Porosity Distribution Research Articles

Microfluidic Bubble-Templating Multinozzle 3D Printing for Preparations of Macroporous Hydrogels with Ordered Pores of Large Size Ranges.

A strategy for fabrication of macroporous hydrogels through 3D printing assisted by molding and multiple microfluidic bubble-templating nozzles is proposed here. This approach aims to address the challenges faced by methods for 3D printing macroporous hydrogels, such as difficulties in precisely controlling the spatial distribution of macropores, limited porosity, and low resolution of external boundaries due to the poor mechanical properties of hydrogel solutions as printing ink. In this method, fast-switching microfluidic bubble-templating nozzles of varying sizes allowed for precise control of target pore sizes over a wide range. Cavities of polydimethylsiloxane (PDMS) molds were used to define the overall external shape of the hydrogels and enhance the resolution of the boundaries. During the printing process, the laminar shear force generated by a coaxial microfluidic system generated microbubbles which were then injected into the PDMS mold cavities. The spatial distribution of the microbubble clusters within the cavities was controlled by a 3D motion module. Finally, macroporous hydrogels with a pore size range of 238-930 μm were fabricated while avoiding millimeter round corners on the boundaries. A control precision of pore sizes was about 60 μm. Most importantly, this strategy broke through the space occupancy limitations of pores in traditional 3D printing methods for macroporous hydrogels. The space occupancy of pores could reach a maximum of about 87% within a single layer.

Lateral torsional stability of porous thin-walled I-beams with nonuniform porosity distributions subjected to a uniformly distributed load

Lateral torsional stability of porous thin-walled I-beams with nonuniform porosity distributions subjected to a uniformly distributed load

Construction of granular aggregates with different porosity, shape, and size distributions

Construction of granular aggregates with different porosity, shape, and size distributions

Effect of Composite Scanning Strategy on Forming Quality, Microstructure, and Tensile Properties of Laser Powder Bed Fusion Titanium Alloy

Laser powder bed fusion (LPBF) technology offers significant advantages in manufacturing complex‐shaped titanium alloy components. Traditional scanning strategies, such as zigzag and island scanning, however, often fall short in fabricating parts with variable cross sections. To enhance the forming quality of components featuring combined thin‐walled and bulk structures, a composite scanning strategy is proposed that adapts to the local characteristics of parts. This novel approach is designed to employ both island and zigzag scanning within the same deposition layer, aiming to optimize the balance between porosity and stress distribution. Notably, with a feature transition distance of 4 mm and a scan line offset of 0.67 mm, the specimens achieve a tensile strength of 1311.0 MPa, a yield strength of 1103.0 MPa, and an elongation of 8.8%. This strategy leads to the optimization of defects and a transition in microstructure for combined structural features. These promising outcomes lay the foundation for the intelligent allocation of scanning strategies and the high‐quality formation of complex‐shaped, high‐strength titanium alloy parts.

Deformation prediction of overlying strata in multi-seam mining based on the influence function method-key stratum hybrid model

The extraction of coal seams from subterranean strata has the potential to induce displacement and deformation within the overlying strata. This leads to fractures in the strata and surface subsidence. The repetitive mining of coal seams, especially in scenarios involving multiple coal seams, can reactivate previously stabilized rock strata, causing them to subside and deform again. This exacerbates deformation and compromises the structural integrity of the strata. Consequently, there is an imperative need to thoroughly investigate the patterns of movement exhibited by overlying rock layers as a result of the recurrent extraction of coal seams. In recent years, significant progress in this field has been the effective utilization of the Influence Function Method-Key Stratum (IFM-KS) hybrid model. This model, distinguished by its amalgamation of mechanical and geometric methodologies, has proven to be efficacious in scrutinizing the movement dynamics of overlying strata in the context of single-layer mining operations. Based on that, this paper introduces a novel model for predicting rock layer movement and deformation in multi-seam mining. The calculation results derived from this model reveal the dynamics of rock layers' movement and deformation following secondary disturbances during repetitive mining. The accuracy and reliability of this newly proposed model in predicting rock movement during dual-layer coal mining are validated through both theoretical calculations and UDEC numerical simulations. The results demonstrate a high degree of consistency between the numerically simulated rock displacement and mathematical calculations and the rationality of the model. Based on the rock strata movement calculation results, the corresponding porosity distribution is also derived. Compared to single-layer coal mining, multi-seam mining could cause greater changes in porosity in the goaf area and extensive damage to rock strata. More fracture pathways between coal seams occur and a volume of air-leakage quantity is expected. This computational model provides invaluable insights for predicting rock strata deformation and porosity distribution in repetitive mining scenarios of a similar nature.

