Fused Glass Beads Research Articles

Comparation of Pressed Powder Pellet and Fused Glass Bead Preparation Techniques for Mayor Elements Analysis of Rock Samples using X-Ray Fluorescence (XRF)

The composition of chemical elements in rock samples can be used as a tool for classifying rock types, predicting the depositional environment, rock age, and the tectonic environment in which the rock is formed. X-Ray Fluorescence (XRF) is one of the instruments used to analyze rock samples. The advantage of using XRF analysis is that the analysis process can be done more quickly, easily, accurately, and does not damage the sample. However, before measurements are made using the XRF instrument it needs to be supported by an appropriate sample preparation process. In this study, optimal conditions in the standard (CRM) rock sample preparation of GBW 07105 and JR-1 have been studied. The parameters studied were variations in the composition ratio of Cellulose Mycro Crystalin (CMC) with standard (CRM) samples, namely 1: 4 and 1: 3. The technique of mixing the sample is done by using a mortar grinder and a shaker. Optimization of sample preparation was carried out using pressed powder pellet and fused glass bead techniques. Based on the data obtained from the analysis results the best ratio of binders to standard samples is 1: 3. The pressed powder pellet preparation technique is the best technique in analyzing samples using the X-Ray Fluorescence method. Further studies also need to be carried out to analyze minor elements and traces (REEs).

Particle Induced X-ray Emission spectrometry (PIXE) of Hawaiian volcanics: An analogue study to evaluate the APXS field analysis of geologic materials on Mars

The Alpha Particle X-ray Spectrometer (APXS), a field instrument onboard four martian rovers, measures largely unprepared, in situ samples on Mars. The APXS has high precision that enables the determination of elemental concentrations in a wide range of geologic materials. However, lack of sample preparation can lead to heterogeneous matrix effects, and understanding the associated uncertainty is essential for interpreting APXS data. Here we use Particle Induced X-ray Emission spectrometry (PIXE) to analyze a suite of geologic samples from Hawai'i as an analogue study to better understand APXS analyses of martian samples. Wavelength-Dispersive X-ray Fluorescence (WDXRF) analyses of fused glass beads establish higher-accuracy standards for the Hawaiian samples. Sulfate-silicate mixtures were made to evaluate sulfur analysis by PIXE. Results show that the PIXE concentrations for most major elements have 2–6% accuracy, which is comparable to the APXS. However, the PIXE concentrations are systematically high in Al and low in Mg, resulting in lower accuracy (13% and 20%, respectively). Olivine-phyric lavas and most of their altered products have the largest discrepancies with Al concentrations up to 25% high and Mg up to 35% low. Sulfur is systematically high (up to 30% in a basalt matrix) compared to gravimetric S concentrations in the sulfate-silicate mixtures. These systematic deviations in Mg, Al, and S are linked to heterogeneous matrix effects, because PIXE and APXS analyses assume all atoms in a sample to be homogeneously mixed on the sub-micrometer scale, which is not the case. Two key implications for APXS results are: (1) Olivine-bearing samples likely have reported concentrations of Mg that is too low and Al that is too high. Thus, olivine-phyric basalts in Gusev crater and the basaltic sand and soil at three landing sites may have Mg and Al concentrations closer to those of the olivine-phyric shergottites and modelled martian crust than previously thought. (2) Sulfate-silicate mixtures may have overestimated S concentrations reported, resulting in greater uncertainty in the stoichiometry of Ca-sulfates, which is used to deduce the geochemical associations of sulfur in samples.

Using Ultrasonic Waves to Determine the Microstructure of Snow

High frequency (ultrasonic) acoustic wave propagation measurements were inverted to determine important near-surface parameters of snow including porosity, tortuosity, and pore geometry, using a newly-designed, oblique-reflection transducer apparatus. Acoustic signals interact with the physical structure of porous media, are particularly sensitive to porosity and tortuosity, and can be used to measure physical properties in a non-destructive manner. Given the fragile nature of freshly fallen snow, non-contact, non-destructive characterization methods made possible via acoustic signals, are desirable. High frequency wave propagation methods can be used to determine in situ near surface micro-pore geometry parameters in snow using methods demonstrated on cohesive porous materials including manufactured foams, porous metals, and sintered glass beads. Here, we conducted high frequency, oblique-angle and near vertical reflection measurements on snow samples to demonstrate the feasibility of acoustic response detection. We compared the acoustically derived snow physical parameters, including porosity and tortuosity, with values determined from X-ray micro-computed tomography (μCT) and gravimetric methods. Preliminary results using different snow types, and following the methods from previous work for cohesive porous media (i.e. fused glass beads) show good agreement between values of porosity determined from the acoustic measurements and the values determined from μCT image analysis.

