Non-wetting Surfaces Research Articles

Aerophilic Surfaces for Sustained Corrosion Protection of Metals Underwater

AbstractCorrosion and biofouling are wetting‐related phenomena that limit the effective use of metals in aqueous media. Nonwettable surfaces can mitigate the adverse effects of wetting by minimizing contact with water. However, current achievements in this field fall short of meeting industrial requirements due to the short lifetime of plastrons. This study proposes a method to measure the protective sustainability of plastron. Superhydrophobic (SHS) and aerophilic (APhS) surfaces are constructed on lightweight aluminum and are initially analyzed by conventional goniometry, which show comparable values. However, the plastron that develops underwater is substantially different. While SHS exhibit unevenly broken plastron, APhS show uniform, continuous plastron. As an example of the sustained protective performance of plastron, the corrosion resistance of SHS and APhS is presented. Potentiodynamic polarization, impedance spectroscopy, and long‐term immersion in seawater show a drastic enhancement in corrosion resistance, exclusively for APhS. In fact, almost no electrochemical signals are measurable, and no pitting corrosion is observed after 415 days of immersion in seawater. Conversely, SHS show no noticeable improvement and corrode faster than bare Al due to plastron loss. Since goniometric measurements do not provide information on plastron, it is essential to analyze the plastron for any non‐wettable surface utilized underwater.

Fabrication of non-wetting surface on zinc substrate with coalescence-induced droplet jumping behavior for atmospheric corrosion protection

In this study, a superhydrophobic surface was constructed on the zinc substrate through a two-step process. The research first examined the correlations between the microstructure and coalescence-induced droplet jumping behavior (CIDJB). Then a droplet jumping phase map was created by considering energy principles. Finally, the performance and mechanism CIDJB in enhancing anticondensation effectiveness and salt spray experiments against atmospheric corrosion were revealed. This study provides insights into designing superhydrophobic surfaces with CIDJB for atmospheric corrosion protection. Further verification of the impact mechanism of environmental factors and stability and durability of CIDJB in practical applications will be investigated.

Fabrication and Applications of Nature-Inspired Surfaces with Selective Wettability.

Inspired by the Stenocora beetle, selective wettability surfaces incorporate alternating wettable and nonwettable surface features that have received substantial attention over the past two decades. These surfaces are sought after for their very promising potential to drive progress in numerous application fields, including ecological protection, biomedical sciences, and industrial technologies. However, despite ongoing efforts to produce such surfaces in commercial quantities, understanding their basic fabrication concepts for practical applications can be challenging, especially for novices, given the vast technical literature in this area. This review, therefore, aims to elucidate the principles of wettability, along with the evolution of selective wettability surfaces and their uses. Beginning with a summary of the essential history and theory of wetting, we explore naturally occurring surfaces that have influenced wetting studies. We then detail state-of-the-art methods for fabricating these unique biwetting surfaces and show how contemporary science employs such designs in solving real-world problems. Finally, we offer an outlook for future research prospects on scalable, printing-based fabrication methods.

Predicting plastron thermodynamic stability for underwater superhydrophobicity

Non-wettable surfaces, especially those capable of passively trapping air in rough protrusions, can provide surface resilience to the detrimental effects of wetting-related phenomena. However, the development of such superhydrophobic surfaces with a long-lasting entrapped air layer, called plastron, is hampered by the lack of evaluation criteria and methods that can unambiguously distinguish between stable and metastable Cassie-Baxter wetting regimes. The information to evaluate the stability of the wetting regime is missing from the commonly used contact angle goniometry. Therefore, it is necessary to determine which surface features can be used as a signature to identify thermodynamically stable plastron. Here, we describe a methodology for evaluating the thermodynamic underwater stability of the Cassie-Baxter wetting regime of superhydrophobic surfaces by measuring the surface roughness, solid-liquid area fraction, and Young’s contact angle. The method allowed the prediction of passive plastron stability for over one year of continuous submersion, the impeding of mussel and barnacle adhesion, and inhibition of metal corrosion in seawater. Such submersion-stable superhydrophobicity, in which water is repelled by a stable passive air layer trapped between the solid substrate and the surrounding liquid for extended periods at ambient conditions, opens new avenues for science and technologies that require continuous contact of solids with aqueous media.

Assessment of Hydrophilicity/Hydrophobicity in Mesoporous Silica by Combining Adsorption, Liquid Intrusion, and Solid-State NMR Spectroscopy.

