Direct in situ measurements of labile inorganic and organically bound metal species in synthetic solutions and natural waters using diffusive gradients in thin films

The emerging technique of DGT (diffusive gradients in thin films) is shown to be capable of performing new speciation measurements in situ in natural waters. In DGT, metals are bound to a resin layer after passing through a well-defined diffusion layer. Cd was measured in solutions containing glycine, EDTA, and fulvic (FA) and humic acids (HA) by atomic absorption spectroscopy (AAS), anodic stripping voltammetry (ASV), and DGT. DGT measured similar labile fractions to ASV, with detailed differences being consistent with a thicker diffusion layer allowing more dissociation of labile complexes and a slower diffusion of FA and HA complexes through the gel. When single measurements are made in complex solutions with DGT, precise quantification is impossible due to uncertainties concerning the distribution of species with different diffusion coefficients. A new procedure was proposed based on the advantage of DGT of being able to control the pore size of the diffusive gel layer. Small (inorganic) species diffuse freely through all gels but larger FA and HA (organic) complexes diffuse less freely in more constrained gels. When measurements were made on known solutions of Cu and FA or HA, it was possible to quantify the inorganic and organic species separately. They agreed well with predictions made using the WHAM speciation code. Multiple DGT units were also deployed in situ in a stream with high dissolved organic carbon (14.6 mg/L). The systematic differences between the devices with different gel compositions enabled determination, for the first time, of the in situ concentrations of both labile inorganic and organic species in natural water. A single DGT device with a constrained gel can be used to quantify inorganic species directly, providing absolute accuracy is not required. This ability of DGT to measure well-defined fractions of metals in situ using a simple device gives it considerable potential as a regulatory tool. A direct speciation measurement may be preferable to modeling approaches which require diverse input data that are difficult to determine.     

Trace metal measurements in low ionic strength synthetic solutions by diffusive gradients in thin films

In view of conflicting reports regarding the performance of DGT in low ionic strength solutions (I < 1 mM), further investigations have been carried out. Minimal washing of the diffusive gel and deployment in 1.0 and 10 mM NaNO3 solutions containing Cu and Cd gave the theoretical response of 1 for [C](DGT)/[C](SOLN), where [C](DGT) is the concentration of metal measured by DGT and [C](SOLN) is the concentration of metal measured directly in the solution by an appropriate analytical method. Erroneously high values for [C](DGT)/[C](SOLN) were obtained when these same gels were deployed at I = 0.1 mM, presumably due to a net negative charge on the gel, attributable to the presence of initiation products of polymerization. However, washing the diffusive gels completely, where the storage solution pH equaled that of deionized water, gave values of approximately 0.5 for [C](DGT)/[C](SOLN) from deployments at I = 0.1 mM, consistent with the lower measured value of the diffusion coefficients at this ionic strength. These results can be explained by the presence of a net positive charge on the gel when it is exhaustively washed, which reduces the effective diffusion coefficient of metal ions by changing their concentration at the gel-solution interface (Donnan partitioning). Diffusive gel equilibration experiments showed the presence of low capacity sites capable of binding metals irrespective of ionic strength. This binding within the diffusive gel does not affect most DGT measurements, as short (4 h) deployments at concentrations of 10 ppb gave theoretical results. Incomplete washing of the resin-gel caused a 5-15% measurement error and a decrease in precision, even at ionic strengths of 10 mM. A high level of accuracy and precision (typically <5%) was maintained during all aspects of this work, even at ionic strengths of 0.1 mM, in contrast to previous results. This is attributable to three factors: (1) exhaustive washing and conditioning protocols, (2) improvements to the DGT sampling device, and (3) low and reproducible blanks due to ultraclean handling procedures. Provided effective diffusion coefficients measured at the same ionic strength are used, the established DGT theory is obeyed irrespective of ionic strength.     