Analysis of FGM-type porous sandwich plates subjected to free vibration using Student Order Theory

This research adopts a high-order shear deformation theory to explore the free vibration characteristics of square porous sandwich plates constructed from functionally graded materials (FGMs). This theory effectively addresses shear deformation effects without necessitating a shear correction factor. The material properties of the FGMs change continuously throughout the plate's thickness, following a power-law distribution (P-FGM) based on the volume fractions of the component materials. The outer layers are metal, while a uniform ceramic layer constitutes the core. The governing equations are derived from the principle of virtual displacements. This study examines three different porosity distributions. By utilizing Hamilton's principle, the equations of motion were formulated. The numerical analysis demonstrates how the distribution of materials, geometry, and porosity affect the free vibration responses of FGM sandwich plates. The validity of the present theory is studied by comparing certain results obtained with other results published in the literature.

Gel Cleaning in Heritage: Comparison of the Water Release among Gels and Traditional Pads

Water release is a crucial aspect when considering cleaning effects on water-sensitive materials. In conservation practice, a water-based cleaning method which limits water release is very often needed. Unfortunately, this is not accompanied by an appropriate measure of the effectively released water. In this paper, water release has been measured by comparing traditional cleaning formulations, such as paper pulp and sepiolite, with several gar gel formulations, used by both Italian and European conservators. The assessment has been carried out by the gravimetric method, using three different stone material specimens as reference: Noto calcarenite, Manciano sandstone and Black Bergamo limestone, whose porosity values and distributions are known. Moreover, water distribution has been evaluated by portable NMR tests. Different commercial agar gel products (Bresciani, CTS, Sigma), having different concentrations (3, 4, and 5%), application modes (rigid at room T or fluid warm gels, with and without inserting Japanese tissue paper), and geometry (horizontal in gravity force direction or vertical), have been compared to obtain a full scenario among different water release mechanisms present in real conservation works. The paper faces the important issue of preparing reproducible chemical or water pads as well, useful for further research aimed at comparing cleaning effects in heritage conservation. The most interesting quantitative results can be summarized as follows. The water release measured from paper pulp and sepiolite was found to be 2 to 4 times higher than from any tested agar gel. Water release decreases by increasing agar concentration; an increase in the agar concentration by 1% induces a decrease in water release in the range 16.98–66.88 g depending on the stone; the increase from 4% to 5% is more obvious with respect to that from 3% to 4%. It is possible to assess the effect of the presence of Japanese paper, which is able to reduce the water release from 18 to 76%, depending on the stone and on the agar used. The gravimetric results were also used in the preliminary calibration tests of a contact probe named System Unit Salinity Index (SUSI), recently patented and useful in providing humidity and salinity indexes in a given porous material.

Quantifying aquifer heterogeneity using superparamagnetic DNA particles

Identifying and determining hydraulic parameters of physically heterogeneous aquifers is pivotal for flow field analysis, contaminant migration and risk assessment. In this research, we applied a novel uniquely sequenced DNA tagged superparamagnetic silica microparticles (SiDNAmag) to quantify hydraulic parameters and associated uncertainties of a heterogeneous sand tank. In the sand tank with lens shaped heterogeneity, we conducted three sets of multi – point injection experiments in unconsolidated (1) homogeneous (zone 0), (2) heterogeneous with a no-conductivity-zone (zone 1), and (3) heterogeneous with a high-conductive-zone (zone 2). From the breakthrough curves (BTC), we estimated the parameters distributions of hydraulic conductivity (k), effective porosity (ne), longitudinal dispersivity (αL), transverse vertical (αTV), and transverse horizontal dispersivities (αTH) applying Monte Carlo simulation approach for BTC fitting. The estimated parameters and associated uncertainties for each of the heterogeneous sections were further statistically compared (distribution non-specific Mann Whitney U test) these parameter distributions with parameter distributions estimated from the conservative salt tracer. While the time of arrival and time to peak concentration of SiDNAmag and salt in effluent were comparable, peak concentration of SiDNAmag was 1–3 log reduced as compared to the salt tracer due to first order kinetic attachment. Nonetheless, the parameters and associated uncertainty distributions (5 %–95 %) of K, ne, αL, αTV, and αTH, determined from SiDNAmag BTCs were statistically equivalent to the salt tracer in all three experiment systems. Through our experimental and modelling approach, our work demonstrated that in a coarse to very coarse grain sand medium, with lens shaped heterogeneity, the uniquely sequenced SiDNAmag were a promising tool to identify heterogeneity and determine hydraulic parameters and associated uncertainty distributions.