We have developed a comprehensive strategy for quantitatively assessing the hydrophilicity/hydrophobicity of nanoporous materials by combining advanced adsorption studies, novel liquid intrusion techniques, and solid-state NMR spectroscopy. For this, we have chosen a well-defined system of model materials, i.e., the highly ordered mesoporous silica molecular sieve SBA-15 in its pristine state and functionalized with different amounts of trimethylsilyl (TMS) groups, allowing one to accurately tailor the surface chemistry while maintaining the well-defined pore structure. For an absolute quantification of the trimethylsilyl group density, quantitative 1H solid-state NMR spectroscopy under magic angle spinning was employed. A full textural characterization of the materials was obtained by high-resolution argon 87 K adsorption, coupled with the application of dedicated methods based on nonlocal-density functional theory (NLDFT). Based on the known texture of the model materials, we developed a novel methodology allowing one to determine the effective contact angle of water adsorbed on the pore surfaces from complete wetting to nonwetting, constituting a powerful parameter for the characterization of the surface chemistry inside porous materials. The surface chemistry was found to vary from hydrophilic to hydrophobic as the TMS functionalization content was increased. For wetting and partially wetting surfaces, pore condensation of water is observed at pressures P smaller than the bulk saturation pressure p0 (i.e., at p/p0 < 1) and the effective contact angle of water on the pore walls could be derived from the water sorption isotherms. However, for nonwetting surfaces, pore condensation occurs at pressures above the saturation pressure (i.e., at p/p0 > 1). In this case, we investigated the pore filling of water (i.e., the vapor-liquid phase transition) by the application of a novel, liquid water intrusion/extrusion methodology, allowing one to derive the effective contact angle of water on the pore walls even in the case of nonwetting. Complementary molecular simulations provide density profiles of water on pristine and TMS-grafted silica surfaces (mimicking the tailored, functionalized experimental silica surfaces), which allow for a molecular view on the water adsorbate structure. Summarizing, we present a comprehensive and reliable methodology for quantitatively assessing the hydrophilicity/hydrophobicity of siliceous nanoporous materials, which has the potential to optimize applications in heterogeneous catalysis and separation (e.g., chromatography).

Multilevel Micronanoscale Texture Effects on Fly Wing Membrane-Water Droplet Interaction.

The wettable surface or nonwettable surface that is derived from a multilevel micronanoscale structure is abundant in nature and biomimetic commodities. Those hoverflies with the seta-coated wing membrane detached from impacting free-falling raindrops were observed in static states. A hoverfly wing membrane with well-ordered setae was identified as a robust nonwettable surface, and the static water contact angle θ on the wing membrane at the microscopic scale is 136.84 ± 0.98°. Hoverfly wing membrane-water droplet interaction with the actual truth and the theoretical models was discussed and indicated that the theoretical calculation might not state the actual situation, arising from the membrane or seta-drop-bubble interaction and those multilevel micronanoscale structure characteristics on the wing membrane. Detailed investigation on nonwettable surface-wettable surface transformation with surface CaCO3 accumulation in a carbonization reaction and the characteristic transformation toward the hoverfly wing membrane with the multilevel micronanoscale structure was carried out. Then, the CaCO3 accumulation on PDMS texture films was carried out and the static water contact angle θ was tested. Those observations offer ideas to fabricate artificial films with a multilevel micronanoscale structure that could obtain some characteristics, i.e., nonwettable surface-wettable surface transformation.

Engineered Sustainable Omniphobic Coatings to Control Liquid Spreading on Food-Contact Materials.

The adhesion of sticky liquid foods to a contacting surface can cause many technical challenges. The food manufacturing sector is confronted with many critical issues that can be overcome with long-lasting and highly nonwettable coatings. Nanoengineered biomimetic surfaces with distinct wettability and tunable interfaces have elicited increasing interest for their potential use in addressing a broad variety of scientific and technological applications, such as antifogging, anti-icing, antifouling, antiadhesion, and anticorrosion. Although a large number of nature-inspired surfaces have emerged, food-safe nonwetted surfaces are still in their infancy, and numerous structural design aspects remain unexplored. This Review summarizes the latest scientific research regarding the key principles, fabrication methods, and applications of three important categories of nonwettable surfaces: superhydrophobic, liquid-infused slippery, and re-entrant structured surfaces. The Review is particularly focused on new insights into the antiwetting mechanisms of these nanopatterned structures and discovering efficient platform methodologies to guide their rational design when in contact with food materials. A detailed description of the current opportunities, challenges, and future scale-up possibilities of these nanoengineered surfaces in the food industry is also provided.

