On the Scale of Diffusion Lengths Observable by Neutron Reflection: Application to Polymers.
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ON THE SCALE OF DIFFUSION LENGTHS OBSERVABLE BY NEUTRON REFLECTION: APPLICATION TO POLYMERS.
A. Karim, A. Mansour and G.P. Felcher Argonne National Laboratory, Argonne, IL 60439, T.P. Russell, IBM Research Division, Almaden Research Center, San Jose, CA 95120.
ABSTRACT A systematic approach has been applied to neutron reflectivity data to study interdiffusion across an interface. It is shown that with this technique it is possible to probe interface broadening from -10A to upward of 200 A, the upper limit being already within the range of observation of other techniques such as Rutherford backscattering spectrometry (RBS), forward recoil spectrometry (FRES) and secondary ions mass spectroscopy (SIMS). As example is analyzed the interdiffusion of a bilayered polymer system: a deuterated polystyrene (d-PS) layer on protonated polystyrene (h-PS). Introduction Neutron reflectivity 1 has emerged as a useful technique to probe interface broadening caused by diffusion across an interface on the nanometer scale. The technique relies on the contrast of the neutrons scattering length densities of the layers facing the interface. The quantity obtained in such experiments is the reflectivity (R) of the sample: this is a function of the component of the neutron momentum perpendicular to the surface, kz. R(kz) is an optical transform of the scattering length density, as measured as a function of the depth z from the surface. In this paper, we define the range of interface widths measurable by neutron reflectivity, by taking the reflectivity curves at subsequent times of the diffusion process. Limits of resolution Consider a homogeneous layer of thickness z2 on an infinitely thick lower layer;, the indices 1,2 and 3 indicate the vacuum above, homogeneous material and lower layer respectively. Then the reflectivity R is given by 2,3 R= r?2+Pa3 +2r 1 2P23cos 2k 2 z2 l+r22P23+2r12P23COs 2k 2z2
(1)
The reflectance at the i,i+l boundary is: rii+l=4 k l ki+ki+l
(2)
where ki = [k 2 - (b/v)i]112 and (b/v)i = scattering length density of the ith layer, while k=27tSin0/?A. If symmetric interdiffusion 4 ,5 occurs between layers 2 and 3,
Mat. Res. Soc. Symp. Proc. Vol. 171. ©1990 Materials Research Society
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P23=r23exp -2k 2k 3a 2
(3)
In conventional diffusion, the diffusion coefficient D is a function of the width 02(t) and of time, 6 namely D =&(t) 4t
(4)
In this paper we would like to examine how to obtain u2 systematically from the experimental data in order to obtain the measurable range of diffusion coefficients. The z2 averaged expression for the reflectivity [ eq(l)], is: p2
P 2
-r2+P232rP2P23 1.r2P223
(5)
which for large values of k can be approximated by the expression: - r12+3 -
[X 2 /k 1 4 ] [(b/v) 2 2 + ((b/v) 3 - (b/v) 2 )2 exp - 4k1 2 02]
(6)
Thus Rk4 vs k tends, for large values of k, to an asymptotic value. This value is different for an infinitely sharp 2,3 interface compared to when some interdiffusion has occurred across such a boundary; the decay to the new asymptote reaches a value l/e for k1 =( 1
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