Analysis of the reservoir properties of rocks of the Lower Kura depression (Azerbaijan)

The article analyzes the features characterizing the Lower Kura depression (LKD). A number of hydrocarbon deposits have been discovered in this oil and gas region. The Kalmas underground gas storage facility, which has the largest volume in Azerbaijan, is also located in this area. In the LKD, all PS formations can be traced throughout the entire well section. The reservoir properties of the rocks were considered to study the features of the area. In order to study the reservoir properties in the well section, the field applies data obtained by using a complex of well logging (WL) methods (laterolog, spontaneous potential logging, induction logging, gamma logging, neutron logging, acoustic logging) and analyses of rock samples. The graphs were constructed to evaluate the relationship between petrophysical parameters (porosity, permeability, etc.) determined in the rocks involved in the well sections of the object of research. In the article, maps of porosity distribution, permeability and permeability of sandy siltstone rocks and fractions greater than 0.1 of the Productive Series of the South-Eastern Gobustan and Baku Archipelago, which are part of the Low Kura Depression, are plotted and analysed. Plots showing the porosity distribution of the Low Kura Depression by depth were also constructed and some results were obtained. Based on reservoir properties, the region is divided into 6 groups, and in each group intermediate values of the porosity and permeability coefficient are determined. In general, the porosity coefficient ranges within 5-25%, permeability – 1-250 mD. The reservoir properties of rocks vary over area from worse to better. The capacitance-filtration properties of rocks were studied. The rock samples analysed identified rocks with good and low reservoir characteristics. To study the variation in the determining factors, maps of the distribution of reservoir properties indicators (fractions greater than 0.1 mm; porosity and permeability in sand-siltstone-containing rocks) were compiled by area and their causes were determined by comparison with other oil and gas fields. It was found that with distance from the mouth of the Kura River (towards the right and left banks), the reservoir properties of the rocks deteriorate. This is due to an increase in the amount of clay in rocks. It is also clarified that the change in porosity coefficient does not depend on depth. The content of sand-siltstone composition in the rocks along the section is 20-50%.

A Higher-Order Theory for Nonlinear Dynamic of an FG Porous Piezoelectric Microtube Exposed to a Periodic Load

This paper investigates the nonlinear dynamic deflection, natural frequency, and wave propagation in functionally graded (FG) porous piezoelectric microscale tubes under periodic load, hygrothermal conditions, and an external electric field. The piezoelectric material used to make the smart microtubes has pores that may be smoothly changed or uniformly distributed over the tube wall. Here, three types of porosity distribution are taken into consideration. The nonlinear motion equations are constructed using a novel shear deformation beam theory and the modified couple stress theory (MCST). The nonlinear motion equations are solved using the fourth-order Runge–Kutta technique and the Galerkin approach. The effects of various geometric parameters, porosity distribution type, porosity factor, periodic load amplitude and frequency, material length scale parameter, moisture, and temperature on the nonlinear dynamic deflection, natural frequency, and wave frequency of FG porous piezoelectric microtubes are explored through a number of parametric investigations.

Free Vibration of Axially Functionally Graded Porous Beams with Varying Cross-Section

This study aims to determine the natural frequencies of axially functionally graded porous material (FGPM) beams with non-uniform cross-sections under a variety of boundary conditions. Two types of pore distribution were used: even and uneven. Initially, an analytical method was used to determine the natural frequencies of FGPM beams with different cross-sections. The Fredholm integral approach was used to obtain the characteristic equations. Furthermore, the collected results are confirmed and compared to the existing literature. The artificial neural network (ANN) technique is then used to predict the fluctuations in the natural frequency of a clamped–simply supported axially FGPM beam with a non-uniform cross-section. The prediction of natural frequencies by ANN is based on a large dataset of over 1[Formula: see text]100 data points acquired from the analytical solution obtained in this study and data available in the literature. This research conducts a parametric analysis to evaluate the influence of numerous aspects, including as beam characteristics, material properties, geometric details, gradient parameters, and porosity distribution, on functionally graded (FG) beam vibration behavior. Finally, both ANN and computational analysis demonstrate that porosity distributions and non-uniform cross-sectional area have a substantial effect on the natural frequencies of axially FG beams.