Behavior of Compound Materials on Superhydrophobic Cylinders: Effects of Droplet’s Size and Interface Angle

Compound droplets can consist of two or more immiscible substances sharing an interface. Among such droplets, the low-viscosity component of Janus droplets can exhibit peculiar bouncing behavior on nonwettable surfaces. There have been recent advances in droplet control technologies, however the impact dynamics of droplets on complex surfaces, and strategies to control their behavior, have not been extensively studied. This study employs the volume of fluid method to analyze the effects of Janus droplet size and the initial interface angle on the dynamics of the two fluidic components in droplets on superhydrophobic cylinders. Janus droplets are composed of low-viscosity (W-) and high-viscosity liquid (G-component). The dynamic characteristics of Janus droplets are investigated as a function of Weber number (&lt;i&gt;We&lt;/i&gt;), initial interface angle, the ratio of the droplet’s diameter to the cylinder’s diameter, and viscosity ratio (α). Numerical models provide a regime map of the separation ratio of Janus droplets based on We and α, and the influence of droplet size on asymmetric bouncing is discussed. This study also examines the threshold We at which separation begins after impact, varying with droplet size and α. In addition, the shape evolutions of the droplets are discussed for various initial interface angles to understand the bouncing behavior and separation efficiency. This study is expected to provide valuable strategies for controlling droplet behavior and separation in applications such as liquid purification, rheology, and solidification.

Contact-point analysis of attached-wall cavitation evolution on chemically patterned surfaces using the lattice Boltzmann method

In this study, we aim to investigate the contact-point characteristics of cavitation bubble evolution on different chemically patterned surfaces using the thermal pseudopotential lattice Boltzmann method (LBM). The local temperature variation is considered in the contact-point dynamics simulation. Force analysis is performed on the contact points, and two main controlling factors for contact-point dynamics are identified, namely, the local pressure difference and the unbalanced Young’s force. A dimensionless temperature is used to describe the heat exchange efficiency between the wall and fluids for different wall wettability and temperatures. The attached-wall bubble on the wet wall has an unbalanced Young’s force, which is directed toward the inside of the bubble throughout the entire growth and collapse stages at the contact point, resulting in a smaller contact radius and stronger collapse intensity and making the wet wall more suitable for wall cooling. Significant pressure differences are observed at the contact points owing to local momentum concentration. In contrast, the non-wetting wall surface is more suitable for wall heating because of its smaller collapse intensity and larger contact radius on the wall.

Non-axisymmetric modes in ultrathin annular liquid films coating a cylindrical fiber

This paper presents a detailed analysis of the three-dimensional stability properties of an annular liquid film coating a cylindrical fiber in the presence of van der Waals (vdW) interactions, whose influence depends on the wettability of the solid by the liquid. Under wetting conditions, vdW interactions can stabilize a uniform annular film when its thickness is smaller than a critical value that depends only on the fiber radius, the Hamaker constant, and the surface tension coefficient [Quéré et al., Science 249(4974), 1256–1260 (1990)]. In contrast, under non-wetting conditions, both surface tension and vdW forces contribute to destabilize the interface, and non-axisymmetric modes may become dominant depending on the thickness of the film and the relative strength of the surface tension and vdW forces. We perform temporal stability analyses of both the Stokes and lubrication equations of motion, allowing us to reveal the dominant azimuthal mode, as well as the optimal axial wavenumber and the corresponding temporal growth rate, as a function of the relevant governing parameters.

Preparation and Applications of Superhydrophobic Coatings on Aluminum Alloy Surface for Anti-Corrosion and Anti-Fouling: A Mini Review

Aluminum alloy is widely used in many fields for its excellent performance. However, in practical application, aluminum alloy is easy to become corroded and be invalidated, owing to the influence of environmental factors. Some pollutants or corrosive media on its surface adhesion will further deepen the failure rate of aluminum alloy, resulting in a large amount of workforce waste, waste of material and financial resources, and energy consumption. So the anti-corrosion and anti-fouling treatment of aluminum alloy surfaces to expand their practical application range and reduce energy loss are of great significance. In this paper, we first summarize the types of anti-corrosion and anti-fouling coatings on aluminum alloy surfaces. We found that superhydrophobic coating has attracted widespread attention recently because of its surface’s non-wetting and low surface energy characteristics. The superhydrophobic coating refers to a coating with a water contact angle on its surface greater than 150° and a sliding angle of less than 10°. Preparing superhydrophobic coatings on the surface of aluminum alloy can improve the corrosion resistance and anti-pollutant adhesion ability at the same time. Therefore, in the following part of the review, the preparation methods of superhydrophobic coatings on aluminum alloy surfaces for anti-corrosion and anti-fouling and the significant problems encountered in the practical application of the superhydrophobic coating on aluminum alloy surfaces, such as poor anti-corrosion durability and poor mechanical stability, and the existing methods to solve these critical problems, are summarized and discussed in detail. Finally, we put forward some new solutions to solve these vital issues.

From Bioinspired Topographies toward Non-Wettable Neural Implants.

