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Interdiffusion at the interface of polymeric bilayers: evidence for reptation?
Journal of Non-Crystalline Solids 131-133 (1991) 703-708 North-Holland
703
Interdiffusion at the interface of polymeric bilayers" evidence for reptation? G . P . F e l c h e r a n d A. K a r i m Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
T.P. R u s s e l l IBM Research Division, Almaden Research Center, San Jose, CA 95120, USA
Neutron reflection is used to study the interdiffusion in equal molecular weight polystyrene bilayer melts with a spatial resolution of 10 ,~. Interracial widths and concentration profiles at the bilayer interface are obtained for annealing times up to and beyond the reptation time, Td. For t < T0, the reptation model predicts a mean square displacement of monomers whose time dependence is a power law, the exponent of which changes with time. For the relatively lightweight polymers of M - 233000, the mean square displacements of monomers is in general agreement with predictions; the permanence of a discontinuity at the interface - which also follows from the reptation model - is observed only for molecular weights M - 10000001
1. Introduction There is considerable interest in the basic mechanism of diffusion in polymer melts: by contrast with atoms or simple molecules, the long chains of m o n o m e r s forming the polymer molecules become entangled along their length. Of the various models proposed [1], the most successful is the reptation model, first proposed by de Gennes [2] and later developed by Edwards [3]. In the reptation model the polymer molecules diffuse by sliding along their contour within a hypothetical tube, which is defined by the locus of intersections of the molecule with its neighboring molecules, The time taken by the molecule to move out of its initial tube is the reptation time, Td. The main success of this theory consists in giving the correct scaling of the bulk self-diffusion coefficient, D * , with molecular weight, M: D * - M -2. This dependence has been confirmed by a variety of different techniques: infrared spectroscopy [4], forward recoil spectrometry [5] and secondary ion mass spectrometry [6]. The reptation theory gives further predictions
for the m o t i o n over distances shorter than the size of the molecule, which might be identified with the radius of gyration, gR" The m e a n square displacement of m o n o m e r s is predicted [3,7] to follow a time dependent power law: 0 2 = t ~, where a < 1 for t < 'I'd, becoming 1 at t = ~'d, in the limit of conventional diffusion. T o observe such displacement requires a technique with a spatial resolution less than the size of the diffusing molecule, and unfortunately the above-mentioned techniques have a resolution of 100 ,~, to be c o m p a r e d with a radius of gyration gR -- 150 ,~, for a molecule with a typical weight of M = 300 000. With the powerful technique of n e u t r o n spin-echo [8], the time dependence of the correlation function has been determined, but only up to times that were a fraction of ~'d" The diffusion in p o l y m e r melts can also be studied by observing the time evolution of the profile at the interface of two p o l y m e r (say, A and B) layers put in contact at time zero. This is exactly the kind of measurement d o n e b y neutron reflection, with a resolution that m a y a p p r o a c h 10 ,~. Actually, neutron reflection can give the de-
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
704
G.P. Felcher et al. / lnterdiffusion at the interface of polymeric bilayers: evidence for reptation?
tailed shape of the concentration profile, which in the reptation theory has some unique characteristics [9,10]. The initial discontinuity of concentration of A (or B) at the interface is predicted to persist up to ~'d" This 'concentration gap' is a direct consequence of reptation, since only the polymer molecules that have their ends close to the interface can cross it, while the rest of the molecules close to the interface simply reptate along their contour length and do not cross the interface. In this paper, we present the results of a series of neutron reflection measurements on bilayers of polystyrene of equal weights, M, ranging from 233 000 to 1 000000, as is needed to test the existence of the concentration gap better; the two layers were distinguished by isotope labelling, by selectively deuterating one of the two.
In a neutron reflection experiment a well collimited neutron beam is brought onto a very flat sample surface at a glancing angle, O; the number of neutrons reflected from the sample is recorded as a function of the component of the neutron momentum perpendicular to the surface, kz0 = 2,~ sin 8/A. (A. is the neutron wavelength). The reflectivity, R ( k 2 o ), is an optical transform of b ( g ) / V ( z ) , the nuclear scattering amplitude density at a distance, z, from the surface of the sample. The contrast between the two polymers composing a bilayer is achieved by deuterating one of the species, Let us recall some notions of neutron reflectivity [11,12] relevant to the interdiffusion process, The reflectivity from a single layer of thickness, dl, on a substrate is given by (1)
where - kz,~+ kz,i+ 11 ri.i+l = kz, k~,ii +
kz, i = ¢(k20 - 4 ~ r b / V ) .