A simple determination of whole-rock major- and trace-element composition for peridotite by micro-XRF spectrometer and ICP-MS using fused-glass bead

A simple determination of whole-rock major- and trace-element composition for peridotite by micro-XRF spectrometer and ICP-MS using fused-glass bead

Quantitative verification of 1 : 100 diluted fused glass beads for X-ray fluorescence analysis of geological specimens

This paper proposed a semi-automated small sample (30 mg) preparation method. The accuracy and precision of this new method were verified by measuring two CRMs among XRF, ICP-OES (fusion), and ICP-OES (acid attack) tools.

Internal standard fused glass beads for high silicon content sample analysis by laser-induced breakdown spectrometry

The direct analysis of high silicon content samples by Laser-Induced Breakdown Spectrometry (LIBS) is a good alternative, since the decomposition of samples with refractory characteristics is not an easy task.

Comparison of Fused Glass Beads and Pressed Powder Pellets for the Quantitative Measurement of Al, Fe, Si and Ti in Bauxite by Laser‐Induced Breakdown Spectroscopy

Due to matrix interference and sample particle size effects, some of the most important and difficult issues in laser‐induced breakdown spectroscopy (LIBS) analysis are the calibration and quantitative measurement of a complex matrix. This study proposes the use of borate fusion as an alternative sample preparation procedure for the quantitative measurement of Al, Fe, Si and Ti in bauxite by LIBS. Analytical calibration curves were made using bauxite certified reference materials (CRM), and the precision and accuracy of the methods were evaluated by analysing an additional bauxite CRM, using two different approaches: pressed powder pellets and fused glass beads. The borate fusion method was the most suitable sample preparation technique, since particle size effects and matrix interference could be minimised, obtaining better linearity on the analytical calibration curves (r2), and more accurate and more precise results for bauxite analysis.

Photocatalytic degradation of 4-chlorophenol in a photochemical reactor with an interconnected microchannel and a light guide network

Abstract Photocatalytic microchannel reactor with a novel light guide of fused glass beads was developed. Anatase nanoparticle photocatalyst was deposited on the inner surface of the porous glass monolith of the reactor. Activity of the reactor was evaluated based on photocatalytic degradation of 4-chlorophenol. All the glass beads in a glass column were fused with each other and with the inner peripheral of the column to form a glass network, which construct a network of light guide to deliver the incident light from side wall to each bead in the column. In contrast, almost all the voids among the beads connect with each other to form a network of microchannels for substrate solution, where many parallel flow of the substrate solution can pass through the reactor. Microchannel reactors composed of 5 mm or 1 mm glass beads in diameter were investigated, so that the dimension of the channels are evaluated as approximately 750 μm and 150 μm, respectively. The glass web monolith serves as a support of TiO 2 photocatalyst and a light guide to deliver the incident light to activate the photocatalyst on each bead. Photocatalytic activity of the reactor constructed with the 1 mm or the 5 mm glass beads was improved 13 and 3 times as much as that of the simple glass column without the beads, respectively. The ratio in the reactivity was proportional to the ratio in internal surface area of the reactors, supporting the effect of the glass web as light guide and the network of the voids as microchannel.

Trace element analysis of whole-rock glass beads of geological reference materials by Nd:YAG UV 213 nm LA-ICP-MS

