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Raman depth-profiling characterization of a migrant diffusion in a polymer
Journal of Membrane Science 375 (2011) 165–171
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Raman depth-profiling characterization of a migrant diffusion in a polymer M. Mauricio-Iglesias, V. Guillard, N. Gontard, S. Peyron ∗ UMR 1208 IATE (Agropolymer Engineering and Emerging Technology), University of Montpellier 2, CIRAD, INRA, Montpellier Supagro, CC023, Pl. E Bataillon, F-34095 Montpellier FR, France
a r t i c l e
i n f o
Article history: Received 7 June 2010 Received in revised form 21 December 2010 Accepted 19 March 2011 Available online 29 March 2011 Keywords: Confocal Raman microscopy Depth-profiling Refraction Diffusion in polymers
a b s t r a c t Raman depth-profiling microspectroscopy provides rich information on chemical/physical characterization in a non-destructive mode with micrometric resolution. However, refraction causes distortions to the data obtained thereby. A method to determine the diffusivity of an additive in low linear density polyethylene (LLDPE) with Raman depth profiling is proposed, combining the latest developments on data treatment of refraction distorted profiles. The method is compared with the results obtained analysing the cross section of the sample, with a maximum 32% relative error between both methods. The main benefits, characteristics of this method, a discussion of the experimental errors, as well as perspectives for future work are highlighted. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Raman microspectroscopy is an outstanding tool for its high performance on physical/chemical characterization and accurate spatial resolution (down to 1 m approx.). Furthermore, it is possible to gather information of the inside of the sample in a nondestructive mode, which has applications on depth-profiling in polymer science [1], identifying inclusions within minerals, assessing the adsorbed layers by an electrode [2,3] or investigating drug penetration in membrane [4]. However the presence of air/sample interfaces (or any substance of different refractive index) will distort the confocal properties of the microscope resulting in a misinterpretation of data as extensively documented by Everall [5,6]. Whereas a measurement setup using immersion objective in oil matching the refractive index of the polymeric sample can prevent optical distortions, such approach is limited to the analysis of thin samples (e.g. minor than the focal distance of objective). On the contrary, the use of a larger optic distance is well-adapted to a broader variety of samples but, in turn, the data obtained thereby must be corrected. With a simple ray-tracing model, Everall [6] explained semi-quantitatively the increase in the depth-of-focus and the focal volume. This simple model has been successively refined in order to reproduce experimental values: Baldwin and Batchelder [7] showed that the confocal aperture limits the collection volume and that the refractive distortion is not so severe as
∗ Corresponding author. Tel.: +33 4 67 14 38 91. E-mail address: [email protected] (S. Peyron). 0376-7388/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2011.03.039
Everall thought; Bruneel et al. [8] corrected with a semiempirical approach Baldwin and Batchelder’s model in which the intensity of the Raman scattering decreased too drastically with the depth in the sample depth; Tomba and Pastor [9] and Tomba et al. [10] recently obtained the first results reasonably close to the experimental observations, after including a factor taking into account both diffraction and instrumental factors involved in distortion. This factor was determined with the results previously obtained from a silicon waffer. Based on a completely different approach, Sourisseau and Maraval [11] have carried out a rigorous theoretical treatment of these effects with a vectorial electromagnetic treatment. This approach has been described as “optimum” [12] but required an extremely complex mathematical treatment and computing resources (it takes about 1 day to perform the calculations of a depth profile). Likewise, other authors have followed different approaches to solve the problem presented in Everall’s pioneer work. Macdonald and Vaughan [13] used a simple model based on geometrical optics but needed to fit four parameters (three of them having a physical meaning) to reproduce experimental data. Gallardo et al. [14,15] proposed that the intensity resulting of a measure in the material core was the sum of the contributions of all adjacent points. Transforming the sum in an integral, the experimental intensity could be expressed as a Freudholm integral. The main drawback of their method is that it needed four unbounded arbitrary parameters to fit the experimental data. Aside from Sourisseau and Maraval’s [11] approach which is unpractical for most applications, there is no agreement to date on which strategy should be used to correct the refractive and instrumental effects.
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To get rid of the refractive distortions, mass transfer profiles in polymers has been studied cutting or microtoming a cross section and then moving the laser probe on the cut surface [16]. In a previous publication [17] we have used this method to successfully determine the diffusivity of a polymer additive (Uvitex OB) in linear low density polyethylene (LLDPE). However, one of the main advantages of Raman microspectroscopy (its ability to perform non-destructive analyses) is lost with such a procedure since the sample must be first cut to be analysed. The aim of this publication is to use Raman depth-profiling spectroscopy to determine the diffusivity of an additive (Uvitex OB) in a polymer (LLDPE). The method illustrated here can be used to study mass transfer in polymers in a non destructive way, which has applications in polymer coatings, food packaging, membrane science. Also, a discussion of the errors arising from this method is presented. Even if Raman depth-profiling of polymers have been recently tackled [1], this is, to the best of our knowledge, the first application taking into account refraction and instrumental-induced distortion, and based on the last significant progress in the field [10]. The method will be validated by comparison with the analysis of the cross-section of the sample in combination with Fourier transform infrared (FTIR) spectroscopy to determine the initial concentration. 2. Experimental 2.1. Chemicals 2,5-Bis-(5-tert.-butyl-benzoxazol-2-yl)-thiophen (Uvitex OB, 430.6 g mol−1 ) was purchased from Fluka. Olive oil was purchased in a local supermarket. LLDPE pellets were purchased from Sigma–Aldrich. 2.2. Films fabrication LLDPE pellets (refractive index, n1 = 1.51) were mixed with Uvitex OB at 140 ◦ C (50 rpm) during 5 min. The dough material obtained after mixing was then thermoformed (hot press) at 150 bar during 5 min at 140 ◦ C. The nominal concentration of the films was 0.2% (w/w). The films were prepared of three different nominal thicknesses (175 m, 250 m and 380 m). The actual film thickness was measured by using a micrometer (Braive Instruments, Chécy, Fr) in quintuplicate. 2.3. Partial desorption of the additive LLDPE film samples were fully immersed in 6 ml olive oil at 60 ◦ C during 60 min. The samples were then removed from the olive oil and thoroughly wiped in a reproducible way with a precision wipe. 2.4. FTIR measurements LLDPE film samples were analysed by transmission FTIR. Spectra were recorded using a Nexus 5700 spectrometer (ThermoElectron Corp.) equipped with HeNe beam splitter and a cooled MCT detector. Spectral data were accumulated from 128 scans with a resolution of 4 cm−1 in the range 800–4000 cm−1 . Three samples were employed for the measure and three spectra were recorded for each sample. All spectra pre-treatments were performed using Omnic v7.3 and TQ Analyst v7.3 softwares (ThermoElectron). Processing included: (1) a multipoint linear baseline correction and (2) a normalization according to the area of the LLDPE doublet (1369–1378 cm−1 ) due to the CH3 symmetric deformation vibration.
