Dev, B N

Dev, S

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RAPID COMMUNICATIONIndian J. Phys. 70A (2), 189-195 (1996)I J P  A— an international |ournalG razing in cid en ce X -ray stand in g  w ave : a novel technique for m easu rin g  u ltrasm all d iffusion  coefficients in  polym ersS Dev(nee Sen)* and B N Dev**Institute of Physics, Sachivalaya Marg, Bhubaneswar-751 (X)5, India Received II November 1995, accepted 16 November 1995Abstract : The grazing incidence X-ray standing wave technique has recently been used to locate heavy ions in a Langmuir-Blodgett  multilayer with a precision of 2 A  In this communication, we show how this technique can be used for measuring ultiasmall diffusion coefficients (~ 10- ’ 1 civr/sec) in thin polymer lilms into which a slowly diffusing species has been introducedKeywords : X-ray standing waves, diffusion in polymers PACS Nos. : 07 85 Yk, 61 41 +e. 68 35 Fx1. In troductionThin polymer films on solid surfaces are of great importance in technology due to their wide variety of applications. Diffusion in such polymeric materials is even more important because of various applications ranging from development of photoresists for very large scale integrated (VLSI) circuit fabrication to the performance of membranes for separation processes.A large number of methods, e.g. weight gain |1], transient permeation 12], nuclear magnetic resonance [3], radioactive tracing [4], etc. are available for measuring diffusion. Almost all of these measurements do not sense directly the concentration gradient of the diffusing species making it difficult to extract meaningful information from systems which exhibit non-Fickian behaviour. Non-Fickian behaviour is common for many polymer systems [5], Among other methods used, are scanning infrared microdensitometryPresent-address * European Molecular Biology Laboratory, EMBL C/o DESY, Notkestrasse 85, 22603 Hamburg, Germany** Present address : HASYLAB at DESY, Notkestrasse 85, 22603 Hamburg, Germany.© 1996 I ACS190 S Dev (nee Sen) and B N Devtechnique [6], electron microprohe techniques [7], radiation damage method [8] etc. These methods arc destructive. Another good technique is the ion-beam technique or rather the Rutherford backscattering spectrometry (RBS) technique which is well suited for slowly diffusing species. With the RBS technique concentration profiles of the diffusing species arc measured with submicroineur depth icsolution [9], However, we will show that by exploiting the subnanometer depth resolution of the grazing incidence X-ray standing wave (XSW) technique, one can measure ultrasmall diffusion coefficients D ~ 10~21 crrf/sec. This non-destructive technique can be used in measuring the concentration profiles of small molecules which diffuse slowly into polymers.2. X-ray standing wavesAn X-ray standing wave can be generated by causing interference between two coherently related X-ray beams. Conventional X-ray standing wave measurements use dynamical Bragg diffraction [1()| from nearly perfect single crystals for generating X-ray standing wave probes with a periodicity around 1--4 A. This technique was applied to study the lattice location of impurity atoms in crystals with a precision of ~ 0.02 A [11-131. It reached a new level of interest when it was realized that the standing wave extends above the surface of the crystal in which it is generated and therefore, could be used for examining adsorbed layers on crystal surfaces [14-17].X-ray standing waves with a variable period (usually long), can be generated using total external reflection of a monochromatic X-ray beam from a mirror-polished surface. By generating X-ray standing waves above a gold mirror surface, Bedzykef al [18] located a zinc ion layer, which was embedded -  200 A above the mirror surface in a Langmuir- Blodgett multilayer, with a precision of 2 A.Grazing incidence X-ray standing waves :For X-rays the (real part of) index of refraction of any solid or liquid medium is less than unity. Thus, it is possible to obtain total external reflection when an X-ray beam is incidentsolidsubstrateFigure 1. Schematic diagram showing total external reflection when the angle of incidence is less than the critical angle 9C. Eq and E# represent the wave amplitudes of the incident and the reflected beam respectively.on the sample