Performance study of diffusive gradients in thin films for 55 elements

The technique of diffusive gradients in thin films (DGT) is a fairly new and useful tool for in situ measurements of labile metal ions in water. The applicability of DGTs was investigated by comparing independently determined or estimated diffusion coefficients with DGT effective diffusion coefficients (D(DGT)) for 55 elements. The DGTs were exposed at a controlled fluid velocity of 0.1 m s(-1) and a concentration of 1 ng mL(-1) at four pH levels between 4.7 and 6.0, and the D(DGT) values were determined from the uptake by the sampler. The measured D(DGT) values for the elements Co, Ni, Cu, Zn, Cd, Pb, Al, Mn, and Ga were close to previously published values with some deviations for Pb and Zn. The uptake of V, Cr, Fe, U, Mo, Ti, Ba, and Sr varied with pH, and there were some experimental problems that require further investigations. A novel set of D(DGT) values for the lanthanides (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Tb, Yb, Lu, Y) was established. The D(DGT) values for these were about 10-15% lower than for free ions in water and indicate that diffusion coefficients of metal ions in the agarose polyacrylamide diffusive hydrogel are 10-15% lower than in water. The high consistency of the data for the lanthanides establishes these elements as new performance test metals for the DGT sampler. The accumulation of the elements Li, Na, K, Rb, Mg, Ca, B, Tl, P, S, As, Bi, Se, Si, Sn, Sb, Te, Zr, Nb, Hf, Ta, W, Th, and Ag was low (D(DGT) lower than 10% of theoretical values). A more efficient elution procedure using concentrated nitric acid for the absorbent gel was established, with elution efficiencies between 95 and 100% for most metals. For deployment times of 24 h, detection limits from 0.001 to 1 ng mL(-1) were achieved with moderate precautions to prevent contamination.     

Perfluorosulfonated ionomer-modified diffusive gradients in thin films: tool for inorganic arsenic speciation analysis

A new concept in speciation analysis based on the diffusive gradients in thin films (DGT) technique is described. By use of two sets of DGT devices, one set with perfluorosulfonated ionomer (Nafion) diffusive membranes and the other with polyacrylamide, anionic and uncharged analytes can be fractionated on the basis of charge. The dual device method is applied to speciation analysis of dissolved inorganic arsenic species. Over the environmentally significant pH range, inorganic As(III) exists as neutral H(3)AsO(3), whereas As(V) is present as anionic H(2)AsO(4)(-) and HAsO(4)(2-). The measured diffusion coefficient of As(III) through the negatively charged Nafion membrane is significantly larger than that of the As(V) species, whereas diffusion rates are similar through polyacrylamide diffusive gels. Hence, after simultaneously deploying DGT devices with and without Nafion membranes, measurement of the amount of accumulated As in each type of device enables the concentration of both oxidation states to be determined.     

Application of a poly(4-styrenesulfonate) liquid binding layer for measurement of Cu2+ and Cd2+ with the diffusive gradients in thin-films technique

A diffusive gradients in thin films (DGT) technique for measuring Cd and Cu is described using, for the first time, a liquid phase binding layer and a dialysis membrane diffusive layer. The binding phase was a 0.020 M solution of poly(4-styrenesulfonate) (PSS) polyelectrolyte using a specially designed DGT deployment device. The binding properties of Cd2+, Cu2+, and a range of alkali and alkaline earth metal ions to the PSS solution were characterized. The PSS behaved like a cation exchanger with preferential binding to Cd2+ (6.0 micromol mL(-1), log K = 9.0) and Cu2+ (2.5 micromol mL(-1), log K = 8.1) under competitive binding conditions. PSS had a substantial binding capacity at pH > 3 and at competitive Na+ concentrations up to 1.0 M. The DGT devices were successfully validated for Cd2+ (accumulated mass vs time r2 = 0.969, recovery compared with predicted values = 98%) and Cu2+ (r2 = 0.980, recovery = 98%) in synthetic lake water (Windermere). Validation was also undertaken for Cu in a spiked local lake water (Parkwood Pond) (r2 = 0.981, recovery = 46%). The low recovery here was due to complexation of Cu by natural organic matter (14 mg C L(-1)). Field deployments of the DGT devices were successful at measuring Cu concentrations of 0.031-0.63 microg L(-1) in local fresh and salt waterways. These DGT-labile measurements were 0.05-39% of the 0.45-microm-filtered Cu values.     