Fig. 1. Preparation of a sample for the Raman analysis. The slice is cut in order that the thickness (2L) becomes the width of the sample.
2.5. Raman spectroscopy Raman spectra were recorded between 95 and 3500 cm−1 Raman shift using a confocal Raman microspectrometer Almega (ThermoElectron) with the following configuration: excitation laser He–Ne 0 = 633 nm, grating 500 grooves/mm, pinhole 25 m, objective 50×, focal distance (f) 380 m, numerical aperture (NA) 0.8. The collection time was about 50 s. All spectra pre-treatments were performed with Omnic v7.1 (ThermoElectron). Processing included: (i) a multipoint linear baseline correction and (ii) normalization according to the area of the LLDPE specific band at 1129 cm−1 representing the symmetric C–C stretching of all-trans PE chains. The relative content of Uvitex OB was assessed using the area of the specific doublet (1569–1614 cm−1 ) assigned to the aromatic C C and C N bands respectively. 2.5.1. Spectra in z-direction (depth-profile) The samples were mounted on an x,y motorized stage with vertical (z) displacement automatically controlled. The spectra used to determine the profiles were obtained at the following nominal depths (): For the 175 m thick samples. From −7.5 m (just above the surface of the sample) to 97.5 m below the surface of the sample with a point measured every 7.5 m For the 250 m thick samples. From −10 m (just above the surface of the sample) to 150 m deep in the sample with a point measured every 10 m For the 380 m thick samples. From −10 m (just above the surface of the sample) to 190 m deep in the sample with a point measured every 10 m The distance between nodes is smaller for the thinnest samples in order to gather a similar number of points as the thicker samples. The nominal depth response of the system (in air) was measured by recording the intensity of the 520 cm−1 band of a silicon wafer as it translated along the z-axis (depth) through the focus of the objective according to Tomba et al. [10]. All the analyses were done in triplicate. 2.5.2. Spectra on the cross section As reported in a previous publication [17] thin slices of LLDPE were prepared using a razor blade and stuck on a microscope slide (Fig. 1). Measurements were carried out in the surface of the sample with a step size of 10 m (for the 175-m and 250-m thick samples) or 15 m (for the 380-m thick samples) in order to cover the whole thickness of the samples. 2.6. Theoretical analysis The analysis presented here has been developed by other authors in previous publications [5–10] and is just briefly outlined here for the sake of completeness.
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Fig. 2. Ray tracing analysis. In absence of refraction all the rays from the lens (here represented by an hemisphere) are focused in point P1 . However, refraction causes the rays to be focused deeper in the sample (P2 ). Note that for a paraxial ray (r = 0), P1 and P2 are coincident and that they reach the maximum separation for a marginal ray r = rmax . Adapted from Everall et al. [6].
Fig. 3. Ray tracing analysis. The arrows indicate the direction of the rays. Not all the Raman scattering from point P is detected. Only the rays that pass through the confocal aperture (0 ) are detected. As a consequence the focal volume is smaller than predicted by Everall [6] and the collection efficiency drastically decreases throughout the depth of the sample. Adapted from Baldwin and Batchelder [7].
Since the refractive index of the sample (in this case LLDPE, n1 = 1.51) is not the same as the air, the rays falling on the sample will be deviated according to Snell’s law, n1 sin( t ) = n2 sin( i ), where n1 and n2 are the refractive index of air (∼1) and the sample respectively; and i and t are the incident and refraction angles respectively (Fig. 2). From the simple ray tracing analysis shown in Fig. 2, Everall [5,6] was able to determine the position of the actual point illuminated by a marginal ray (zk ) and the depth of focus (range of laser local positions within the sample):
is too deviated and is not collected. As a general rule, if a point is located much deeper than the focal plane, a less significant fraction of the Raman scattering produced will be detected and, therefore, its contribution to the detected signal will be less important. Bruneel et al. [8] brought together the different elements previously presented and came to the following expression for the total Raman intensity of an experiment performed at nominal depth :
z=
rk2 NA2 (n2 − 1) 2 (1 − NA2 ) rmax
d.o.f. =
1/2
(1)
1/2
I
zmax
() = zmin
+ n2
NA2 (n2 − 1) + n2 (1 − NA2 )
Ap
−n
(2)
where is the nominal depth, NA is the numerical aperture, rk is the distance to the centre of the lens and n is the ratio of refractive index (equal to that of the polymer for the refractive index of the air is 1). An important conclusion is apparent from Eqs. (1) and (2). Not only the point of focus (z) but also the depth of focus varies with . Therefore, the deeper the measure is taken, the larger the depth of focus will be and consequently the resolution will decrease steadily throughout the depth of the sample. Baldwin and Batchelder [7] refined the model of Everall taking into account the effect of the confocal aperture on the collection. Actually, not all the Raman scattered beam is collected, but only the fraction that passes through the confocal aperture (a disk of radius 0 located at the focal distance) is detected. As an illustration, in Fig. 3 the ray from rk falls onto the point P, located z– deeper than the focal plane. Raman scattering from P point is emitted in all directions but not all reach the detector. In effect, only the rays that pass through the focal plane at a distance limited by the radius of the confocal aperture is detected (less than 0 ). In Fig. 3 this statement is represented by the rays emitted with angles 1 and 2 . The first passes through the confocal aperture whereas the second
r rmax
I0 exp
−2
r2
2 rmax
dr dz
˝(z)
dz
(3)
where I0 is the intensity of the paraxial ray, r is related to z by Eq. (1) and ˝ is the solid angle collected. Tomba et al. [10] improved Eq. (3) but introduced in turn empirical factors. They defined the response of a profile as a contribution between instrumental factors that may distort the signal and the contribution of the signal itself as:
∞
RInst (z) =
RSignal (z)LInst (z − z) dz
(4)
0
where RSignal is the response given by the polymer (a constant value given that its concentration remains virtually constant) or the additive (proportional to its concentration profile) and LInst is a Lorentzian function that takes into account the broadening of the instrumental factors that may distort the signal as diffraction, mismatch of the apertures, etc. This function is supposed to be invariant for a given set of instrumental conditions and focusing depth. Then, the strategy proposed by Tomba et al. [10] is to determine LInst as the best fit of the silicon wafer intensity profiles (intensity of the 520 cm−1 band). Finally, the apparent intensity obtained with a depth profile experiment for each nominal depth point () will be given by:
zmax
I Ap ()=
RInst (z) zmin
r rmax
I0 exp
−2
r2 2 rmax
dr dz
˝(z) dz
(5)
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Fig. 4. Example of Uvitex OB concentration profiles obtained with depth-profiling for a 226-m thick sample. (A) The normalized intensity, i.e. ratio to the maximum intensity registered, of the specific doublet (1569–1614 cm−1 ) assigned to Uvitex OB. (B) The result after normalisation by the initial concentration profile. Dots represent the experimental points and the continuous line the best fit.
