves the integral of an infinite series of error functions. To simplify matters somewhat, we consider the following system, whose behavior will be identical with that considered above for sufficiently short diffusion times. Just before diffusion begins ( t = 0), the magnetization is dis-
Macromolecules, Vol. 18, No. 9, 1985
1672 Havens and VanderHart
tributed such that C = Co for all x 5 0 and C = 0 for all x > 0. Under these conditions the solution to the diffusion equation is given by53
C(x,t) = o.5c0erfc x/2(Dt)l12 where the error-function complement, erfc z, is defined by erfc z = 1 - erf z = (2/n112)c(lme-"'dq Now the integral representing the total noncrystalline magnetization becomes more tractable:
AMNC(t)= J m C ( x , t ) dx = Co(Dt/a)'I2 0
From this result it is clear that for short diffusion times (before the magnetization fronts feel the finite sizes of the domains) the noncrystalline magnetization grows with t1I2. A remaining question is how long this t1I2dependence can be expected to apply in a system with finite domain sizes, such as the one-dimensional, crystalline/noncrystalline model. A reasonable approximation is that the t112 behavior will be observed until two magnetization fronts diffusing in opposite directions meet in the center of either the crystalline or noncrystalline domains. (Unfortunately, the magnetization front is a somewhat nebulous quantity, since it becomes increasingly diffuse as time progresses. This tends to make the deviation from t112behavior a gradual one, but we feel that the meeting of fronts offers a useful guide to the point at which this deviation becomes considerable.) The front position can be defined by the root mean square displacement, ( x 2 ) l 1 2 , which is in turn defined by ( x 2 ) = L m x 2 Cd n / I 0m C dx
Using the above expression for C in terms of the errorfunction complement and again being restricted to short diffusion times, we can integrate by parts and obtain (x2)
= (4/3)Dt
This result can be compared with the well-known solution for an instantaneous &function source, which has been used in previous spin diffusion studies:26B28 ( x 2 ) = 2Dt
Estimates of finite domain size based on this latter formula are probably too large. Using the first formula and assuming a diffusion constant of 5.0 X cm2/s, we calculate that in 4 ms the magnetization front will move roughly 1.6 nm, while in 20 ms of diffusion it will move -3.7 nm. These dimensions are approximately half those of the thinnest dimension in samples C and N (see Table VI). Therefore, we can expect that the magnetization recovery in the noncrystalline regions will be linear in t1/2 for a t least the first 4 ms with sample C and for -20 ms with sample N. After these times the magnetization fronts should meet, and the rate of growth of AMNc will be reduced. 3. Experimental Results. Plots of magnetization in the noncrystalline domains vs. t112are shown in Figure 13 for two PET samples. The horizontal dashed lines indicate the equilibrium level of noncrystalline magnetization. The experimental points appear to be linear with t112out to roughly 10 ms, at which point the rate of growth slows. The broadening of the magnetization fronts with increased diffusion time is probably responsible for the absence of sharp deviations from linearity after 4 and 20 ms of diffusion, as expected from the calculations in the previous
2
-
C!=V53b,
6
4
- b4E
1:: IT-:
I