(3)
For an arbitrary assembly of layers, at large k the average reflectivity reduces to [13] Rk 4
~ [(b) = ~ r 2 i•
_ (b) i
"V
]2
(2)
(4)
i+1
where the summation is limited to the sharp interfaces of the system. If an interface is rough or diffuse, so that the laterally averaged density depth profile can be described by an error function with mean square displacement, (z2), the reflectance of that interface is modified by a Debye-Waller factor (ri.i+ 1)eft = (l'i,i+l)ideal exp( -- 2kiki+
2. Experimental
R = rgl + r?2 + 2r°1r12 c ° s ( 2 k z l d l ) 1 + r~r122 + 2r01r12 cos(2k.ldl) '
and
1(22)).
(5)
The reflectance of a smeared interface rapidly approaches zero when k is increased and as a result the oscillations of the reflectivity of a single layer, eq. (1), are damped out and their average value is depressed. For a bilayer of polymers deposited on silicon, there are three interfaces and the reflectivity contains three oscillation periods. However, in practice the largest term in the reflectivity is due to the deuterated layer; this is because ( b / V ) d _ p S = 6.5 X 10 -6 A -2, (b//V)hpos = 1.5 × 10-6 ~ - 2 , and ( b / V ) s i = 2.08 × 10-6-A -2. Figures l(a)-(c) show the reflectivity of a bilayer of M -- 233 000 before and after annealing for subsequent times. The samples investigated were bilayers of polymet on polished silicon substrates (5 cm in diameter, 0.5 cm thickness). The first layer was prepared by coating the substrate with a solution of the polymer in toluene and then spinning it at 2000 rpm to evaporate most of the solvent, producing a uniform film with thickness greater than 1500 ,~,. The specimen was then annealed at 120 ° C for 48 h to remove residual solvent and relax the film. The second layer was prepared separately by spin coating a 600-800 A film onto a 5.0 cm × 7.0 cm microscope slide. The sides of the slide were scored with a razor blade and the film was floated off onto a pool of de-ionized water and then transferred on to the top of the first layer. The bilayer was placed in vacuum for at least 48 h to remove
G.P. Felcher et al. / lnterdiffusion at the interface of polymeric bilayers: evidence for reptation?
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residual solvent, water and possible air bubbles trapped between the layers. Interdiffusion of the deuterated (d-PS) and normal polystyrene (h-PS) layers was achieved by placing the specimen in a preheated oven (controlled within 0.5°C) under helium for a predeterrnined time and then rapidly coiling it. In all cases, the sum of heating and cooling time ( - 4 5 s) was small by comparison with the annealing times. To what extent do these artificially prepared bilayers describe the interdiffusion through an ideal surface? To partially check this point, a bilayered sample was prepared, where both the bottom layer and that transposed to the top were d-PS. For the as-prepared sample, the neutron reflectivity curve showed frequency oscillations corresponding to the top layer thickness, as if the top layer were separated from the bottom polymer layer by an intervening medium with lower refractive index. Selective deuteration of the water in the pool used to float off the second layer confirmed that a 9 A thick water layer was present at the interface. Annealing of the sample for just 2 rain at the glass transition temperature (100°C) greatly diminishes the reflection from the interface, which, however, persists up to the Rouse time.
3. lnterdiffusion in polystyrene × 0.~ i_~
o ~
~--a~
~
o
~:
0.006
t
I
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I
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0.018 k0tA-~ )
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Fig. 1. R(k._)k4..o as a function of k. o for a bilayered s a m p l e of d-PS ( M = 233 000) on top of h-PS ( M = 203 000) before heating (a) and after heating at 1 3 5 ° C for 22 rain (b) and 53 rain (c). The open circles are the e x p e r i m e n t a l data, whereas the solid lines in (a), (b) a n d (c) were calculated u s i n g a bilayered model with a s y m m e t r i c w i d t h of 10, 135 a n d 200 ,A, respec-
tively.
The plots in figs. l(a)-(c) show that the reflectivity loses progressively all its distinguishing oscillation features: the technique of neutron reflection becomes insensitive to diffusion over lengths longer than 200 ,~. In monitoring the diffusion of polymers, it is desirable to have (by the same technique) measurements over a range of diffusion lengths which encompasses the radius of gyration: this, for M = 233000, is - 130 A. The conventional (or Fickian) diffusion equation is
r 0~ |D Ot - ~z [ -~-z] '
1.