surface at a grazing angle of incidence (0) less than the critical angle 0C (Figure 1).Grazing incidence X-ray standing wave : a novel technique etc 191The refractive index of the material is in general complex and is given byn  = \ -  8  -  i(5. (1)In cq. (1), 8 = reX2N  / 2 ^  and fi = A/i /An ,  where re is the classical electron radius, A is the wavelength of the incident X-rays, N  is the total number of electrons per unit volume and jU is the linear absorption coefficient [191.Let us consider the case of a-polarization, i.e., the electric field vectors £ 0 and E H, associated with the incident and the reflected beam respectively, arc perpendicular to the scattering plane. The wave-amplitude ratio for the vacuum/solid interface can be obtained from Fresnel coefficient of reflection [20].ii* = = e - ( 0 2 - 2 S - 2 i P ) ^E0 En e + id 1 - 2 S - 2 i P ) in ( }where v is the phase of the complex £-field ratio EH/E{). The reflectivity R{ 9) is given by|2r  (3)EoThe interference between the two (the incident and the reflected) coherent travelling plane waves generates a standing wave, the electric-field intensity of which at a distance c from the surface into vacuum is given by [18]1 ( 0 , z )  = |E0|2 [ i + *  + 2V e  cos ( v - 2 k Q z )], (4)where Q -  2 sin Q/X. D ~ \/Q  is the periodicity of the standing wave and thus depends on ft the grazing angle of incidence (Figure 1). For 0= Qc (sin Gc ~ 6C, since ft is very small) the critical period isE>c = j L  = ( V ^ /2  ) [ N a r e{ Z  +  f ' j \ U \  (5)where re is the classical electron radius, Na,Z  and/"are atomic density, atomic number and anomalous dispersion correction for the material of the reflecting surface respectively. For /'=  0, D c is an energy independent parameter, which is 200 A for silicon. That is at 0 = ft. for a silicon surface, the antinodes extend outward from the surface with a periodicity of 200 A. As the angle of incidence increases from zero towards ft., the positions of the antinodes as well as their periodicity changes. These properties can be exploited to determine the depth distribution of heavy atoms (or molecules containing heavy atoms) confined in a thin film of very low-Z elements (say, a polymer film) on the reflecting surface of a medium or high-Z substrate material. When the embedded heavy atoms are at a given height from the substrate surface, the height can be determined as follows. X-rays will cause photoelectric effect, the cross section of which is proportional to the intensity of X-rays at the centre of an atom (in the dipole approximation). Fluorescence follows the ejection of a photoelectron. Thus by measuring the fluorescence yield, the X-ray (£-field) intensity on the atom can be estimated. By varying the positions of antinodes above the substrate surface through the variation of ft one can determine when the antinodes have70A (2)-2192 S Dev (nee Sen) and B N Devpassed through the atomic layer The intensity variation of the fluorescence yield as a function of 6 contains the information about the height of the fluorescing atoms from the substrate surface. If the fluorescing atoms have a concentration distribution, contribution from the concentration at each height will add up to provide an effective fluorescence yield profile. From the analysis of this profile, the concentration distribution can be extracted.45Reflectivity4 z-02-200353 f  1|"25 /•!0.50 0.05 0 1 015 0 2 0.25 0 3 0 35 0 4Incident ang le  (in de g)Figure 2. Reflectivity and E-field intensities on the surface (r = 0) and at 200 A above the surface of a silicon substratePolystyrenef i l ml - >Figure 3. A polystyrene thin film of thickness ; () on a silicon substrate Before diffusion all absorbed molecules are at ro- Altei diffusion into the film depth distribution of molecules are shown by a half- Gaussian of standard deviation aFigure 2 shows the reflectivity and the field intensities at z = 0 A and z = 200 A for a silicon substrate surface. These would be the measured fluorescence yield profile from an atomic or molecular layer on the substrate surface (z = 0),and at a height of 200 A, respectively. The results presented in Figure 1 are for X = 0.709 A (Mo Kal X-rays : 17.479 keV). Here we will restrict to this value of X for all calculations. Photons of this energy are suitable tor causing /^-fluorescence upto