A trace element composition of rock samples is difficult to determine when they contain refractory minerals that are hardly dissolved with conventional acid digestion techniques. Fused glass beads of rock samples could be an adequate target to circumvent this problem. We here report inductively coupled plasma-mass spectrometry (ICP-MS) results for geological reference materials (GRMs) prepared to normal- (sample/flux = 1:5) and low-dilution (sample/flux = 1:2) glass beads by using Nd:YAG UV 213-nm laser ablation system. Concentrations of 24 trace elements (Ba, Hf, Nb, Rb, Sr, Ta, Th, U, Y, Zr, and 14 rare earth elements (REEs)) were analyzed for three USGS GRMs (G-3 granite, AGV-2 andesite, and BHVO-2 basalt). Each analysis for the GRM beads was performed as spot (ca. 55 μm diameter) analysis with 120-s ablation time. The depth-to-diameter ratio of the laser spot was low (<4) enough to prevent significant elemental fractionation. The NIST612 glass and 29Si were employed as the external standard and the internal standard element, respectively. When the middle half of time-integrated data was taken to minimize the fractionation effect, the low-dilution fused glasses yielded reproducible and accurate results for all analyzed elements. In the case of normal-dilution fused glasses, comparable precision and accuracy were obtained only for elements with concentrations higher than 1 μg g−1, likely resulting from higher dilution ratios. Low-dilution glass beads can be an adequate target to analyze trace element composition of rock samples by using the laser ablation ICP-MS. This simple and rapid technique can be applied directly to the same glass beads prepared for major elemental analysis using the XRF.

ガラスビード試料を用いたレーザーアブレーション誘導結合プラズマ質量分析計（LA-ICP-MS） による全岩微量元素分析

ガラスビード試料を用いたレーザーアブレーション誘導結合プラズマ質量分析計（LA-ICP-MS） による全岩微量元素分析

규산염 암석의 알칼리 용융 유리원판에 대한 LA-ICP-MS 전암 미량원소 분석법 평가

알칼리 용융법으로 제조한 규산염 암석 참조물질(JR-3, JG-3, JGb-1, JB-1b) 유리원판에 대하여 레이저 삭마-유도결합 플라즈마 질량분석법(LA-ICP-MS)으로 미량원소를 분석한 결과의 신뢰도를 평가하였다. 희토류 원소를 포함하는 28개 원소 분석결과시 암석내 농도가 10 ppm 이상인 경우에는 측정의 상대표준편차(RSD)가 7% 이하이었으며, 10 ppm이하의 농도에서는 RSD가 다소 증가하기도 하지만 25%를 넘지는 않았다. 동일한 암석 참조물질로부터 이전에 보고된 분석자료의 평균과 비교하면 대부분의 경우 최대 10% 내외의 편차로 잘 일치하는 만족스러운 결과를 나타낸다. 알칼리 용융 유리원판을 이용한 LA-ICP-MS 분석법은 용액 ICP-MS 분석법에 비하여 전처리 과정을 대폭 줄이고 빠른 시간 내에 비교적 높은 정밀도로 고철질에서 규장질까지 대부분 암석의 미량원소를 분석할 수 있는 방법으로 평가된다. Using laser-ablation inductively coupled plasma mass spectrometry (LA-ICP-MS), we evaluated the reliability of trace element abundance data measured from the silicate rock references of JR-3, JG-3, JGb-1 and JB-1b using glass discs made by alkali fusion. For 28 elements including rare earth elements, relative standard deviations (RSD) are better than 7% in case when the concentrations of the elements in the rock samples are greater than 10 ppm. However, RSD shows somewhat increased values for the concentrations less than 10 ppm, but never exceeds 25%. Compared with previously reported averages of the compiled abundance data, our data display satisfactory results for the most cases with differences less than 10%. We suggest that LA-ICP-MS analysis using fused glass beads is a reliable, precise and time-saving method of trace element analysis for the silicate rocks spanning from mafic to felsic compositions.

High-resolution line-scan analysis of resin-embedded sediments using laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS)

Laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS) line-scanning is a promising technique for producing high-resolution (μm-scale) geochemical records on resin-embedded sediments. However, this approach has not yet been thoroughly tested on sediment samples of known elemental compositions. Here, we address this through the analysis of resin-embedded quartz, calcite, and clay (montmorillonite) sediments spiked with Al, V, Mo, and Ba across a range of concentrations. LA-ICP-MS spot analyses on these samples were compared to data independently obtained by conventional techniques: solution nebulization-ICP-optical emission spectroscopy (SN-ICP-OES), SN-ICP-MS, and X-Ray Fluorescence on fused glass beads. Data were reported as (log-)ratios of the analyte elements normalized to Ca, Si, and Al for calcite, quartz and clay respectively to correct for varying ablation yields. Our LA-ICP-MS data demonstrate close agreement to within 6% of the reference values determined by the conventional techniques, with high correlation coefficients (R2≈1.0) across the full range of concentrations, and high precision (<3% relative standard deviations after three repeated analyses). Barium was hosted in both aluminosilicates and carbonates in the clay matrix, giving different yields (elemental fractionation), and leading to variable accuracy (mean deviation of 15%). A selection of the spiked sediments was used to produce artificial laminated sediments to further test the effects of line scanning on LA-ICP-MS accuracy and precision in resin-embedded samples. It appears that LA-ICP-MS line-scan analyses have good accuracy (deviations from the reference values generally <5%) and high precision (relative standard deviations of repeated analyses on the same sample generally <5%) for our target elements. Moreover, the LA-ICP-MS line-scanning closely records the alternating geochemical profiles of our artificial laminations. However, calcium showed a clear tailing signal at the transition from a calcite to a quartz layer, indicating that geochemical signals can be smeared at transitions to sediment layers devoid of analyte element(s). We also analyzed resin-embedded natural sediments for comparison of LA-ICP-MS line-scan data to parallel sub-samples measured by SN-ICP-OES and SN-ICP-MS. In general, the results from LA-ICP-MS line-scans of natural sediments correspond well with reference values and with high reproducibility, corroborating the results of the artificial laminations. The results for Ca, V, Mn, Sr, Mo, Ba, and U (in log-ratios to Al) suggest that these elements can be analyzed semi-quantitatively, showing correlation coefficients (R2) ranging from 0.59 to 0.85 compared to conventional analytical techniques. However, we show that using non-matrix matched calibration standards such as NIST SRM 610 may induce deviations from the LA-ICP-MS values to the reference values. Consequently, we recommend a simple protocol for further data correction in which binned-mean LA-ICP-MS values are calibrated to a parallel series of discrete samples analyzed by these conventional techniques.

Performance Criteria for Chemical Analysis of Hydraulic Cements by X-Ray Fluorescence Analysis

Abstract Bulk oxide determinations from two pairs of ASTM C150/C150M-12 portland cements are used to calculate precision and accuracy values for X-ray fluorescence (XRF) by both the fused glass bead and the pressed powder sample preparation methods. The first pair of cements are ordinary portland cement (OPC) with 45 participants, while the second pair contains ca. 3 % limestone, with 24 labs participating. Each lab provided results from six replicates analyzed in duplicate, covering fifteen analytes, CaO, SiO2, Al2O3, Fe2O3, SO3, MgO, Na2O, K2O, TiO2, P2O5, Mn2O3, SrO, ZnO, Cr2O3, and Cl, with the laboratories roughly split between the two different sample preparations. Chemical data using traditional chemical analyses (the Reference Methods per ASTM C150) from the Cement and Concrete Reference Laboratory (CCRL) proficiency test program were included for comparison to the XRF results. Precision measures for within- and between-laboratory performance are presented as 1σ and 95 % limits (ASTM d2s). Accuracy criteria are based upon a two-sided 95 % prediction interval for the mean of two separate test results, defining the range of values one might expect for each analyte relative to a certified value of a reference material. Comparing to current C114 limits, based upon classical wet chemical analysis, within-lab precision for CaO is larger, wheras SiO2precision is similar to existing limits. Al2O3 and Fe2O3 precisions are substantially better, whereas for the remaining analytes, the precision is generally similar to ASTM C114 limits. A comparison of the ASTM accuracy criteria shows that CaO accuracy is poorer than the current limit, SO3 is larger, SiO2 and Na2O are similar, while the remaining accuracy limits for XRF are generally better than the ASTM C114 criteria. The powder method shows a positive relative bias for CaO for the cements containing less than 5 % limestone, suggesting a need for more limestone-containing reference materials for powder method calibration for these cements.