(6)
G.P. Felcher et al. / lnterdiffusion at the interface of polymeric bilayers: evidence for reptation?
706
when D is constant, the solution [14] to eq. (6) is ~(z, t) = 0.5 e r f c ( ~ - - ~ ).
(7)
The expected reflectivities can now be calculated with the aid of eqs. (2) and (5). The solid lines in figs. l(a)-(c) show that conventional diffusion describes well the experimental reflectivities. The best fitted values of the diffusion coefficient, D, are plotted against time in fig. 2. Here, D is normalized by the self-diffusion coefficient, D*, at 120°C o f P S o f M = 2 3 3 0 0 0 . Figure 2 shows that D / D * is a function of time, varying rapidly for short times but then reaching unity in the proximity of the reptation time [3]. This is in full agreement with existing experimental [4-6] and theoretical knowledge [3,14] obtained for t > ~'d. For shorter times it may be surmised from fig. 2 that ( z 2) = t ~, where a < 1. For very short times (shorter than the Rouse time) the experimental result might be dependent on the sample preparation. The results above contain an apparent paradox: on the one hand, the reflectivity can be described in terms of a conventional diffusion profile, yet the time dependence of the mean square thickness of the interface does not show the linearity predicted by eq. (7). In terms of a
5 ~. . . . . . . . . . . . . . . . . 4 3
t
~Itt! •
°z, . . . . . 1'00
a 1000
tt T~
2 1
l
.......
~ '5000
aTt (rain) Fig. 2. The diffusion coefficient, D, measured from the reflectivity data divided by the self-diffusion coefficient, D *, of PS ( M = 233000) in the bulk at 120°C as a function of reduced time, aTt. For t >_~'d,D = D*.
~
9-ROUSE BROADENING
~x, ?PTATION WIDTH -~'q---Fig. 3. The density profile at an interface according to the simple reptation model, at a time 0 < t < ~'d; and the same profilewhensmearedby segmentalmotion.
reptation model the paradox is only apparent. The reptative motions leave a sharp density gap at the interface (fig. 3), which decreases to the point of disappearing at Td. However, if segmental motions are taken into account the sharp features of this profile are smeared out by a resolution function, or 'Rouse width' of the order of - 5 0 A. The resulting profile is dissimilar from an error function only if the characteristic lengths (Rouse width, reptation length)are sufficiently different. For a polymer with M - 233000 the calculated difference is only marginal: the reptation model can be tested only by using heavier (and larger) molecules. The experimental reflectivity from the system of pure homopolymer melts d-PS(M 1 150000)/h-PS(M - 1 O00000)/Si after annealing for 15 min at 155°C is shown in fig. 4; the continuous lines in figs. 4(a) and 4(b) correspond to reflectivities calculated from conventional diffusion (eq. (7)) for two different interface o. Unlike the M - 233000 system, here the conventional error function profile does not fit the data: 0 = 8 0 A fits the data only for k>0.016 ,~-1, while o = 175 ,~ fits the data only for k < 0.011 ,~-1. In real these results mean that the gradient at the interface is sharper than predicted by an error function of the width needed to account for the extent of interdiffusion. In terms of the reptation model, the fit obtained is shown in fig. 4(c). Profiles taken at subsequent annealing times can be interpreted in terms of a decreasing concentration gap; an increasing reptation width
G.P. Felcheret al. / Interdiffusion at the interface of polymeric bilayers: evidencefor reptation?
707
and a Rouse width which is basically time-independent.