where RInst is the response to instrumental distortion defined in Eq. (4), and the second term in the integral was defined in Eq. (3). 2.7. Diffusivity identification In the case of a sheet of polymer immersed in a liquid of infinite volume and constant concentration, the evolution of additive content with time is given by [18]:
× exp
n=0
−(2n + 1)2 2 D.t 4L2
cos
LInst (z) = 1.08 × 105
16.6 16.6 + z 2
(7)
3.2. Characteristics of depth-profiling results
C(z, t) − Cin 4 (−1)n =1− CL,∞ − Cin 2n + 1 ∞
RSignal (z, t) =
to determine the characteristic LInst function of the experimental configuration, which, according to Tomba et al. [10] is invariant for a given apparatus and numerical aperture. According to the best fit results, the LInst function is given by Eq. (7):
(2n + 1)z 2L
(6)
where C(z,t) is the concentration in diffusing substance in the packaging at time t and depth z, Cin is the initial concentration in additive and CL,∞ is the concentration of the diffusing substance on the surface of the polymer required to maintain equilibrium with the concentration of this substance, here considered to be null due to the large volume of olive oil used. The strategy followed to determine the diffusivity of the additive in the polymer consisted on (i) building a concentration profile (for a given diffusivity with Eq. (6)), (ii) obtaining RSignal for the concentration profile calculated, and (iii) calculating IAp with Eq. (5) to compare with the experimental data. The additive diffusivity was identified minimizing the root mean square error (RMSE) between experimental and predicted data (Eq. (5)) using an optimization method (Levenberg–Marquardt algorithm predefined in Matlab (Mathworks, USA) software). 3. Results and discussion Two methods based on Raman microspectroscopy were used to determine the profile of Uvitex OB in LLDPE: analysis of the crosssection and depth-profiling. As presented in a previous study [17], the profile of Uvitex OB was accessible after cutting the sample and analysing the surface of the cross-section. As this is a destructive method, it had to be combined with FTIR to accurately determine the additive concentration prior to the analysis. On the contrary, Raman depth-profiling is a fully non-destructive technique and the concentration may be determined with the same method before and after the test. 3.1. Results with silicon wafer. Obtention of LInst One must determine the Lorentzian function that describes the distortions caused by diffraction and instrumental factors before using Eqs. (4) and (5). The results with the silicon wafer allowed
The distortion of the signal leads to a steady decrease of the detected intensity of Raman scattering which limits its applications. Fig. 4A shows an example of the fitted profile of a 240-m thick sample after immersion in olive oil for 60 min at 60 ◦ C. It is quite illustrative to see the trends of the decreasing intensity of the Uvitex OB band throughout the thickness of the sample. This decrease is only partially due to the variation of Uvitex OB concentration in the sample, which decreases from = 70 m approx., but mainly to the attenuation of the signal as predicted by Everall [5,6] and Baldwin and Batchelder [7]. The reasons of this attenuation may have a clearer interpretation if the Eq. (4) is examined. This equation states that the intensity that should be obtained without refraction in the sample is the convolution of a Lorentzian function (monotonically decreasing above for z > 0) and the concentration profile (which reaches a maximum at the half thickness of the sample). If the signal obtained in Fig. 4A is normalized by the profile corresponding to the initial concentration of Uvitex (experimentally obtained), the profile in Fig. 4B is obtained. It is important to note that the decrease of intensity evidenced in Fig. 4A is no longer visible after normalization in Fig. 4B. However, since the intensity decreases steadily, the noise/signal ratio increases as the focus goes deeper in the sample which, consequently, represents a practical limit to the use of depth-profiling with thick samples. An attempt to characterize the dependency between the thickness of the sample and performance of depth-profiling was carried out with simulated data (Fig. 5). Arbitrary limit of quantification and initial concentration were set so that the maximum intensity value of the initial concentration profile accounted for a hundred times the limit of quantification. In order to compare different sample thicknesses, let us define the dimensionless time as = Dte−2 where D stands for the diffusivity, t for the time and e for the thickness of the sample. With these considerations, Fig. 5 represents the region of the sample (i.e. a fraction of the normalized depth) that lies above the limit of quantification as a function of the dimensionless time. It is clear from Fig. 5 that the characterization of mass transfer may be much more comprehensive for thin samples than for thick samples. As an example, let us inspect what happens for a value of dimensionless time equal to one ( = 1). This value is reached for a normalized depth of 0.66 for the 150 m sample, 0.51 for the
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Table 1 Comparison of diffusivity values obtained by depth-profiling and cross section analysis. Samples were immersed in olive oil for 60 ◦ C at 60 min. Sample
1 2 3 4 5
Fig. 5. Depth of the sample above a certain limit of quantification for samples of different thicknesses in function of the dimensionless time (simulated data for an arbitrary initial concentration set one hundred times higher than the limit of quantification).