Terra-cotta figurines from the Roman theatre of Malaga (Spain): An archaeometric study

A series of 22 figured Phoeno-Punic and Roman terra-cottas, of high relevant archaeological and patrimonial values, have been studied. The samples were recovered in the Roman Theatre of Malaga (Spain). A thorough analysis including stereomicroscopy, elemental analysis by WDXRF, mineralogical analysis by XRPD coupled with the Rietveld method, and SEM, has been carried out. Highly diluted fused glass beads probed to be enough for the X-ray fluorescence analyses, minimizing the damage to the samples. For selected samples, in addition to the classical use of the Rietveld method, the G-factor external-standard approach has been employed to get full quantitative crystalline and amorphous phase analysis. The analytical results allow proposing a local/regional provenance for the main part of the samples, being a relevant data for several artifacts, not contextualized due to stratigraphic alterations in the site. In addition, some relevant samples, for instance a theatrical mask fragment (TRC005), are likely foreign. The overall amorphous is not negligible in phase quantification, accounting for up to 50 wt %. Technologically, all terra-cotta samples are similar, made of calcareous clays, fired at ~700-950 oC.

Glass bead with minimized amount (11 mg) of sample for X‐ray fluorescence determination of archaeological ceramics

By using very small amount – 11‐mg – of sample powder, major oxides (Na2O, MgO, Al2O3, SiO2, P2O5, K2O, CaO, TiO2, MnO, and total Fe2O3) in ancient pottery (and igneous rocks) were determined with X‐ray fluorescence spectrometry. This minimized amount of sample was used to prepare a fused glass bead with 300 times the weight of lithium tetraborate as an alkali flux. Calibration standards were obtained by compounding chemical reagents (Na2CO3, MgO, Al2O3, SiO2, Na4P2O7, K2CO3, CaCO3, TiO2, MnO2, and Fe2O3) and the flux. Fewer 11 mg of reagents as oxides were able to give reliable calibration curves with good linearity (correlation coefficient: r &gt; 0.995). Fewer 11 mg of sample was able to give reliable analytical results with good precision (relative standard deviation: &lt;3% for more than 10.0 mass% of analyte, &lt;10% for 1.0–10.0 mass% of analyte, and &lt;20% for 0.1–1.0 mass% of analyte). Lower limits of detection were roughly a sub‐percentage of analyte in an unprepared sample (e.g. 0.3 mass% for Na2O, 0.5 mass% for MgO, 1.0 mass% for Al2O3, and 0.01 mass% for MnO). Composition of major oxides in artificial and natural aluminosilicate materials (including rock, stone, sand, sediment, and clay; and their products) should be fundamental information to be considered in detail. The present X‐ray determination based on very small amount of sample might be made readily accessible for destructive analysis of precious samples for archaeology (and geochemistry). Copyright © 2011 John Wiley &amp; Sons, Ltd.

Micrometer silicon isotope diagnostics of soils by UV femtosecond laser ablation

This study presents the first Si isotope data of the principle Si pools in soils determined by a UV femtosecond laser ablation system coupled to a multicollector inductively coupled plasma mass spectrometer (MC-ICP-MS). This method reveals accurate and precise Si isotope data on bulk materials, and at high spatial resolution, on the mineral scale. The following Si pools have been investigated: a) the Si source to soils on all major silicate minerals on thin sections from bedrock fragments in the soil profiles; b) bulk soils (particle size < 2 mm) after fusion to glass beads with an iridium-strip heater or pressed into powder pellets; c) separated clay fractions as pressed powder pellets and e) separated phytoliths as pressed powder pellets. Multiple analyses of three rock standards, BHVO-2, AGV-1 and RGM-1 as fused glass beads and as pressed powder pellets, reveal δ 30Si values within the expected range of igneous rocks. The MPI-DING reference glass KL2-G exhibits the same Si isotope composition after remelting by an iridium-strip heater showing that this technique does not alter the isotope composition of the glass. We used this approach to investigated two immature Cambisols developed on sandstone and paragneiss in the Black Forest (Germany), respectively. Bulk soils show a largely uniform Si isotope signature for different horizons and locations, which is close to those of primary quartz and feldspar with δ 30Si values around − 0.4‰. Soil clay formation is associated with limited Si mobility, which preserves initial Si isotope signatures of parental minerals. An exception is the organic horizon of the paragneiss catchment where intense weathering leads to a high mobility of Si and significant negative isotope signatures as low as to − 1.00‰ in bulk soils. Biogenic opal in the form of phytoliths, exhibits negative Si isotope signatures of about − 0.4‰. These results demonstrate that UV femtosecond laser ablation MC-ICP-MS provides a tool to characterize the Si isotope signature of the principle Si pools left behind after weathering and Si transport have altered soils. These results can now serve as a fingerprint of the residual solids that can be used to explain the isotope composition of dissolved Si in soil solutions and river water, which is mostly enriched in the heavy isotopes.