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Discussion
The results on the relatively lightweight system M - 233000 establishes the physical significance of the reptation time, ~'d, as the onset of conventional diffusion, confirming one of the basic Ansatz of the reptation model. While for t < T0 the diffusion of polystyrene is definitely faster, an accurate time dependence of the interface width in this time region is still under investigation. Direct evidence for reptation was not found in the density profiles of polystyrene of light molecular weight; instead, such a model explains well the unusual profiles for M = 1 000000. The question then arises if the reptation model is also unique in explaining the observed features of the interdiffusion of polystyrene of high molecular weight. Deuterium labelling of molecules is a powerful method for providing contrast between chemically similar polymer molecules. However, it has recently been shown in a series of elegant experiments by Bates and Wignall [15] that for high molecular weight polymers, this labelling itself could lead to non-ideal mixing. In such systems, Green and Doyle [16] have experimentally shown that the mutual diffusion coefficient, D M , is reduced from its ideal value as a consequence of a finite segmental interaction parameter (X) between the deuterated and protonated monomers. For two polymers of equal degree of polymerization, N, DM(,) = D'N[1
0.01
~ lS 22 26 30 l cekth'b Fig. 4. R(k~)k4o as a function of k~0 for a bilayered sample of d-PS (M=1150000) on top of h-PS (M=1000000) after annealing for 15 min at 155°C. The open circles are experimental data, while the solid lines in (a) and (b).correspond to conventional fits with widths of 80 and 175 A, respectively. The solid line in (c) corresponds to a profile with a reptation width of 250 .~ and an interface gap of 0.3 in d-PS volume J
10
±
14
fraction, convoluted with a Rouse width of 50 ,~.
- 2,(1 -,)X].
(8)
The m a x i m u m reduction of D M o c c u r s at , = 0.5, which corresponds to the composition at the interface in pure bilayer melts of deuterated and normal polymer. The thermodynamic slowdown diminishes the mutual diffusion coefficient to 0.1 of the self-diffusion coefficient, D * , for M 1 000 000, while it is almost negligible for the lighter weight polystyrene. In view of this decrease, the calculated interface remains sharp; the profile obtained by numerically integrating the diffusion equation with D given by eq. (8) fits the
708
G.P. Felcher et al. / Interdiffusion at the interface of polymeric bilayers: evidence for reptation?
experimental data of fig. 4 as well as the reptation model.
5. Conclusions
To verify partially the effects of the thermodynamic slowdown, neutron reflectivity experiments were performed on bilayers of polystyrene of M 1 000 000. One of the layers was protonated, while the second layer was a blend of 10% deuterated polystyrene or, in a second sample, 50% deuterated polystyrene. The thermodynamic slowdown (which conventionally is observed in a time range well exceeding the reptation time) should be drastically different for these couples. By contrast, their profiles as obtained by neutron reflection were virtually identical, and quite similar values of (z 2) were obtained as a function of time. Is this body of information sufficient to declare the basic validity of the reptation theory? Possibly it will, after further experiments will be run to test and confirm the consistency of the physical evidence. The authors would like to thank Professor E.J. Kramer for his valuable comments and R.J. Goyette for his scientific assistance. The work at Argonne was supported by the U.S. Department of Energy, Basic Energy Sciences, under Contract No. W-31-104-ENG-38.
References [1] H.H. Kausch and M. Tirrell, Annu. Rev. Mater. Sci. 19 (1989) 341. I21 P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press, Ithaca, NY, 1979).
[31 S.F.
Edwards and M. Doi, The Theory of Polymer Dynamics (Oxford Science Publishers, Oxford, 1986). 14] J. Klein and B.J. Briscoe, Proc. R. Soc. London, Ser. A365 (1979) 53.
[51 P.F. Green, PhD thesis, Cornell University (1985). [6] s.J. Whitlow and R.P. Wool, Macromolecules,22 (1989) 2648. [7] K. Kremer, G.S. Grest and I. Carmesin, Phys. Rev. Lett. 61 (1988) 566. [81 D. Richter, B. Farago, L.J. Fetters, J.S. Huang, B. Ewen and C. Lartigue, Phys. Rev. Lett. 64 (1990) 1389. ]91 S. Prager and M.J. Tirrell, J. Chem. Phys. 75 (1983) 5953. ]101 H. Zhang and R.P. Wool, Macromolecules 22 (1989) 3018. [11] T.P. Russell, A. Karim, A. Mansour and G.P. Felcher, Macromolecules 21 (1988) 1890. [12] M.L. Fernandez, J.S. Higgins, J. Penfold, R.C. Ward, C. Shackelton and D.J. Walsh, Polymer 29 (1988) 1923. [13] R.J. Composto, R.S. Stein, R.A.L. Jones, E.J. Kramer, G.P. Felcher, A. Karim and A. Mansour, Physica B156& 157 (1989) 434. [14] J. Crank, The Mathematics of Diffusion, 2nd Ed. (Clarendon, Oxford, 1983). ]15] F.S. Bates and G.D. Wignall, Phys. Rev. Lett. 57 (1986)
12. [16] P.F. Green and B.L. Doyle, Phys. Rev. Lett. 57 (1986) 19.