250 m and only 0.38 for the 350 m. This roughly means, that for the same conditions of desorption and initial conditions, only 38% of the sample would be analyzable for the 350 m sample compared to a 66% for the 150 m sample at this point (note that this is just an approximation since the paraxial focus, n, is not exactly equivalent to the true depth, z). An important factor that should be taken into account when performing a depth-profile of a polymer sample is the selection of the right numerical aperture. An objective with a high numerical aperture (0.9–0.95) will have a high resolution and will be suitable to get a detailed profile close to the interface. However, refraction effects will be more severe for high numerical apertures (NA) and the decrease of signal will be more drastic as we get deeper in the sample. This is clearly seen in Eqs. (1) and (2), and elegantly illustrated by Baldwin and Batchelder [7] in Figure (8) of their study. Consequently, there must be an optimal numerical aperture for samples of different thicknesses i.e. high numerical aperture objectives would arguably be suitable for thin samples whereas lower numerical aperture objectives would be more adapted to thicker samples. As a perspective of future work, it would be interesting to carry out a formal quantitative assessment so that the right numerical aperture could be applied for a given thickness of sample. 3.3. Comparison and validation of depth-profiling method To validate depth-profiling as a suitable method to determine the diffusivity of an additive in a polymer sheet, the results obtained thereof were compared to the values of diffusivity obtained by the cross-section method (i.e. cutting the sample and directly
Thickness (m)
176 226 240 375 399
Diffusivity (×10−13 m2 s−1 ) Depth-profiling
Cross section
7.3 8.5 8.0 10.2 29.0
7.0 12.5 8.1 12.4 28.0
Relative error (%)
3.0 32.2 2.1 22.2 3.4
scanning the surface of the normal section which represents actually the thickness). Since the performance of depth-profiling is dependent of the thickness of the sample (Section 3.2), the comparison of results was done for several samples of different thicknesses (Table 1) for similar time of contact with olive oil. An essential point is to consider how depth the profile is taken into account in the determination of the diffusivity. Indeed, to be sure that the profile of concentration (and the diffusivity determined thereof) is representative of the sample, it is suitable to use the whole profile of concentration obtained. However, as it was seen in Section 3.2, the spectral data obtained deep in the sample have a larger noise/signal ratio and are of a poorer quality than those obtained closer to the interface and should not influence equally the parameter obtained. Since concentration profiles are symmetric in respect to the half-thickness of the sample, it seemed reasonable to take into account, at least half the profile of concentration. Following this reasoning, the nodes taken into account for the determination of diffusivity were those located between the interface (z = 0) and the first node after the half-thickness (considering the half-thickness as indicated by the paraxial focus, z = n). This is illustrated in Fig. 6, where only the first 8 points of the concentration profile are considered in the determination of diffusivity. All the diffusivity values obtained by depth-profiling in Table 1 were determined following this protocol. The comparison of the results between cross-section and depthprofiling (Table 1) leads to some remarkable conclusions. The relative error on the obtained diffusivities ranges from low (3%) to larger values (22% and 32%) taking the cross-section results as the reference. Interestingly, sample 5 (Table 1) shows an unexpectedly high value of diffusivity (maybe caused by a problem during the processing of the material). It is remarkable that this value is correctly registered by both methods. Unfortunately, the limited number of samples analysed here prevent us from carrying out a sound statistic-based validation of the method. A Student’s t-test, at a signification level of ˛ = 0.05, only states that the rel-
Fig. 6. Example of Uvitex OB concentration profiles obtained with depth-profiling (A) and cross section analysis (B) for a 240 m sample. Note that the horizontal axes are different: the nominal depth () for depth-profiling and the distance to the interface for cross section analysis. Squares represent the experimental points and the continuous line the best fit of data. Empty squares (A) were not taken into account for the fitting.
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ative error between the two methods is less than 34%, which is arguably not low enough for a quantitative method. It must be borne in mind, nonetheless, that diffusivity in a polymer is a parameter that spreads by several orders of magnitude. As a consequence, its determination usually bears high uncertainties, in particular if different methods are used. As an example, Dole et al. [19] recollect a number of diffusion coefficients gathered with different methods showing relative errors up to 100% whereas the maximum relative error found here stands at 32%. In conclusion, depth-profiling appears as a promising method allowing a fast, non-invasive and reasonably accurate determination of diffusivity. Furthermore, the fact that a complete profile of concentration can be determined in one measurement avoids long experimental trials, in particular for slow diffusion processes. This example is more readily applied to the study of migration of undesirable substances from packaging to foodstuff (essential for food safety assessment) but can be easily adapted to the study of polymer coatings, membranes, etc. Future work with more numerous samples and different configurations (different numerical apertures, thicknesses, confocal aperture) should be accomplished to ultimately validate this promising method and find the most suitable configuration for each application. The main differences between depth-profiling and cross section can be observed comparing Fig. 6A and B. Regarding the deviation between the experimental and the fitted data, it clearly appears that the determination of diffusion based on the half concentration profile in depth-profile results in the most accurate value. In Fig. 6, the concentration maximum is reached at 70 m at depth-profiling whereas it is located at 120 m for cross-section (as expected at the half-thickness). It is interesting to note in Fig. 6A that the values of concentration seem to stop decreasing from 120 m approx. This effect is due to the fact that at 120 m, the focal volume already contains part of the air below the sample. The extension of the focal volume is usually defined as the distance in which the light intensity falls to 50%. As an illustration, for 120 m nominal depth, the actual focal depth is located at 216 m to the interface and extends 52 m. Consequently, when focusing at 120 m, the actual measure extends from 190 m to 242 m (which lies out of the sample) and the specific signal of Uvitex OB (which is normalized by the intensity of the LLDPE band) tends towards a constant value. In comparison, the thickness of the sample does not influence the performance of the cross-section analysis method, but, cutting and manipulating the slice of polymer may become very difficult if not impossible for very thin samples, representing as well a limit to the use of cross-section analysis. In such cases, depth-profiling would be better adapted.
Acknowledgments
4. Conclusion
References
Raman depth-profiling microspectroscopy was successfully used, adapting the approach presented by Tomba et al. [10] to characterize the release of an additive (Uvitex OB) in LLDPE. The method was tentatively validated by comparison with the results obtained analysing the cross section of the sample. Even if more trials would be needed to carry out a sound validation of the depthprofiling method, it was illustrated how this method allowed a fast, non-invasive and reasonably accurate determination of diffusivity. Furthermore as a non-destructive method, depth-profiling represents an evident advantage over cross-section analysis. Comparisons were drawn between the two methods, in particular regarding the influence of the thickness of the sample on their performance. It turned out that depth-profiling is better adapted for thin samples whereas cross-section would be rather used for thicker samples.