Impact of biofouling on diffusive gradient in thin film measurements in water

The technique of diffusive gradient in thin film (DGT) is commonly used to assess metal contamination in natural waters. In this paper, we assess the effect of biofouling on DGT measured labile concentrations in water and investigate whether an additional nuclepore polycarbonate membrane on the surface of DGT devices can limit biofilm growth. Simultaneous field deployments of DGT equipped with and without the additional membrane in a canal receiving wastewater were compared. The effect of the biofilm was also assessed in controlled laboratory experiments, completed by the experimental determination of several metals diffusion coefficients in the hydrogel and membrane systems. The biofilms effect was problematic only from the 10th day of accumulation. Accumulation of some elements is highly biased by the presence of a thick biofilm (Zn, Ni, Cd). The polycarbonate membrane improved the quantification of Cd and Ni but adversely affects the quantification of Cr and Co. A kinetic model is proposed to explain the biofilm role on the DGT measurement. Depending on the metals of interest, it is possible to limit bias due to biofilms by using an additional polycarbonate membrane.     

Performance of a mixed binding layer for measuring anions and cations in a single assay using the diffusive gradients in thin films technique

The performance of a mixed binding layer (MBL) for use in diffusive gradients in thin films (DGT) was investigated. The MBL consisted of ferrihydrite and Chelex-100 cation-exchange resin combined together in a binding gel in an attempt to allow measurement of anions and cations in a single assay. Results from the MBL were compared to experiments performed using individual Chelex gels and ferrihydrite gels that have been shown to work successfully for DGT methodology. To facilitate combined analysis of P and cations by ICP-MS, HCl (1 M) was used for gel elution to minimize interferences from 14N16OH or 15N16O on 31P. All elements tested (Cd, Cu, Mn, Mo, P, and Zn) were bound successfully to the MBL. An elution efficiency of 0.92 was obtained for all elements, apart from Mo (0.79). This is higher than the elution efficiencies obtained previously for pure Chelex or ferrihydrite gels using HNO3 (1 M) as the eluent. Uptake of cations by DGT using the MBL was consistent across the pH range 5-9, which compares well with results using pure Chelex. Below pH 5, accumulated masses were lower for Mn, Cu, and Zn. Uptake of P and Mo was unaffected by pH in the range 3-8, and the amount absorbed compared well with results obtained previously for pure ferrihydrite gels. Performance of the MBL at different ionic strengths (0.001, 0.01 M) was comparable to performance using the pure Chelex gel. DGT measurements obtained using the MBL on agricultural soils correlated well (r2 = 0.95) with separate measurements obtained using either pure Chelex or ferrihydrite binding layers. This suggests that the MBL could be used for simultaneous measurement of cationic and anionic element availability in soils.     

Development of the diffusive gradients in thin films technique for the measurement of labile gold in natural waters

Gold is a precious metal that exists in most soils, sediments, and natural waters at extremely low concentrations (<1 μg/kg). The diffusive gradients in thin films (DGT) technique, used extensively for measuring trace metal concentrations in soils, sediments, and waters, has potential for geochemical exploration for gold, but has not been developed for this metal. This work investigates the possibility of measuring labile gold using DGT by introducing a new binding layer based on activated carbon. The performance of this new technique was assessed using gold(III) chloride in solution by: (1) determining the diffusion coefficient of gold(III) in hydrogels; (2) determining the uptake of gold(III) chloride by the new activated carbon binding layer; (3) determining an elution methodology for the binding layer and evaluating its efficiency; (4) assessing the capacity of the activated carbon binding layer to adsorb gold; (5) determining the effect of pH and ionic strength (as NaCl) on performance, and (6) assessing the selectivity of the new binding layer for gold. It was found that the diffusion coefficient of gold(III) increased as solution pH decreased. The diffusion coefficient also increased at high ionic strength (≥0.1 M NaCl). Accounting for these phenomena, the DGT technique behaved predictably under all tested conditions. The technique can potentially be used as a geochemical exploration tool for gold in soils and in aqueous environments, with method detection limits as low as 0.9 ng/L for a 7-day deployment.     

In situ measurements of labile Al and Mn in acid mine drainage using diffusive gradients in thin films

The technique of diffusive gradients in thin films (DGT) can be used for in situ measurements of labile metal species in water, but the application for this method on acid mine drainage (AMD) is complicated due to reduced sampler adsorption of metals at low pH. This study evaluates the use of DGT on labile Al and Mn in AMD (pH 3.1-4.2). DGT measurements were performed both in standard solutions in the laboratory and in situ in the field. Laboratory results show that DGT can be used in water with pH as low as 3.0 for Al and 4.0 for Mn without correcting for reduced adsorption. Below pH 4.0, the adsorption of Mn showed a linearly decrease with pH to approximately 55% at pH 3.0. Taking this correction into account revealed that 84-100% of the total dissolved Al and Mn measured in the field was DGT-labile. Measurements using DGT agreed well with predictions using the speciation program WHAM VI. This study shows that the use of DGT can be extended below the previously reported pH working range for Al, and for Mn using a simple linear correction with respect to pH, and demonstrates that the technique can be applied for monitoring time-integrated labile metal concentrations at AMD sites.     