703
Interdiffusion at the interface of polymeric bilayers" evidence for reptation? G . P . F e l c h e r a n d A. K a r i m Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
T.P. R u s s e l l IBM Research Division, Almaden Research Center, San Jose, CA 95120, USA
Neutron reflection is used to study the interdiffusion in equal molecular weight polystyrene bilayer melts with a spatial resolution of 10 ,~. Interracial widths and concentration profiles at the bilayer interface are obtained for annealing times up to and beyond the reptation time, Td. For t < T0, the reptation model predicts a mean square displacement of monomers whose time dependence is a power law, the exponent of which changes with time. For the relatively lightweight polymers of M - 233000, the mean square displacements of monomers is in general agreement with predictions; the permanence of a discontinuity at the interface - which also follows from the reptation model - is observed only for molecular weights M - 10000001
1. Introduction There is considerable interest in the basic mechanism of diffusion in polymer melts: by contrast with atoms or simple molecules, the long chains of m o n o m e r s forming the polymer molecules become entangled along their length. Of the various models proposed [1], the most successful is the reptation model, first proposed by de Gennes [2] and later developed by Edwards [3]. In the reptation model the polymer molecules diffuse by sliding along their contour within a hypothetical tube, which is defined by the locus of intersections of the molecule with its neighboring molecules, The time taken by the molecule to move out of its initial tube is the reptation time, Td. The main success of this theory consists in giving the correct scaling of the bulk self-diffusion coefficient, D * , with molecular weight, M: D * - M -2. This dependence has been confirmed by a variety of different techniques: infrared spectroscopy [4], forward recoil spectrometry [5] and secondary ion mass spectrometry [6]. The reptation theory gives further predictions
for the m o t i o n over distances shorter than the size of the molecule, which might be identified with the radius of gyration, gR" The m e a n square displacement of m o n o m e r s is predicted [3,7] to follow a time dependent power law: 0 2 = t ~, where a < 1 for t < 'I'd, becoming 1 at t = ~'d, in the limit of conventional diffusion. T o observe such displacement requires a technique with a spatial resolution less than the size of the diffusing molecule, and unfortunately the above-mentioned techniques have a resolution of 100 ,~, to be c o m p a r e d with a radius of gyration gR -- 150 ,~, for a molecule with a typical weight of M = 300 000. With the powerful technique of n e u t r o n spin-echo [8], the time dependence of the correlation function has been determined, but only up to times that were a fraction of ~'d" The diffusion in p o l y m e r melts can also be studied by observing the time evolution of the profile at the interface of two p o l y m e r (say, A and B) layers put in contact at time zero. This is exactly the kind of measurement d o n e b y neutron reflection, with a resolution that m a y a p p r o a c h 10 ,~. Actually, neutron reflection can give the de-
0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
704
G.P. Felcher et al. / lnterdiffusion at the interface of polymeric bilayers: evidence for reptation?
tailed shape of the concentration profile, which in the reptation theory has some unique characteristics [9,10]. The initial discontinuity of concentration of A (or B) at the interface is predicted to persist up to ~'d" This 'concentration gap' is a direct consequence of reptation, since only the polymer molecules that have their ends close to the interface can cross it, while the rest of the molecules close to the interface simply reptate along their contour length and do not cross the interface. In this paper, we present the results of a series of neutron reflection measurements on bilayers of polystyrene of equal weights, M, ranging from 233 000 to 1 000000, as is needed to test the existence of the concentration gap better; the two layers were distinguished by isotope labelling, by selectively deuterating one of the two.
In a neutron reflection experiment a well collimited neutron beam is brought onto a very flat sample surface at a glancing angle, O; the number of neutrons reflected from the sample is recorded as a function of the component of the neutron momentum perpendicular to the surface, kz0 = 2,~ sin 8/A. (A. is the neutron wavelength). The reflectivity, R ( k 2 o ), is an optical transform of b ( g ) / V ( z ) , the nuclear scattering amplitude density at a distance, z, from the surface of the sample. The contrast between the two polymers composing a bilayer is achieved by deuterating one of the species, Let us recall some notions of neutron reflectivity [11,12] relevant to the interdiffusion process, The reflectivity from a single layer of thickness, dl, on a substrate is given by (1)
where - kz,~+ kz,i+ 11 ri.i+l = kz, k~,ii +
kz, i = ¢(k20 - 4 ~ r b / V ) .