Authors would like to kindly acknowledge financial support from the Commission of the European Communities, Framework 6, Priority 5 ‘Food Quality and Safety’, Integrated Project NovelQ FP6-CT-2006-015710. Note: The authors have tried to keep, as much as possible, the same notation as used in the main papers on Raman depth-profiling.] Nomenclature c Cin CL,∞
concentration initial concentration concentration on the surface of the polymer in equilibrium with the concentration in the solution D diffusivity d.o.f. depth of focus (range of laser local positions within the sample) f focal distance I intensity of the Raman signal I0 intensity of the Raman signal for the paraxial ray apparent intensity detected for a given IAp IInst intensity of the response taking into account the instrumental deviation for a given z L half thickness of the sample LInst Lorentzian function n refraction index NA numerical aperture RInst response taking into account the instrumental deviation for a given z response; proportional to the concentration of the RSignal substance distance to the centre of the lens rk z depth of the point actually illuminated by the incident ray zmax , zmin respectively the deepest and the shallowest point illuminated by the incident ray (obtained in Eq. (3) for rk = 0 and rk = rmax ) nominal depth, i.e. change of position of the microscope in relation to the surface of the sample 0 radius of the confocal aperture angle between the surface and the ray dimensionless time ˝ solid angle collected by the detector
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Raman depth-profiling characterization of a migrant diffusion in a polymer M. Mauricio-Iglesias, V. Guillard, N. Gontard, S. Peyron ∗ UMR 1208 IATE (Agropolymer Engineering and Emerging Technology), University of Montpellier 2, CIRAD, INRA, Montpellier Supagro, CC023, Pl. E Bataillon, F-34095 Montpellier FR, France
a r t i c l e
i n f o
Article history: Received 7 June 2010 Received in revised form 21 December 2010 Accepted 19 March 2011 Available online 29 March 2011 Keywords: Confocal Raman microscopy Depth-profiling Refraction Diffusion in polymers
a b s t r a c t Raman depth-profiling microspectroscopy provides rich information on chemical/physical characterization in a non-destructive mode with micrometric resolution. However, refraction causes distortions to the data obtained thereby. A method to determine the diffusivity of an additive in low linear density polyethylene (LLDPE) with Raman depth profiling is proposed, combining the latest developments on data treatment of refraction distorted profiles. The method is compared with the results obtained analysing the cross section of the sample, with a maximum 32% relative error between both methods. The main benefits, characteristics of this method, a discussion of the experimental errors, as well as perspectives for future work are highlighted. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Raman microspectroscopy is an outstanding tool for its high performance on physical/chemical characterization and accurate spatial resolution (down to 1 m approx.). Furthermore, it is possible to gather information of the inside of the sample in a nondestructive mode, which has applications on depth-profiling in polymer science [1], identifying inclusions within minerals, assessing the adsorbed layers by an electrode [2,3] or investigating drug penetration in membrane [4]. However the presence of air/sample interfaces (or any substance of different refractive index) will distort the confocal properties of the microscope resulting in a misinterpretation of data as extensively documented by Everall [5,6]. Whereas a measurement setup using immersion objective in oil matching the refractive index of the polymeric sample can prevent optical distortions, such approach is limited to the analysis of thin samples (e.g. minor than the focal distance of objective). On the contrary, the use of a larger optic distance is well-adapted to a broader variety of samples but, in turn, the data obtained thereby must be corrected. With a simple ray-tracing model, Everall [6] explained semi-quantitatively the increase in the depth-of-focus and the focal volume. This simple model has been successively refined in order to reproduce experimental values: Baldwin and Batchelder [7] showed that the confocal aperture limits the collection volume and that the refractive distortion is not so severe as
∗ Corresponding author. Tel.: +33 4 67 14 38 91. E-mail address: [email protected] (S. Peyron). 0376-7388/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2011.03.039
Everall thought; Bruneel et al. [8] corrected with a semiempirical approach Baldwin and Batchelder’s model in which the intensity of the Raman scattering decreased too drastically with the depth in the sample depth; Tomba and Pastor [9] and Tomba et al. [10] recently obtained the first results reasonably close to the experimental observations, after including a factor taking into account both diffraction and instrumental factors involved in distortion. This factor was determined with the results previously obtained from a silicon waffer. Based on a completely different approach, Sourisseau and Maraval [11] have carried out a rigorous theoretical treatment of these effects with a vectorial electromagnetic treatment. This approach has been described as “optimum” [12] but required an extremely complex mathematical treatment and computing resources (it takes about 1 day to perform the calculations of a depth profile). Likewise, other authors have followed different approaches to solve the problem presented in Everall’s pioneer work. Macdonald and Vaughan [13] used a simple model based on geometrical optics but needed to fit four parameters (three of them having a physical meaning) to reproduce experimental data. Gallardo et al. [14,15] proposed that the intensity resulting of a measure in the material core was the sum of the contributions of all adjacent points. Transforming the sum in an integral, the experimental intensity could be expressed as a Freudholm integral. The main drawback of their method is that it needed four unbounded arbitrary parameters to fit the experimental data. Aside from Sourisseau and Maraval’s [11] approach which is unpractical for most applications, there is no agreement to date on which strategy should be used to correct the refractive and instrumental effects.
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To get rid of the refractive distortions, mass transfer profiles in polymers has been studied cutting or microtoming a cross section and then moving the laser probe on the cut surface [16]. In a previous publication [17] we have used this method to successfully determine the diffusivity of a polymer additive (Uvitex OB) in linear low density polyethylene (LLDPE). However, one of the main advantages of Raman microspectroscopy (its ability to perform non-destructive analyses) is lost with such a procedure since the sample must be first cut to be analysed. The aim of this publication is to use Raman depth-profiling spectroscopy to determine the diffusivity of an additive (Uvitex OB) in a polymer (LLDPE). The method illustrated here can be used to study mass transfer in polymers in a non destructive way, which has applications in polymer coatings, food packaging, membrane science. Also, a discussion of the errors arising from this method is presented. Even if Raman depth-profiling of polymers have been recently tackled [1], this is, to the best of our knowledge, the first application taking into account refraction and instrumental-induced distortion, and based on the last significant progress in the field [10]. The method will be validated by comparison with the analysis of the cross-section of the sample in combination with Fourier transform infrared (FTIR) spectroscopy to determine the initial concentration. 2. Experimental 2.1. Chemicals 2,5-Bis-(5-tert.-butyl-benzoxazol-2-yl)-thiophen (Uvitex OB, 430.6 g mol−1 ) was purchased from Fluka. Olive oil was purchased in a local supermarket. LLDPE pellets were purchased from Sigma–Aldrich. 2.2. Films fabrication LLDPE pellets (refractive index, n1 = 1.51) were mixed with Uvitex OB at 140 ◦ C (50 rpm) during 5 min. The dough material obtained after mixing was then thermoformed (hot press) at 150 bar during 5 min at 140 ◦ C. The nominal concentration of the films was 0.2% (w/w). The films were prepared of three different nominal thicknesses (175 m, 250 m and 380 m). The actual film thickness was measured by using a micrometer (Braive Instruments, Chécy, Fr) in quintuplicate. 2.3. Partial desorption of the additive LLDPE film samples were fully immersed in 6 ml olive oil at 60 ◦ C during 60 min. The samples were then removed from the olive oil and thoroughly wiped in a reproducible way with a precision wipe. 2.4. FTIR measurements LLDPE film samples were analysed by transmission FTIR. Spectra were recorded using a Nexus 5700 spectrometer (ThermoElectron Corp.) equipped with HeNe beam splitter and a cooled MCT detector. Spectral data were accumulated from 128 scans with a resolution of 4 cm−1 in the range 800–4000 cm−1 . Three samples were employed for the measure and three spectra were recorded for each sample. All spectra pre-treatments were performed using Omnic v7.3 and TQ Analyst v7.3 softwares (ThermoElectron). Processing included: (1) a multipoint linear baseline correction and (2) a normalization according to the area of the LLDPE doublet (1369–1378 cm−1 ) due to the CH3 symmetric deformation vibration.