Nanomaterial Texture-Based Machine Learning of Ciprofloxacin Adsorption on Nanoporous Carbon.

Drug substances in water bodies and groundwater have become a significant threat to the surrounding environment. This study focuses on the ability of the nanoporous carbon materials to remove ciprofloxacin from aqueous solutions under specific experimental conditions and on the development of the mathematical model that would allow describing the molecular interactions of the adsorption process and calculating the adsorption capacity of the material. Thus, based on the adsorption measurements of the 87 carbon materials, it was found that, depending on the porosity and pore size distribution, adsorption capacity values varied between 55 and 495 mg g-1. For a more detailed analysis of the effects of different carbon textures and pores characteristics, a Quantitative nano-Structure-Property Relationship (QnSPR) was developed to describe and predict the ability of a nanoporous carbon material to remove ciprofloxacin from aqueous solutions. The adsorption capacity of potential nanoporous carbon-based adsorbents for the removal of ciprofloxacin was shown to be sufficiently accurately described by a three-parameter multi-linear QnSPR equation (R2 = 0.70). This description was achieved only with parameters describing the texture of the carbon material such as specific surface area (Sdft) and pore size fractions of 1.1-1.2 nm (VN21.1-1.2) and 3.3-3.4 nm (VN23.3-3.4) for pores.

Characterization of early fatigue damage in CFRP unidirectional laminates based on Micro‐CT

AbstractFatigue damages of Carbon Fiber Reinforced Polymer (CFRP) laminates are not obvious. Obtaining the evolution of the internal fatigue damage in CFRP laminates before visible cracks occur is of vital importance for ensuring structural reliability. Micro X‐ray computed tomography (Micro‐CT) can provide micron scale nondestructive imaging of internal structure of materials. In this paper, a preliminary study was conducted on the early fatigue damage evolution of 0° UD laminates. Axial tension‐tension fatigue tests of [0]16 CFRP laminates were carried out, and the evolutionary behavior of damages was confirmed by fatigue dissipated energy. Then, the internal damages of fatigued specimens were observed and reconstructed using Micro‐CT, and statistical parameters such as damage sphericity, length and angle were calculated. The spatial distribution characteristics of cracks were further acquired by surface porosity and frequency distribution of damage positions in ROI. Early damages from tensile‐tensile fatigue in 0° UD laminates extend along the fiber direction with original defects in the interlayer resin enrichment areas as the nucleus, and tend to concentrate near the width edge of the specimens. Finally, a “sandwich” structural model was developed to analyze the internal stresses of laminates under the influence of interlayer resin enrichment. The FEA results revealed that the internal stresses σy and σz concentrate near the width edges of the specimen, resulting in cracks tend to evolve more in the regions close to edges.Highlights Imaging matrix cracks of specimens by Micro‐CT at 12.91 μm voxel size. Demonstrating evolution patterns of cracks with its morphology statistics. Revealing crack distributions in ROIs with variations of surface porosity. Analyzing internal stress distribution by “sandwich” model created from reality.

A review on the mechanical and biocorrosion behaviour of iron and zinc-based biodegradable materials fabricated using powder metallurgy routes

Abstract Over the past two decades, research on alloys and composites based on Mg, Fe, and Zn has focused on biodegradable orthopaedic implants. Mg-based materials face issues like excessive corrosion rates and hydrogen gas evolution, while Fe and Zn-based materials show lower corrosion rates. However, these rates are slower than the optimal rate, which can be modified using powder metallurgy (PM) manufacturing. The PM process offers precise control over porosity distribution which in turn affects the mechanical and corrosion properties of the fabricated specimen. The highest rate of corrosion i.e. 0.944 mmpy was observed with the alloying of 2 wt% Pd in Fe and by using conventional sintering technique. Similarly, Zn-based samples fabricated by conventional sintering was found to exhibit higher corrosion rate as compared to microwave and spark plasma sintered specimen. PM-fabricated Fe and Zn-based bone scaffolds have been investigated for in-vitro corrosion and osseointegration. A higher porosity in the Fe and Zn scaffolds (>60 %) resulted in high corrosion rate which adversely impacted the cell proliferation. This timely review critically assessed PM-fabricated Fe and Zn-based materials that have the potential to transform regenerative medicine and patient care by redefining the field of biodegradable implants.