(3)
For an arbitrary assembly of layers, at large k the average reflectivity reduces to [13] Rk 4
~ [(b) = ~ r 2 i•
_ (b) i
"V
]2
(2)
(4)
i+1
where the summation is limited to the sharp interfaces of the system. If an interface is rough or diffuse, so that the laterally averaged density depth profile can be described by an error function with mean square displacement, (z2), the reflectance of that interface is modified by a Debye-Waller factor (ri.i+ 1)eft = (l'i,i+l)ideal exp( -- 2kiki+
2. Experimental
R = rgl + r?2 + 2r°1r12 c ° s ( 2 k z l d l ) 1 + r~r122 + 2r01r12 cos(2k.ldl) '
and
1(22)).
(5)
The reflectance of a smeared interface rapidly approaches zero when k is increased and as a result the oscillations of the reflectivity of a single layer, eq. (1), are damped out and their average value is depressed. For a bilayer of polymers deposited on silicon, there are three interfaces and the reflectivity contains three oscillation periods. However, in practice the largest term in the reflectivity is due to the deuterated layer; this is because ( b / V ) d _ p S = 6.5 X 10 -6 A -2, (b//V)hpos = 1.5 × 10-6 ~ - 2 , and ( b / V ) s i = 2.08 × 10-6-A -2. Figures l(a)-(c) show the reflectivity of a bilayer of M -- 233 000 before and after annealing for subsequent times. The samples investigated were bilayers of polymet on polished silicon substrates (5 cm in diameter, 0.5 cm thickness). The first layer was prepared by coating the substrate with a solution of the polymer in toluene and then spinning it at 2000 rpm to evaporate most of the solvent, producing a uniform film with thickness greater than 1500 ,~,. The specimen was then annealed at 120 ° C for 48 h to remove residual solvent and relax the film. The second layer was prepared separately by spin coating a 600-800 A film onto a 5.0 cm × 7.0 cm microscope slide. The sides of the slide were scored with a razor blade and the film was floated off onto a pool of de-ionized water and then transferred on to the top of the first layer. The bilayer was placed in vacuum for at least 48 h to remove
G.P. Felcher et al. / lnterdiffusion at the interface of polymeric bilayers: evidence for reptation?
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residual solvent, water and possible air bubbles trapped between the layers. Interdiffusion of the deuterated (d-PS) and normal polystyrene (h-PS) layers was achieved by placing the specimen in a preheated oven (controlled within 0.5°C) under helium for a predeterrnined time and then rapidly coiling it. In all cases, the sum of heating and cooling time ( - 4 5 s) was small by comparison with the annealing times. To what extent do these artificially prepared bilayers describe the interdiffusion through an ideal surface? To partially check this point, a bilayered sample was prepared, where both the bottom layer and that transposed to the top were d-PS. For the as-prepared sample, the neutron reflectivity curve showed frequency oscillations corresponding to the top layer thickness, as if the top layer were separated from the bottom polymer layer by an intervening medium with lower refractive index. Selective deuteration of the water in the pool used to float off the second layer confirmed that a 9 A thick water layer was present at the interface. Annealing of the sample for just 2 rain at the glass transition temperature (100°C) greatly diminishes the reflection from the interface, which, however, persists up to the Rouse time.
3. lnterdiffusion in polystyrene × 0.~ i_~
o ~
~--a~
~
o
~:
0.006
t
I
0.012
I
t
0.018 k0tA-~ )
I
I
0.024
I
0.030
Fig. 1. R(k._)k4..o as a function of k. o for a bilayered s a m p l e of d-PS ( M = 233 000) on top of h-PS ( M = 203 000) before heating (a) and after heating at 1 3 5 ° C for 22 rain (b) and 53 rain (c). The open circles are the e x p e r i m e n t a l data, whereas the solid lines in (a), (b) a n d (c) were calculated u s i n g a bilayered model with a s y m m e t r i c w i d t h of 10, 135 a n d 200 ,A, respec-
tively.
The plots in figs. l(a)-(c) show that the reflectivity loses progressively all its distinguishing oscillation features: the technique of neutron reflection becomes insensitive to diffusion over lengths longer than 200 ,~. In monitoring the diffusion of polymers, it is desirable to have (by the same technique) measurements over a range of diffusion lengths which encompasses the radius of gyration: this, for M = 233000, is - 130 A. The conventional (or Fickian) diffusion equation is
r 0~ |D Ot - ~z [ -~-z] '
1.