medium-Z atoms and Lrfluorescence in heavier atoms.3. Determination of diffusion coefficientsLet us take a simple model, where a thin polystyrene film is deposited on a silicon surface. The polystyrene film is exposed to bromine vapour to adsorb a molecular (or atomic) layer of bromine. Then, we let these bromine molecules diffuse into the film at a given temperature for a given duration of time. We represent the concentration distribution ofbromine by a delta function before diffusion and a half-Gaussian after the diffusion has taken place (Figure 3).The second distribution is given byG(z) = (6)Grazing incidence X-ray standing wave : a novel technique etc 193where d  is the standard deviation. The factor 2 in the numerator of eq. (6) is to ensure that all the molecules are contained in this half-Gaussian; Zo represents the thickness of the polymer film on top of which the initial 5-function-like distribution is located.Eq. (4) provides the X-ray intensity at a distance z as a function of the angle of incidence. For the distribution in eq. (6) the intensity is1(9) = P  I(0,z)G(z)dz.  (7)JoWe explore the behaviour of 1(9) for a specific case : a 200 A polystyrene film deposited on a mirror-polished silicon substrate; a monolayer Film of molecular bromine is then deposited on the surface of the polymer layer. /(0), the X-ray intensity, or the bromine K(X fluorescence yield for this case, is shown in Figure 4.0 005 01 015 02  025 03  035 04Incident angle (in deg)Figure 4. The X-ray intensity 1(0) or the fluorescence yield as a function of 0 for the distribution in eq (6) and Figure 3Incident angle (in deg)Figure 5 .1(0) vs 0 plot for two values of cr differing by 10 A Changes in peak (or dip) positions as well as in intensity modulation are easily distinguishablePrior to diffusion, the bromine distribution should realistically be a Gaussian (rather than a delta function) because the polymer surface on which bromine is deposited would certainly have a surface roughness, some of which might arise from the roughness of the silicon surface on which the polymer film is deposited. The smallest roughness expected is of the order of a few angstrom [21]. That is why we have actually evaluated 1(9) for a Gaussian with a = 4  A. The diffused profile corresponds to cr = 60 A. It is seen from the figure (Figure 4) that the fluorescence yield profiles are drastically different.Figure 5 shows 1(9) for two different a  values (20 A and 30 A) differing by 10 A, It is clear that the two curves are distinguishable in intensity as well as peak (or dip) position shifts. A larger value of o  brings more bromine molecules closer to the silicon surface. Thus, more molecules see the antinodes at slightly larger ft It may be noted that when the tluorescing atoms are at the substrate surface (z = 0), they do not sec an antinode until 9 = is reached. This is evident from the peak in the 1(6) vs ©curve in Figure 2.Noting that, in the Fickian solution to the diffusion problem, o ~  CDt)m , where D is the diffusion coefficient and t is the time duration over which the diffusion process has194 S Dev (nee Sen) and B N Devoccurred, we can estimate the smallest order of magnitude of D that can he measured using the grazing incidence X-ray standing wave technique. As shown in Figure 5, a (Dr)U2 value of 10 A is easily measurable. It may be noted that the technique has a position resolution of ~ 2 A [ 18]. First, how large a t is practicable? Romanelli et al |9 | studied iodine diffusion in polymer (polyimidc) by Rutherford buckscattcring spectrometry (RBS). They used a diffusion time of 19 days. They estimated that for a diffusion time of one month a D-value of about 1(F17 cm2/sec could be measured by the RIBS technique. With the grazing incidence XSW technique t = I month (~ 2.6 x 106 sec) would enable one to measure a D as small as 10"21 cnvVsec \(Dt)m = 10 A, D = 3.8 x l O'21 cm2/sec].The particular case we have discussed here, may not give rise to exactly a half- Gaussian profile aftei diffusion has occurred. However, it does not affect the points of our discussion. If another layer of polymer is deposited above the bromine layer, this situation may be described as diffusion into a semi-infinite solid from a thin planar source. The solution to this problem is known [22] and the concentration profile of