Transient freezing–thawing phenomena in water-filled cohesive porous materials

Abstract An oscillator circuit-based capacitive method is used to study the ice/water phase transition in cohesive porous materials like water-filled fused glass beads. It straightforwardly gives the temperature domain of supercooling, freezing, or melting. It also provides an estimation of the ice content time-evolution during the transient stage of solidification and melting. This is done by calibration tests at 20 °C on a progressively dried sample and by an up-scaling dielectric model. The latter allows taking account of the temperature and frequency dependence of the water dielectric constant as well as the slight difference between the dielectric constants of water vapour (≃ 1) and ice (≃ 3.2). From the ice content-versus-time curve, the water-to-ice phase transition dynamics is found to follow the Avrami law with an exponent close to 0.5. This suggests that the ice/liquid interface is planar so that the liquid and ice pressures are equal, which is confirmed by the Thomson–Gibbs and Young–Laplace equations. The resulting pore pressure can then be predicted in the framework of linear poroelasticity. The analysis reveals a three-step time-history pressure: an increase at the onset of stable ice nuclei, then a relaxation induced by the unfrozen water Poiseuille-type flow and finally a further pressure increase until the end of crystallization. In all cases, the pressurization remains very low (0.1 MPa) at the 0 °C-isothermal transient stage of solidification.

A broadband model of sandy ocean sediments: Biot–Stoll with contact squirt flow and shear drag

Unlike the application of the Biot model for fused glass beads, which was conclusively demonstrated by Berryman [Appl. Phys. Lett. 37(4), 382–384 (1980)] using the experimental measurements by Plona [Appl. Phys. Lett. 36, 259–261 (1980)], the model for unconsolidated water-saturated sand has been more elusive. The difficulty is in the grain to grain contact physics. Unlike the fused glass beads, the connection between the unconsolidated sand grains is not easily modeled. Measurements over a broad range of frequencies show that the sound speed dispersion is significantly greater than that predicted by the Biot–Stoll model with constant coefficients, and the observed sound attenuation does not seem to follow a consistent power law. The sound speed dispersion may be explainable in terms of the Biot plus squirt flow (BISQ) model of Dvorkin and Nur [Geophysics 58(4), 524–533 (1993)]. By using a similar approach that includes grain contact squirt flow and viscous drag (BICSQS), the observed diverse behavior of the attenuation was successfully modeled.

An investigation of a dc dielectric barrier discharge using a disc ofglass beads

We discuss the properties of a dc discharge which occurs atthe interfaces between a disc of fused glass beads and parallel electrodes.The discharge consisted of a large number of microdischarges. The dischargecurrent had a large number of pulses superimposed on a dc backgroundcurrent. The background current increased with decreasing bead size anddecreased as the separation between the disc and the electrode increased.The current had an extremely slow and nonlinear transient behaviour when thedc high-voltage source was switched on. The frequency of the current pulsesdecreased with increasing bead size and disc-electrode separation. Theaverage amplitude of the pulses, in contrast, increased with increasing beadsize and disc-electrode separation. A significant dependence of theproperties of the current pulses on the polarity of the applied voltage wasobserved, although the dc current was almost independent of the polarity. Itwas found that the enhancement of the electric field near the contact pointsbetween the electrode and the beads, where the current concentrates,initiates the discharge. The density of ozone produced by the discharge wasalso measured. Damage to the discs by the discharge was also observed,thereby altering the properties of the discharge.

Low-frequency AC electrokinetics

By using a low-frequency AC method based on lock-in amplification, electrokinetic response in porous rock can be accurately measured with high sensitivity. With this method, we have measured streaming potentials as small as 300 nV and electroosmotic pressures as low as 10 Pa. We describe our technique, and discuss the results from a suite of 12 brine-saturated samples, consisting of six sandstones, two carbonates, and four fused glass bead specimens. We show that the permeability of the rock may be derived exactly from the electrokinetic transport coefficients and sample conductivity. This result is verified experimentally for all NaCl concentrations. The electrokinetic response as a function of concentration and temperature has been measured and is compared with predictions from the Debye–Hückel model and Helmholtz–Smoluchowski equation in the thin-layer limit.