(6)
G.P. Felcher et al. / lnterdiffusion at the interface of polymeric bilayers: evidence for reptation?
706
when D is constant, the solution [14] to eq. (6) is ~(z, t) = 0.5 e r f c ( ~ - - ~ ).
(7)
The expected reflectivities can now be calculated with the aid of eqs. (2) and (5). The solid lines in figs. l(a)-(c) show that conventional diffusion describes well the experimental reflectivities. The best fitted values of the diffusion coefficient, D, are plotted against time in fig. 2. Here, D is normalized by the self-diffusion coefficient, D*, at 120°C o f P S o f M = 2 3 3 0 0 0 . Figure 2 shows that D / D * is a function of time, varying rapidly for short times but then reaching unity in the proximity of the reptation time [3]. This is in full agreement with existing experimental [4-6] and theoretical knowledge [3,14] obtained for t > ~'d. For shorter times it may be surmised from fig. 2 that ( z 2) = t ~, where a < 1. For very short times (shorter than the Rouse time) the experimental result might be dependent on the sample preparation. The results above contain an apparent paradox: on the one hand, the reflectivity can be described in terms of a conventional diffusion profile, yet the time dependence of the mean square thickness of the interface does not show the linearity predicted by eq. (7). In terms of a
5 ~. . . . . . . . . . . . . . . . . 4 3
t
~Itt! •
°z, . . . . . 1'00
a 1000
tt T~
2 1
l
.......
~ '5000
aTt (rain) Fig. 2. The diffusion coefficient, D, measured from the reflectivity data divided by the self-diffusion coefficient, D *, of PS ( M = 233000) in the bulk at 120°C as a function of reduced time, aTt. For t >_~'d,D = D*.
~
9-ROUSE BROADENING
~x, ?PTATION WIDTH -~'q---Fig. 3. The density profile at an interface according to the simple reptation model, at a time 0 < t < ~'d; and the same profilewhensmearedby segmentalmotion.
reptation model the paradox is only apparent. The reptative motions leave a sharp density gap at the interface (fig. 3), which decreases to the point of disappearing at Td. However, if segmental motions are taken into account the sharp features of this profile are smeared out by a resolution function, or 'Rouse width' of the order of - 5 0 A. The resulting profile is dissimilar from an error function only if the characteristic lengths (Rouse width, reptation length)are sufficiently different. For a polymer with M - 233000 the calculated difference is only marginal: the reptation model can be tested only by using heavier (and larger) molecules. The experimental reflectivity from the system of pure homopolymer melts d-PS(M 1 150000)/h-PS(M - 1 O00000)/Si after annealing for 15 min at 155°C is shown in fig. 4; the continuous lines in figs. 4(a) and 4(b) correspond to reflectivities calculated from conventional diffusion (eq. (7)) for two different interface o. Unlike the M - 233000 system, here the conventional error function profile does not fit the data: 0 = 8 0 A fits the data only for k>0.016 ,~-1, while o = 175 ,~ fits the data only for k < 0.011 ,~-1. In real these results mean that the gradient at the interface is sharper than predicted by an error function of the width needed to account for the extent of interdiffusion. In terms of the reptation model, the fit obtained is shown in fig. 4(c). Profiles taken at subsequent annealing times can be interpreted in terms of a decreasing concentration gap; an increasing reptation width
G.P. Felcheret al. / Interdiffusion at the interface of polymeric bilayers: evidencefor reptation?
707
and a Rouse width which is basically time-independent.
4.
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.