Fig. 1. Preparation of a sample for the Raman analysis. The slice is cut in order that the thickness (2L) becomes the width of the sample.
2.5. Raman spectroscopy Raman spectra were recorded between 95 and 3500 cm−1 Raman shift using a confocal Raman microspectrometer Almega (ThermoElectron) with the following configuration: excitation laser He–Ne 0 = 633 nm, grating 500 grooves/mm, pinhole 25 m, objective 50×, focal distance (f) 380 m, numerical aperture (NA) 0.8. The collection time was about 50 s. All spectra pre-treatments were performed with Omnic v7.1 (ThermoElectron). Processing included: (i) a multipoint linear baseline correction and (ii) normalization according to the area of the LLDPE specific band at 1129 cm−1 representing the symmetric C–C stretching of all-trans PE chains. The relative content of Uvitex OB was assessed using the area of the specific doublet (1569–1614 cm−1 ) assigned to the aromatic C C and C N bands respectively. 2.5.1. Spectra in z-direction (depth-profile) The samples were mounted on an x,y motorized stage with vertical (z) displacement automatically controlled. The spectra used to determine the profiles were obtained at the following nominal depths (): For the 175 m thick samples. From −7.5 m (just above the surface of the sample) to 97.5 m below the surface of the sample with a point measured every 7.5 m For the 250 m thick samples. From −10 m (just above the surface of the sample) to 150 m deep in the sample with a point measured every 10 m For the 380 m thick samples. From −10 m (just above the surface of the sample) to 190 m deep in the sample with a point measured every 10 m The distance between nodes is smaller for the thinnest samples in order to gather a similar number of points as the thicker samples. The nominal depth response of the system (in air) was measured by recording the intensity of the 520 cm−1 band of a silicon wafer as it translated along the z-axis (depth) through the focus of the objective according to Tomba et al. [10]. All the analyses were done in triplicate. 2.5.2. Spectra on the cross section As reported in a previous publication [17] thin slices of LLDPE were prepared using a razor blade and stuck on a microscope slide (Fig. 1). Measurements were carried out in the surface of the sample with a step size of 10 m (for the 175-m and 250-m thick samples) or 15 m (for the 380-m thick samples) in order to cover the whole thickness of the samples. 2.6. Theoretical analysis The analysis presented here has been developed by other authors in previous publications [5–10] and is just briefly outlined here for the sake of completeness.
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Fig. 2. Ray tracing analysis. In absence of refraction all the rays from the lens (here represented by an hemisphere) are focused in point P1 . However, refraction causes the rays to be focused deeper in the sample (P2 ). Note that for a paraxial ray (r = 0), P1 and P2 are coincident and that they reach the maximum separation for a marginal ray r = rmax . Adapted from Everall et al. [6].
Fig. 3. Ray tracing analysis. The arrows indicate the direction of the rays. Not all the Raman scattering from point P is detected. Only the rays that pass through the confocal aperture (0 ) are detected. As a consequence the focal volume is smaller than predicted by Everall [6] and the collection efficiency drastically decreases throughout the depth of the sample. Adapted from Baldwin and Batchelder [7].
Since the refractive index of the sample (in this case LLDPE, n1 = 1.51) is not the same as the air, the rays falling on the sample will be deviated according to Snell’s law, n1 sin( t ) = n2 sin( i ), where n1 and n2 are the refractive index of air (∼1) and the sample respectively; and i and t are the incident and refraction angles respectively (Fig. 2). From the simple ray tracing analysis shown in Fig. 2, Everall [5,6] was able to determine the position of the actual point illuminated by a marginal ray (zk ) and the depth of focus (range of laser local positions within the sample):
is too deviated and is not collected. As a general rule, if a point is located much deeper than the focal plane, a less significant fraction of the Raman scattering produced will be detected and, therefore, its contribution to the detected signal will be less important. Bruneel et al. [8] brought together the different elements previously presented and came to the following expression for the total Raman intensity of an experiment performed at nominal depth :
z=
rk2 NA2 (n2 − 1) 2 (1 − NA2 ) rmax
d.o.f. =
1/2
(1)
1/2
I
zmax
() = zmin
+ n2
NA2 (n2 − 1) + n2 (1 − NA2 )
Ap
−n
(2)
where is the nominal depth, NA is the numerical aperture, rk is the distance to the centre of the lens and n is the ratio of refractive index (equal to that of the polymer for the refractive index of the air is 1). An important conclusion is apparent from Eqs. (1) and (2). Not only the point of focus (z) but also the depth of focus varies with . Therefore, the deeper the measure is taken, the larger the depth of focus will be and consequently the resolution will decrease steadily throughout the depth of the sample. Baldwin and Batchelder [7] refined the model of Everall taking into account the effect of the confocal aperture on the collection. Actually, not all the Raman scattered beam is collected, but only the fraction that passes through the confocal aperture (a disk of radius 0 located at the focal distance) is detected. As an illustration, in Fig. 3 the ray from rk falls onto the point P, located z– deeper than the focal plane. Raman scattering from P point is emitted in all directions but not all reach the detector. In effect, only the rays that pass through the focal plane at a distance limited by the radius of the confocal aperture is detected (less than 0 ). In Fig. 3 this statement is represented by the rays emitted with angles 1 and 2 . The first passes through the confocal aperture whereas the second
r rmax
I0 exp
−2
r2
2 rmax
dr dz
˝(z)
dz
(3)
where I0 is the intensity of the paraxial ray, r is related to z by Eq. (1) and ˝ is the solid angle collected. Tomba et al. [10] improved Eq. (3) but introduced in turn empirical factors. They defined the response of a profile as a contribution between instrumental factors that may distort the signal and the contribution of the signal itself as:
∞
RInst (z) =
RSignal (z)LInst (z − z) dz
(4)
0
where RSignal is the response given by the polymer (a constant value given that its concentration remains virtually constant) or the additive (proportional to its concentration profile) and LInst is a Lorentzian function that takes into account the broadening of the instrumental factors that may distort the signal as diffraction, mismatch of the apertures, etc. This function is supposed to be invariant for a given set of instrumental conditions and focusing depth. Then, the strategy proposed by Tomba et al. [10] is to determine LInst as the best fit of the silicon wafer intensity profiles (intensity of the 520 cm−1 band). Finally, the apparent intensity obtained with a depth profile experiment for each nominal depth point () will be given by:
zmax
I Ap ()=
RInst (z) zmin
r rmax
I0 exp
−2
r2 2 rmax
dr dz
˝(z) dz
(5)
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Fig. 4. Example of Uvitex OB concentration profiles obtained with depth-profiling for a 226-m thick sample. (A) The normalized intensity, i.e. ratio to the maximum intensity registered, of the specific doublet (1569–1614 cm−1 ) assigned to Uvitex OB. (B) The result after normalisation by the initial concentration profile. Dots represent the experimental points and the continuous line the best fit.