In situ, High-Resolution Measurement of Dissolved Sulfide Using Diffusive Gradients in Thin Films with Computer-Imaging Densitometry

The technique of diffusive gradients in thin films (DGT) has been developed for the measurement of dissolved sulfide. Sulfide species from the sampled waters diffuse through a polyacrylamide hydrogel and then react with pale yellow AgI((s)), incorporated at the surface of a second gel, to form black Ag(2)S((s)). The accumulated sulfide can be measured with a conventional purge-and-trap method followed by colorimetry (methylene blue). This enables the dissolved-sulfide concentration to be calculated under suitable conditions. Alternatively, the color change in the accumulating gel can be used to measure sulfide. A conventional flat-bed scanner, allied to imaging software, provided a densitometric measurement that was quantitatively related to the amount of sulfide accumulated. DGT measurements on synthetic solutions accurately determined the sulfide concentration (95% recovery), thereby confirming the unobstructed diffusion of HS(-) through the gel. The accumulated mass was inversely proportional to the diffusion-layer thickness as theoretically predicted. With the selected geometry, the limit of detection of the densitometric procedure for a 24-h deployment was 0.13 μmol L(-)(1), and the maximum concentration measurable was 60 μmol L(-)(1). When used in anoxic lacustrine waters, DGT provided sensible concentrations. It was also used to measure depth profiles at submillimeter resolution in estuarine surface sediments.     

Analysis of polyacrylamide gels for trace metals using diffusive gradients in thin films and laser ablation inductively coupled plasma mass spectrometry

A simple method for the analysis of polyacrylamide diffusive gradients in thin film (DGT) gels by laser ablation inductively coupled plasma mass spectrometry (LA-ICPMS), employing a novel use of (115)In internal standardization, has been developed. This method allows the determination of Co, Ni, Cu, Zn, Cd, and Pb concentrations (at the DGT filter face) or fluxes in sediments at a spatial resolution of 100 microm. Single-layered gels, using an optimized laser defocus of 4000 microm at 400 mJ power, showed high precision (generally approximately 10%) and a linear response during solution deployment. Of the elements Sc, In, Ba, La, Ce, and Tb, Ba most closely tracked variations in laser energy and showed the highest analytical precision but could not be used as an internal standard due to its elevated presence in natural sediments. Therefore, internal standardization, necessary to normalize data collected on different days, was carried out using (115)In contained within a second layer of backing gel and dried along with the analyte layer as a dual-gel disk. This multilayered gel standard required a laser defocus setting of 1000 microm and a laser power of approximately 800 mJ. Analytical precision for a 64-spot ablation grid at 100-microm spacing was approximately 10%. Verification of this method was carried out on DGT sediment probes deployed in Priest Pot (English Lake District). Results obtained by conventional slicing techniques and aqueous elution agreed with laser ablation results when the different sampling areas were considered. The elution results varied by a factor of <2, whereas the laser ablation technique showed a variability of approximately 4, indicating localized elevated concentrations of Co. This higher resolution LA-ICPMS method could ultimately lead to an improved understanding of the geochemical processes responsible for metal uptake and release in sediments.     

Effect of organic P forms and P present in inorganic colloids on the determination of dissolved P in environmental samples by the diffusive gradient in thin films technique, ion chromatography, and colorimetry

The speciation of P in environmental samples is operationally defined, since it depends on the analytical method used. In this study, we compared four methods to measure P in solution: ion chromatography (IC), the malachite green colorimetric method (CM), the diffusive gradient in thin films technique (DGT) and, for total dissolved P, optical inductively coupled plasma (ICP). These methods were compared on three sets of solutions (filtered over <0.45 μm): solutions with model organic P compounds, suspensions of synthesized inorganic Fe and Al colloids loaded with P, and environmental samples. The environmentally relevant organic P compounds were only marginally detected by CM and IC. Substantial fractions of certain organic P compounds contributed to the DGT measurement. Colorimetric analysis of DGT eluates detected in general less P than ICP analysis, indicating that these organic P compounds sorbed on the zero sink layer. Phosphorus associated with inorganic colloids was completely recovered by CM, but not by IC and least by DGT. Measurements on a wide set of 271 environmental samples (soil pore waters, groundwaters, and surface waters) suggest that surface water P is largely present as orthophosphate and phosphate sorbed onto inorganic colloids, whereas organic P contributes more in groundwaters.     