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Discussion
The results on the relatively lightweight system M - 233000 establishes the physical significance of the reptation time, ~'d, as the onset of conventional diffusion, confirming one of the basic Ansatz of the reptation model. While for t < T0 the diffusion of polystyrene is definitely faster, an accurate time dependence of the interface width in this time region is still under investigation. Direct evidence for reptation was not found in the density profiles of polystyrene of light molecular weight; instead, such a model explains well the unusual profiles for M = 1 000000. The question then arises if the reptation model is also unique in explaining the observed features of the interdiffusion of polystyrene of high molecular weight. Deuterium labelling of molecules is a powerful method for providing contrast between chemically similar polymer molecules. However, it has recently been shown in a series of elegant experiments by Bates and Wignall [15] that for high molecular weight polymers, this labelling itself could lead to non-ideal mixing. In such systems, Green and Doyle [16] have experimentally shown that the mutual diffusion coefficient, D M , is reduced from its ideal value as a consequence of a finite segmental interaction parameter (X) between the deuterated and protonated monomers. For two polymers of equal degree of polymerization, N, DM(,) = D'N[1
0.01
~ lS 22 26 30 l cekth'b Fig. 4. R(k~)k4o as a function of k~0 for a bilayered sample of d-PS (M=1150000) on top of h-PS (M=1000000) after annealing for 15 min at 155°C. The open circles are experimental data, while the solid lines in (a) and (b).correspond to conventional fits with widths of 80 and 175 A, respectively. The solid line in (c) corresponds to a profile with a reptation width of 250 .~ and an interface gap of 0.3 in d-PS volume J
10
±
14
fraction, convoluted with a Rouse width of 50 ,~.
- 2,(1 -,)X].
(8)
The m a x i m u m reduction of D M o c c u r s at , = 0.5, which corresponds to the composition at the interface in pure bilayer melts of deuterated and normal polymer. The thermodynamic slowdown diminishes the mutual diffusion coefficient to 0.1 of the self-diffusion coefficient, D * , for M 1 000 000, while it is almost negligible for the lighter weight polystyrene. In view of this decrease, the calculated interface remains sharp; the profile obtained by numerically integrating the diffusion equation with D given by eq. (8) fits the
708
G.P. Felcher et al. / Interdiffusion at the interface of polymeric bilayers: evidence for reptation?
experimental data of fig. 4 as well as the reptation model.
5. Conclusions
To verify partially the effects of the thermodynamic slowdown, neutron reflectivity experiments were performed on bilayers of polystyrene of M 1 000 000. One of the layers was protonated, while the second layer was a blend of 10% deuterated polystyrene or, in a second sample, 50% deuterated polystyrene. The thermodynamic slowdown (which conventionally is observed in a time range well exceeding the reptation time) should be drastically different for these couples. By contrast, their profiles as obtained by neutron reflection were virtually identical, and quite similar values of (z 2) were obtained as a function of time. Is this body of information sufficient to declare the basic validity of the reptation theory? Possibly it will, after further experiments will be run to test and confirm the consistency of the physical evidence. The authors would like to thank Professor E.J. Kramer for his valuable comments and R.J. Goyette for his scientific assistance. The work at Argonne was supported by the U.S. Department of Energy, Basic Energy Sciences, under Contract No. W-31-104-ENG-38.
References [1] H.H. Kausch and M. Tirrell, Annu. Rev. Mater. Sci. 19 (1989) 341. I21 P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press, Ithaca, NY, 1979).
[31 S.F.
Edwards and M. Doi, The Theory of Polymer Dynamics (Oxford Science Publishers, Oxford, 1986). 14] J. Klein and B.J. Briscoe, Proc. R. Soc. London, Ser. A365 (1979) 53.
[51 P.F. Green, PhD thesis, Cornell University (1985). [6] s.J. Whitlow and R.P. Wool, Macromolecules,22 (1989) 2648. [7] K. Kremer, G.S. Grest and I. Carmesin, Phys. Rev. Lett. 61 (1988) 566. [81 D. Richter, B. Farago, L.J. Fetters, J.S. Huang, B. Ewen and C. Lartigue, Phys. Rev. Lett. 64 (1990) 1389. ]91 S. Prager and M.J. Tirrell, J. Chem. Phys. 75 (1983) 5953. ]101 H. Zhang and R.P. Wool, Macromolecules 22 (1989) 3018. [11] T.P. Russell, A. Karim, A. Mansour and G.P. Felcher, Macromolecules 21 (1988) 1890. [12] M.L. Fernandez, J.S. Higgins, J. Penfold, R.C. Ward, C. Shackelton and D.J. Walsh, Polymer 29 (1988) 1923. [13] R.J. Composto, R.S. Stein, R.A.L. Jones, E.J. Kramer, G.P. Felcher, A. Karim and A. Mansour, Physica B156& 157 (1989) 434. [14] J. Crank, The Mathematics of Diffusion, 2nd Ed. (Clarendon, Oxford, 1983). ]15] F.S. Bates and G.D. Wignall, Phys. Rev. Lett. 57 (1986)
12. [16] P.F. Green and B.L. Doyle, Phys. Rev. Lett. 57 (1986) 19.