Identification of the Gas Explosion Hazardous Area in a Gob by Numerical Simulations-A Case study of the 6310 Working Face of Tangkou Coal Mine.

In the process of coal mining, gas explosion accidents occur frequently, which seriously affect the safety of mine production. Identifying the gas explosion hazardous zone in a gob is of specific guiding significance for the prevention and control of gas explosion in a coal mine. In this work, we analyze the porosity and gas concentration distribution in the gob after the working face was mined by combining the particle flow and finite-element simulation. A trianglular gas explosion hazardous zone is obtained by multielement coupling, and the CO2 injection simulation is then performed to eliminate the hazardous zone. The results show that the upper part of the gob and the edge of the return airway in the lower part are highly porous, and the porosities of other areas are relatively low with a value of 0.4. The gas concentrations in the upper left part and lower right part of the gob are relatively low, and the oxygen concentration distribution is the opposite. The gas explosion hazardous zone is mainly distributed near the oxidation zone. Yet, the simulation results reveal that after an injection of 97.5 m3/h of CO2, there is no area in the goaf that meets the gas explosion concentration conditions, that is, the oxygen concentration is greater than 12% and the methane concentration is greater than 5% but less than 16%, which can effectively eliminate the gas explosion hazardous zone.

Effect of Porosity and Gradient Parameters on Compressive Mechanical Properties of Sheet Gyroid Gradient Porous Structures and Construction of Mechanical Properties Prediction Model

A series of uniform porous structures and radial stepwise and radial linear gradient structures are designed based on sheet‐gyroid to explore the influence mechanism of porosity and gradient parameters on forming quality and compressive mechanical properties. On this basis, mathematical models for predicting mechanical properties of porous structures based on uniform, discrete gradient, and linear gradient porosity distribution are established. The volume fraction deviation of uniform porous structures increases gradually with the increase of porosity. The relative density of porous structures ranges from 96.78 to 98.79%. The influence of porosity on compressive mechanical properties is investigated, and a prediction model of the equivalent elastic modulus and yield strength of porous structures is constructed based on the generalized G‐A model. The elastic modulus of the radial stepwise gradient porous structures is predicted by combining the rule of mixing. The deviation between the predicted results and the experimental results ranges from 0.21 to 2.61%. At the same time, a prediction model of the compressive mechanical properties of the radial linear gradient porous structure is constructed, and it is found that the predicted value is somewhat different from the experimental value, which is related to the synergistic strengthening effect of the gradient porous structure.

Development of Polyvinyl Alcohol Hydrogels for Controlled Glucose Release in Biomedical Applications.

Polyvinyl alcohol (PVA) hydrogels have a wide range of applications in the pharmaceutical and biomedicine fields due to their exceptional biophysical properties. The study focuses on preparing and characterizing capsule-shaped PVA hydrogels to enhance their biocompatibility and porosity for controlled glucose release and cell proliferation. The hydrogels were prepared using different concentrations (Cs) and molecular weights (MWs) of PVA, with two different lengths, A (10 mm) and B (20 mm), to control glucose release over 60 min. The preparation process involved PVA gel preparation and PVA hydrogel formation. A total of 500 µL of glucose was injected into all dehydrated hydrogels in groups A and B. Glucose release was studied by immersing the hydrogels in saline at 37 °C with stirring at 500 rpm. The SUP-B15 cell line was grown in six A1 hydrogels for biocompatibility testing. The results indicate that all hydrogels remained stable at 37 °C without degrading. Those with a higher C and MW exhibited a denser and less porous structure, lower glucose storage capacity, and higher elongation at break. Significant differences in glucose release, diffusion speed, and flux were observed, which were more evident in A1 > A4, B1 > B4, and B1 > A1 over 60 min. A1 and B1 had higher values because their higher porosity distribution allowed glucose to diffuse more easily. B1, being larger, has more glucose due to its increased length. The cell growth response and viability at 48 h in contact with the hydrogels was similar to that of the control (4.5 × 105 cells/mL, 98.5% vs. 4.8 × 105 cells/mL, 99.7% viability), thus demonstrating biocompatibility. The hydrogels effectively released glucose over 60 min, with variations based on porosity, C, MW, and length, and demonstrated good biocompatibility with the cell line.