where RInst is the response to instrumental distortion defined in Eq. (4), and the second term in the integral was defined in Eq. (3). 2.7. Diffusivity identification In the case of a sheet of polymer immersed in a liquid of infinite volume and constant concentration, the evolution of additive content with time is given by [18]:
× exp
n=0
−(2n + 1)2 2 D.t 4L2
cos
LInst (z) = 1.08 × 105
16.6 16.6 + z 2
(7)
3.2. Characteristics of depth-profiling results
C(z, t) − Cin 4 (−1)n =1− CL,∞ − Cin 2n + 1 ∞
RSignal (z, t) =
to determine the characteristic LInst function of the experimental configuration, which, according to Tomba et al. [10] is invariant for a given apparatus and numerical aperture. According to the best fit results, the LInst function is given by Eq. (7):
(2n + 1)z 2L
(6)
where C(z,t) is the concentration in diffusing substance in the packaging at time t and depth z, Cin is the initial concentration in additive and CL,∞ is the concentration of the diffusing substance on the surface of the polymer required to maintain equilibrium with the concentration of this substance, here considered to be null due to the large volume of olive oil used. The strategy followed to determine the diffusivity of the additive in the polymer consisted on (i) building a concentration profile (for a given diffusivity with Eq. (6)), (ii) obtaining RSignal for the concentration profile calculated, and (iii) calculating IAp with Eq. (5) to compare with the experimental data. The additive diffusivity was identified minimizing the root mean square error (RMSE) between experimental and predicted data (Eq. (5)) using an optimization method (Levenberg–Marquardt algorithm predefined in Matlab (Mathworks, USA) software). 3. Results and discussion Two methods based on Raman microspectroscopy were used to determine the profile of Uvitex OB in LLDPE: analysis of the crosssection and depth-profiling. As presented in a previous study [17], the profile of Uvitex OB was accessible after cutting the sample and analysing the surface of the cross-section. As this is a destructive method, it had to be combined with FTIR to accurately determine the additive concentration prior to the analysis. On the contrary, Raman depth-profiling is a fully non-destructive technique and the concentration may be determined with the same method before and after the test. 3.1. Results with silicon wafer. Obtention of LInst One must determine the Lorentzian function that describes the distortions caused by diffraction and instrumental factors before using Eqs. (4) and (5). The results with the silicon wafer allowed
The distortion of the signal leads to a steady decrease of the detected intensity of Raman scattering which limits its applications. Fig. 4A shows an example of the fitted profile of a 240-m thick sample after immersion in olive oil for 60 min at 60 ◦ C. It is quite illustrative to see the trends of the decreasing intensity of the Uvitex OB band throughout the thickness of the sample. This decrease is only partially due to the variation of Uvitex OB concentration in the sample, which decreases from = 70 m approx., but mainly to the attenuation of the signal as predicted by Everall [5,6] and Baldwin and Batchelder [7]. The reasons of this attenuation may have a clearer interpretation if the Eq. (4) is examined. This equation states that the intensity that should be obtained without refraction in the sample is the convolution of a Lorentzian function (monotonically decreasing above for z > 0) and the concentration profile (which reaches a maximum at the half thickness of the sample). If the signal obtained in Fig. 4A is normalized by the profile corresponding to the initial concentration of Uvitex (experimentally obtained), the profile in Fig. 4B is obtained. It is important to note that the decrease of intensity evidenced in Fig. 4A is no longer visible after normalization in Fig. 4B. However, since the intensity decreases steadily, the noise/signal ratio increases as the focus goes deeper in the sample which, consequently, represents a practical limit to the use of depth-profiling with thick samples. An attempt to characterize the dependency between the thickness of the sample and performance of depth-profiling was carried out with simulated data (Fig. 5). Arbitrary limit of quantification and initial concentration were set so that the maximum intensity value of the initial concentration profile accounted for a hundred times the limit of quantification. In order to compare different sample thicknesses, let us define the dimensionless time as = Dte−2 where D stands for the diffusivity, t for the time and e for the thickness of the sample. With these considerations, Fig. 5 represents the region of the sample (i.e. a fraction of the normalized depth) that lies above the limit of quantification as a function of the dimensionless time. It is clear from Fig. 5 that the characterization of mass transfer may be much more comprehensive for thin samples than for thick samples. As an example, let us inspect what happens for a value of dimensionless time equal to one ( = 1). This value is reached for a normalized depth of 0.66 for the 150 m sample, 0.51 for the
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Table 1 Comparison of diffusivity values obtained by depth-profiling and cross section analysis. Samples were immersed in olive oil for 60 ◦ C at 60 min. Sample
1 2 3 4 5
Fig. 5. Depth of the sample above a certain limit of quantification for samples of different thicknesses in function of the dimensionless time (simulated data for an arbitrary initial concentration set one hundred times higher than the limit of quantification).