服役条件下方钴矿基热电元件的界面应力分析

热电器件中, 界面可靠性是影响整体稳定和功率输出的关键因素。对于方钴矿(SKD)器件, 热电臂和电极通过扩散阻挡层(DBL)连接。在高温下, DBL与SKD、电极之间会发生反应并生成复杂的界面结构, 导致界面附近的热、电、力学性能发生变化。本研究根据实际界面结构建立了包含微观结构的有限元模型, 并将其用于分析方钴矿基元件的界面应力状态。采用单层模型对DBL材料参数进行了筛选, 发现热膨胀系数(CTE)和弹性模量(E)对第一主应力有显著影响。采用包含界面微结构的多层模型定量模拟了不同老化温度、时间下元件内部的应力分布, 结果表明在SKD/Zr和SKD/Nb中, CoSb₂反应层最为薄弱, 随着老化时间的延长, 反应层的厚度增加, 界面应力变大。同时, 元件的拉伸试验结果与计算结果吻合较好, 验证了模型的准确性与可行性。本研究为提升SKD/DBL元件的结构稳定性提供了指导, 同时也为精确模拟多层结构中的应力状态提供了研究思路。     

Heat transfer enhancement by Marangoni convection in the NH3–H2O absorption process

The objectives of this paper are to quantify the effect of Marangini convection on the absorption performance for the ammonia–water absorption process, and to visualize Marangoni convection that is induced by adding a heat transfer additive, n-octanol. A real-time single-wavelength holographic interferometer is used for the visualization using a He–Ne gas laser. The interface temperature is always the highest due to the absorption heat release near the interface. It was found that the thermal boundary layer (TBL) increased faster than the diffusion boundary layer (DBL), and the DBL thickness increased by adding the heat transfer additive. At 5 s after absorption started, the DBL thickness for 5 mass% NH₃ without and with the heat transfer additive was 3.0 and 4.5 mm, respectively. Marangoni convection was observed near the interface only in the cases with heat transfer additive. The Marangoni convection was very strong just after the absorption started and it weakened as time elapsed. It was concluded that the absorption performance could be improved by increasing the absorption driving potential (x_vb−x_vi) and by increasing the heat transfer additive concentration. The absorption heat transfer was enhanced as high as 3.0–4.6 times by adding the heat transfer additive that generated Marangoni convection.     

Speciation of dissolved inorganic arsenic by diffusive gradients in thin films: selective binding of AsIII by 3-mercaptopropyl-functionalized silica gel

A diffusive gradients in thin films (DGT) technique for selectively measuring As(III) utilizes commercially available 3-mercaptopropyl-functionalized silica gel. Deployment of the new technique alongside the Metsorb-DGT for total inorganic arsenic allows the calculation of As(III) directly and As(V) by difference. Uptake of As(III) by mercapto-silica was quantitative and elution with a mixture of 1 mol L(-1) HNO(3) and 0.01 mol L(-1) KIO(3) gave a recovery of 85.6 ± 1.7%. DGT validation experiments showed linear accumulation of As(III) over time (R(2) > 0.998). Accumulation was unaffected by varying ionic strength (0.0001-0.75 mol L(-1) NaNO(3)) and pH (3.5-8.5). Deployment of mercapto-silica DGT and Metsorb DGT in seawater spiked with As(III) and As(V) demonstrated the ability of the combined approach to accurately quantify both species in the presence of potential competing ions. Ferrihydrite DGT, which has been previously reported for the measurement of total inorganic arsenic, was evaluated in seawater and shown to underestimate both As(III) and As(V) at longer deployment times (72 h). Reproducibility of the new mercapto-silica DGT technique was good (relative standard deviations < 9%), and the average method detection limit was sufficiently low to allow quantification of ultratrace concentrations of As(III) (0.03 μg L(-1); 72 h deployment).     