Effects of Build Direction and Heat Treatment on the Defect Characterization and Fatigue Properties of Laser Powder Bed Fusion Ti6Al4V

Laser powder bed fusion (LPBF) is one of the high-precision additive manufacturing techniques for producing complex 3D components. It is well known that defects appear in additive-manufactured parts, and they deeply affect the fatigue properties; even heat treatment is performed after printing. In order to meet the safe-life design requirements of additive-manufactured aircraft structures, the effects of build direction and heat treatment on defects and fatigue properties need to be quantified. Hence, Ti6Al4V alloy samples with different build directions were designed and printed by LPBF. X-ray computed tomography was used to quantitatively analyze the defect size, the sphericity, and the defect orientation. And their effects on fatigue properties were studied. An extended effective defect size and a defect-based fatigue anisotropy evaluation process are proposed to qualify the effects of the defect size, sphericity, and defect orientation. It is shown that the build direction can affect the porosity distribution and maximum defect size, while the annealing treatment can cause the coalescence of small defects and higher porosity. The defect orientation exhibited a fluctuating trend of 0°–90°–0°–90°–0° as the volume increased. The elongated lack of fusion defects related to the build direction was the main crack source and could lead to fatigue anisotropy of LPBF Ti6Al4V.

Effect of Microgravity on the Metal Droplet Transfer and Bead Characteristics in the Directed Energy Deposition-Arc Process

Abstract Directed energy deposition-arc (DED-arc) is a viable method of metal 3D printing for manufacturing in-space under microgravity conditions. This study investigates the effect of reduced gravity on the droplet transfer in a gas metal arc welding (GMAW)-based DED-arc process. Single bead deposited using GMAW welding process under microgravity and standard terrestrial gravity (1 g) are compared. Microgravity was simulated in a drop tower where the experimental capsule was subjected to 2.5 s of free-fall. The experimental setup for GMAW welding process, including high-speed cameras and sensors, was present within the experimental capsule. Droplet frequency and diameter were measured and compared between microgravity and 1 g using the images obtained. Further, the impact of reduced gravity on weld bead geometry and the distribution of gas porosity was investigated. Microhardness analysis was also conducted on both 1 g and reduced gravity samples to assess variations in material hardness. A statistically significant difference in droplet diameter and frequency was found. This difference is attributed to the reduction in gravitational force. Upon analyzing the weld bead geometry, noticeable variations are detected in the contact angles and the reinforcement of beads formed under different gravity conditions. These differences are attributed to alterations in convection within the molten weld pool. The blowhole analysis revealed a noticeable trend, wherein reduced gravity facilitated the coalescence of gas porosity, resulting in larger diameters due to alterations in weld pool convection. There were no statistically significant changes observed in both microhardness and surface finish.

Poroelastic size dependent dynamics of viscoelastic microbeams connected via a viscoelastic layer

This article aims to investigate the dynamics of clamped-clamped porous viscoelastic double microbeams interconnected via a viscoelastic layer within the framework of the modified couple stress theory (MCST). Different types of porosity distributions are considered to examine the material imperfection effects on the dynamics of the double microbeam. A closed-cell porosity model is employed to model the variation of the material properties in the thickness plane. The coupled longitudinal and lateral equations of motion are derived using Hamilton’s principle. The governing equations of motion are solved using the modal decomposition method. The equations of motion are verified against the literature for simplified cases (i.e., a non-porous single viscoelastic microbeam and a porous single microbeam resting on a viscoelastic foundation). The numerical results are validated for simpler versions of the system using the finite element method and against the existing literature for a simplified case of a uniformly distributed porous double macrobeam connected via an elastic layer. Influences of viscoelasticity in the double microbeam, viscoelastic layer, porosity, and small-scale influences on the dynamics of the viscoelastic double microbeams are studied. It was found that increasing the material length-scale factor causes the complex fundamental lateral vibrational frequencies to increase for all the porosity patterns.