bromine would be u full Gaussian spreading symmetrically outwards from the initial position with (7= (2 Di)],\  In this situation, unlike Figure 4, the peak positions in the 7(0) vs 0plot would not shift, only the modulation will be reduced as shown in Figure 6. Again a change of <J= 10 A canFigure 6.7(0) vs 9 plot for the case of diffusion into a semi-infinite film from a thin planar sourcebe easily measured. When diffusion process is allowed to proceed while the polymer surface is kept exposed to bromine, the concentration profile would be a Complementary error function, for which 1(0) can be calculated and measured with the same precision. In this case also, like Figure 4 for a larger value of a  (i.e., long exposure time for constant D), the peaks in the 1(6) plot would shift towards larger 0 in addition to a reduction in modulation. In all the cases, experimental data could be fitted to the theoretical models to extract the diffusion coefficients, and also detect possible deviations from Fickian behaviour.For a thicker polymer layer, say 1000 A (= 5 Dc) about five complete peaks would be observed in the 1(0) vs 0 curve. The general arguments about peak position shifts andGrazing incidence X-ray standing wave: a novel technique etc 195modulation changes discussed above for the case of the 200 A film will still be applicable. As the film thickness increases, the number of peaks also increases and the largest usable thickness will be dictated by the precision with which 0can be measured.In computing the contents of Figures 4, 5 and 6, we have actually neglected the presence of the polymer layer as it is composed of low- Z elements, e.g. C and H, and thus a weak scattcrer. This layer can be explicitly included in the computation [20|. However, it would not bring any significant change in the results presented here.4. C onclusionsIn  t h i s  p a p e r ,  w e  h a v e  p r o p o s e d  a  m e t h o d  b a s e d  o n  g r a z i n g  i n c i d e n c e  X - r a y  s t a n d i n g  w a v e s  l o r  t h e  d e t e r m i n a t i o n  o f  u l t r a s m a l l  ( ~  10~21 c m 2/ s c c )  d i f f u s i o n  c o e f f i c i e n t s  i n  p o l y m e r  f i l m s .  It w o u l d  n o t  b e  d i f f i c u l t  t o  p e r f o r m  e x p e r i m e n t s  o n  t h e  s p e c i f i c  s y s t e m s  u s e d  a s  e x a m p l e s  i n  this p a p e r .R e f e r e n c e s[ 1 ] S Prager and F A Long J Am Chem. Soc 73 4072 (1951)12] J Crank and G S Park Diffusion in Polymers (New York) (1968)PJ V J Me Briefly and D C Douglas Phys Rep 63 61 ( 1980)141 Y Kumagai, H Wananabc, K Miyasaka and T Hata J Chem. Jpn 12 1 (1979)151 T Alfrey, E F Gurney and W G I -loyd J Polxm Sc i C l2 249 (1966)|()J J Klein and B J Briscoe Pun Roy Soc London A365 53 (1979)[7] P T Gilmore, R Falabella and R L Lawrence Macromolecules 13 880 (1980)|8] T Venkaiesan, D Edelson and W L Brown Appl Phys Lett. 43 364 (1983)(9) J F Romanelli, J W Mayei and E J Kramer J Polymer Sci Polymer Ph\s edn 24 263 (1986)[10] B W Battermann and H Cole Rev Mod Phys. 36 681 (1964)111] B W Battermann Phys. Rev 133 A759 (1964)[12] J A Golovchenko, B W Battermann and W L Brown Phys Rev B10 4239 (1974)1131 S K Andtrsen, J A Golovchenko and G Mair Phys Rev Lett. 37 1141 (1976)114] PL  Cowan, J A Golovchenko and M F Robbins Phys. Rev. Lett 44 1680 (1980)115] J A Golovchenko, J R Patel, D R Kaplan. P L Cowan and M J Bcdzyk Phys Rev. Lett. 49 560 (1982)[16] BN Dev, G Materlik, F Grey, R L Johnson and M Clausnitzer Phys Rev Lett 57 3058 (1986)117] J Zcgcnhagen Surf Sn. Rep. 18 199 (1993)[18] M J Bcdzyk, G M Bommarito and J S Schildkraut Phys. Rev. Lett 62 1376 (1989)[19] R W James The Optical Principles o f Diffraction of X-rays (Woodridge : Ox Bow) Ch 13 (1982)[20] L G Parrat Phys. Rev. 95 359 ( 1954)[21] P V Satyam, D Bahr, S K Ghose, G Kuri. B Sundaravcl, B Rout and B N Dev Curr. Sci (India) 69 526 (1995)122] R J Borg and G J Dienes An Introduction to Solid State Diffusion (New York Academic) Ch 1 (1988)

Grazing Incidence X-Ray Standing Wave : A Novel Technique for Measuring Ultrasmall Diffusion Coefficients in Polymers

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Grazing Incidence X-Ray Standing Wave : A Novel Technique for Measuring Ultrasmall Diffusion Coefficients in Polymers

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