Development and performance of the diffusive gradients in thin-films technique for the measurement of technetium-99 in seawater

A novel technique for obtaining time-integrated 99Tc concentrations in seawater has been developed, using diffusive gradients in thin films (DGT). The performance of TEVA resin as a binding agent for 99Tc was investigated via laboratory experiments. The accumulated 99Tc activity per unit area of resin-gel was proportional to both the bulk solution activity and the exposure time for deployments of up to 2 weeks. The response of DGT was found to be independent of solution chemistries over the pH range 3-8 and ionic strength range 0.01-1.3 M. Seawater has pH 8 and ionic strength of approximately 0.7 M; therefore, the potential of the technique for field deployment in seawater was demonstrated. Detection limits of 0.05 and 0.025 Bq L(-1), for 2- and 4-week DGT deployments, respectively, were calculated for 99Tc measurement by liquid scintillation spectrometry. Using quadrupole ICPMS to measure bound 99Tc could reduce these detection limits to 0.125 mBq L(-1) for a 4-week deployment. These detection limits are sufficiently low for monitoring contaminated environments, including the Irish Sea. This method is simpler and faster than other 99Tc analysis methods and represents the only means of obtaining time-integrated data.     

Applicability of diffusive gradients in thin films for measuring Mn in soils and freshwater sediments

Manganese (Mn) is an essential plant nutrient, receiving increased attention due to significant deficiency problems in modern crop production. In aquatic sediments, Mn plays an important role in controlling the mobility of other elements due to its high redox sensitivity. Diffusive gradients in thin films (DGT) is recognized as one of the most promising techniques to assess plant availability of nutrients in soils and mobility in sediments. However, the appropriate conditions where DGT can be used to measure Mn in soils and sediments have not been thoroughly investigated. We deployed DGTs in soil, sediment, and solution to investigate the effect of pH and competition from Ca and Fe ions. We found that by using DGT it is possible to accurately measure Mn in soils at pH levels and Ca and Fe concentrations resembling those of normal and fertile agricultural soils. However, in acid soils at pH below 5.5, Mn measurements might be biased due to potential competition effects caused by Ca. Soil deployments showed that changes in soil redox conditions were closely reflected by the DGT based Mn measurements. This might enable a novel approach of using DGT to predict Mn mobility and plant availability in soils. In reducing aquatic sediments, high concentrations of ferrous ions can displace Mn from the resin-gel of the DGT device. We found this to be a significant problem with longer deployment times.     

Boundary conditions for light propagation in diffusive media with nonscattering regions

The diffusion approximation proves to be valid for light propagation in highly scattering media, but it breaks down in the presence of nonscattering regions. We present a compact expression of the boundary conditions for diffusive media with nonscattering regions, taking into account small-index mismatch. Results from an integral method based on the extinction theorem boundary condition are contrasted with both Monte Carlo and finite-element-method simulations, and a study of its limit of validity is presented. These procedures are illustrated by considering the case of the cerebro-spinal fluid in the brain, for which we demonstrate that for practical situations in light diffusion, these boundary conditions yield accurate results.     

A hierarchical multi-scale method to simulate reactive–diffusive transport in porous media

A time-dependent, hierarchical multi-scale model to simulate reactive–diffusive transport in a 3-dimensional, porous structure, using a combination of molecular dynamics (MD), kinetic Monte-Carlo (KMC) and Monte-Carlo diffusion (MCD) methods has been developed. The parameters that are passed from the models at the lower spatial and temporal scales (MD, KMC) to those at the higher scales (KMC, MCD respectively) are identified. The advantages and disadvantages of the different methods are discussed with respect to reactive–diffusive transport simulations of atoms in porous 3D structures. The model is illustrated by simulating hydrogen transport, reactions, inventory and re-emission from a porous hydrocarbon layer.     

Analytical approximate solutions of the time-domain diffusion equation in layered slabs.

Time-domain analytical solutions of the diffusion equation for photon migration through highly scattering two- and three-layered slabs have been obtained. The effect of the refractive-index mismatch with the external medium is taken into account, and approximate boundary conditions at the interface between the diffusive layers have been considered. A Monte Carlo code for photon migration through a layered slab has also been developed. Comparisons with the results of Monte Carlo simulations showed that the analytical solutions correctly describe the mean path length followed by photons inside each diffusive layer and the shape of the temporal profile of received photons, while discrepancies are observed for the continuous-wave reflectance or transmittance.