250 m and only 0.38 for the 350 m. This roughly means, that for the same conditions of desorption and initial conditions, only 38% of the sample would be analyzable for the 350 m sample compared to a 66% for the 150 m sample at this point (note that this is just an approximation since the paraxial focus, n, is not exactly equivalent to the true depth, z). An important factor that should be taken into account when performing a depth-profile of a polymer sample is the selection of the right numerical aperture. An objective with a high numerical aperture (0.9–0.95) will have a high resolution and will be suitable to get a detailed profile close to the interface. However, refraction effects will be more severe for high numerical apertures (NA) and the decrease of signal will be more drastic as we get deeper in the sample. This is clearly seen in Eqs. (1) and (2), and elegantly illustrated by Baldwin and Batchelder [7] in Figure (8) of their study. Consequently, there must be an optimal numerical aperture for samples of different thicknesses i.e. high numerical aperture objectives would arguably be suitable for thin samples whereas lower numerical aperture objectives would be more adapted to thicker samples. As a perspective of future work, it would be interesting to carry out a formal quantitative assessment so that the right numerical aperture could be applied for a given thickness of sample. 3.3. Comparison and validation of depth-profiling method To validate depth-profiling as a suitable method to determine the diffusivity of an additive in a polymer sheet, the results obtained thereof were compared to the values of diffusivity obtained by the cross-section method (i.e. cutting the sample and directly
Thickness (m)
176 226 240 375 399
Diffusivity (×10−13 m2 s−1 ) Depth-profiling
Cross section
7.3 8.5 8.0 10.2 29.0
7.0 12.5 8.1 12.4 28.0
Relative error (%)
3.0 32.2 2.1 22.2 3.4
scanning the surface of the normal section which represents actually the thickness). Since the performance of depth-profiling is dependent of the thickness of the sample (Section 3.2), the comparison of results was done for several samples of different thicknesses (Table 1) for similar time of contact with olive oil. An essential point is to consider how depth the profile is taken into account in the determination of the diffusivity. Indeed, to be sure that the profile of concentration (and the diffusivity determined thereof) is representative of the sample, it is suitable to use the whole profile of concentration obtained. However, as it was seen in Section 3.2, the spectral data obtained deep in the sample have a larger noise/signal ratio and are of a poorer quality than those obtained closer to the interface and should not influence equally the parameter obtained. Since concentration profiles are symmetric in respect to the half-thickness of the sample, it seemed reasonable to take into account, at least half the profile of concentration. Following this reasoning, the nodes taken into account for the determination of diffusivity were those located between the interface (z = 0) and the first node after the half-thickness (considering the half-thickness as indicated by the paraxial focus, z = n). This is illustrated in Fig. 6, where only the first 8 points of the concentration profile are considered in the determination of diffusivity. All the diffusivity values obtained by depth-profiling in Table 1 were determined following this protocol. The comparison of the results between cross-section and depthprofiling (Table 1) leads to some remarkable conclusions. The relative error on the obtained diffusivities ranges from low (3%) to larger values (22% and 32%) taking the cross-section results as the reference. Interestingly, sample 5 (Table 1) shows an unexpectedly high value of diffusivity (maybe caused by a problem during the processing of the material). It is remarkable that this value is correctly registered by both methods. Unfortunately, the limited number of samples analysed here prevent us from carrying out a sound statistic-based validation of the method. A Student’s t-test, at a signification level of ˛ = 0.05, only states that the rel-
Fig. 6. Example of Uvitex OB concentration profiles obtained with depth-profiling (A) and cross section analysis (B) for a 240 m sample. Note that the horizontal axes are different: the nominal depth () for depth-profiling and the distance to the interface for cross section analysis. Squares represent the experimental points and the continuous line the best fit of data. Empty squares (A) were not taken into account for the fitting.
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ative error between the two methods is less than 34%, which is arguably not low enough for a quantitative method. It must be borne in mind, nonetheless, that diffusivity in a polymer is a parameter that spreads by several orders of magnitude. As a consequence, its determination usually bears high uncertainties, in particular if different methods are used. As an example, Dole et al. [19] recollect a number of diffusion coefficients gathered with different methods showing relative errors up to 100% whereas the maximum relative error found here stands at 32%. In conclusion, depth-profiling appears as a promising method allowing a fast, non-invasive and reasonably accurate determination of diffusivity. Furthermore, the fact that a complete profile of concentration can be determined in one measurement avoids long experimental trials, in particular for slow diffusion processes. This example is more readily applied to the study of migration of undesirable substances from packaging to foodstuff (essential for food safety assessment) but can be easily adapted to the study of polymer coatings, membranes, etc. Future work with more numerous samples and different configurations (different numerical apertures, thicknesses, confocal aperture) should be accomplished to ultimately validate this promising method and find the most suitable configuration for each application. The main differences between depth-profiling and cross section can be observed comparing Fig. 6A and B. Regarding the deviation between the experimental and the fitted data, it clearly appears that the determination of diffusion based on the half concentration profile in depth-profile results in the most accurate value. In Fig. 6, the concentration maximum is reached at 70 m at depth-profiling whereas it is located at 120 m for cross-section (as expected at the half-thickness). It is interesting to note in Fig. 6A that the values of concentration seem to stop decreasing from 120 m approx. This effect is due to the fact that at 120 m, the focal volume already contains part of the air below the sample. The extension of the focal volume is usually defined as the distance in which the light intensity falls to 50%. As an illustration, for 120 m nominal depth, the actual focal depth is located at 216 m to the interface and extends 52 m. Consequently, when focusing at 120 m, the actual measure extends from 190 m to 242 m (which lies out of the sample) and the specific signal of Uvitex OB (which is normalized by the intensity of the LLDPE band) tends towards a constant value. In comparison, the thickness of the sample does not influence the performance of the cross-section analysis method, but, cutting and manipulating the slice of polymer may become very difficult if not impossible for very thin samples, representing as well a limit to the use of cross-section analysis. In such cases, depth-profiling would be better adapted.
Acknowledgments
4. Conclusion
References
Raman depth-profiling microspectroscopy was successfully used, adapting the approach presented by Tomba et al. [10] to characterize the release of an additive (Uvitex OB) in LLDPE. The method was tentatively validated by comparison with the results obtained analysing the cross section of the sample. Even if more trials would be needed to carry out a sound validation of the depthprofiling method, it was illustrated how this method allowed a fast, non-invasive and reasonably accurate determination of diffusivity. Furthermore as a non-destructive method, depth-profiling represents an evident advantage over cross-section analysis. Comparisons were drawn between the two methods, in particular regarding the influence of the thickness of the sample on their performance. It turned out that depth-profiling is better adapted for thin samples whereas cross-section would be rather used for thicker samples.
Authors would like to kindly acknowledge financial support from the Commission of the European Communities, Framework 6, Priority 5 ‘Food Quality and Safety’, Integrated Project NovelQ FP6-CT-2006-015710. Note: The authors have tried to keep, as much as possible, the same notation as used in the main papers on Raman depth-profiling.] Nomenclature c Cin CL,∞
concentration initial concentration concentration on the surface of the polymer in equilibrium with the concentration in the solution D diffusivity d.o.f. depth of focus (range of laser local positions within the sample) f focal distance I intensity of the Raman signal I0 intensity of the Raman signal for the paraxial ray apparent intensity detected for a given IAp IInst intensity of the response taking into account the instrumental deviation for a given z L half thickness of the sample LInst Lorentzian function n refraction index NA numerical aperture RInst response taking into account the instrumental deviation for a given z response; proportional to the concentration of the RSignal substance distance to the centre of the lens rk z depth of the point actually illuminated by the incident ray zmax , zmin respectively the deepest and the shallowest point illuminated by the incident ray (obtained in Eq. (3) for rk = 0 and rk = rmax ) nominal depth, i.e. change of position of the microscope in relation to the surface of the sample 0 radius of the confocal aperture angle between the surface and the ray dimensionless time ˝ solid angle collected by the detector
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