The present study investigates different design strategies to produce non-wettable micropatterned surfaces. In addition to the classical method of measuring the contact angle, the non-wettability is also discussed by means of the immersion test. Inspired by non-wettable structures found in nature, the effects of features such as reentrant cavities, micropillars, and overhanging layers are studied. We show that a densely populated array of small diameter cavities exhibits superior non-wettability, with 65% of the cavities remaining intact after 24 h of full immersion in water. In addition, it is suggested that the wetting transition time is influenced by the length of the overhanging layer as well as by the number of columns within the cavity. Our findings indicate a non-wetting performance that is three times longer than previously reported in the literature for a small, densely populated design with cavities as small as 10 μm in diameter. Such properties are particularly beneficial for neural implants as they may reduce the interface between the body fluid and the solid state, thereby minimiing the inflammatory response following implantation injury. In order to assess the effectiveness of this approach in reducing the immune response induced by neural implants, further in vitro and in vivo studies will be essential.

Analysis of freezing of a sessile water droplet on surfaces over a range of wettability

HypothesisNonwetting surfaces, by virtue of their water-repelling trait, offer desirable anti-icing characteristics. Surface roughness, type and wettability are important interfacial characteristics that affect the icing dynamics that can be tailored to achieve desired anti-icing designs. Experiments and SimulationsThe present study systematically explores the effect of surface roughness on the freezing behaviour of water droplets on surfaces ranging in their wettability. Surfaces with tailored textures and wettability were fabricated using chemical etching and electrodeposition by varying the voltage. The surfaces studied include bare copper, five different dry nonwetting copper surfaces, and five different lubricant-infused copper surfaces that ranged in surface texture fractal dimension from nearly 1.0 to 1.92 and wettability measures of average water contact angle from 91° to 162° and sliding angle from less than 3° to greater than 50°. A computational model is developed to simulate the freezing dynamics on the surfaces studied. FindingsWith increasing roughness features, the freezing time increased due to the dual effects of increased contact angle and poor interfacial conductance caused by trapped air or infused liquid within the asperity textures. In general, the nonwetting surfaces increased the freezing time by a factor of at least 1.33 and up to about 3.2 compared to freezing on bare copper surfaces. The computational model shows close agreement with experimental measurements on the freeze front progression as well as freeze time. Design guidelines on the suitability of the different nonwetting surfaces for anti-icing purposes are derived from the systematic study, with the overall design recommendation favoring lubricant infused surfaces.

Droplet attraction and coalescence mechanism on textured oil-impregnated surfaces

Droplets residing on textured oil-impregnated surfaces form a wetting ridge due to the imbalance of interfacial forces at the contact line, leading to a wealth of phenomena not seen on traditional lotus-leaf-inspired non-wetting surfaces. Here, we show that the wetting ridge leads to long-range attraction between millimeter-sized droplets, which coalesce in three distinct stages: droplet attraction, lubricant draining, and droplet merging. Our experiments and model show that the magnitude of the velocity and acceleration at which droplets approach each other horizontally is the same as the vertical oil rise velocity and acceleration in the wetting ridge. Moreover, the droplet coalescence mechanism can be modeled using the classical mass-spring system. The insights gained from this work will inform future fundamental studies on remote droplet interaction on textured oil-impregnated surfaces for optimizing water harvesting and condensation heat transfer.

Long Term Durability of a Lubricant-Infused Surface for Dew Harvesting.

Hydrophobic, lubricant-infused surfaces offer enhanced potential for dew harvesting compared to bare metal substrates due to their water repellent nature. Most of the studies to date examine the condensation effectiveness of the nonwetting surfaces over a short duration and have not considered the durability or performance of the surfaces over extended periods. To address this limitation, the present study experimentally investigates the long-term performance of a lubricant-infused surface subject to dew condensation for 96 h. Condensation rates as well as sliding and contact angles are measured periodically to examine the surface properties and water harvesting potential over time. Due to the narrow time window in which dew harvesting can be conducted in application, the additional collection time gained by shedding droplets at lower nucleation times is explored. It is shown that three phases occur in lubricant drainage, which affect performance metrics relevant to dew harvesting. The first 24 h of condensation induces drainage that has little effect on the adhesion of droplets to the surface and on the additional collection time. The next phase, from about 24 to 72 h, showed steady drainage and a steady decrease in performance. The final 24 h, from about 72 to 96 h of operation, was seen to have little added effect on drainage and therefore on the performance metrics. The study bears significance in the design of surfaces for long-term use in practical water harvesters.

Superhydrophobic aluminum surfaces with nano-micro hierarchical composite structures: A novel and sustainable approach to corrosion protection

The unique properties of aluminum (Al) and its alloys make Al one of the most versatile, economical, and widely-used materials in various industries. However, the corrosion and leaching of Al can cause significant issues for the lifetimes of mechanical structures and machine components as well as environment and health problems. In the present study, a simple, eco-friendly, and rapid fabrication of superhydrophobic Al surfaces was introduced. The superhydrophobic Al surfaces with hierarchical composite structures combining microstructures prepared by laser texturing, the new formation of pseudo-boehmite nanostructures prepared by boiling water treatment, and low surface energy prepared by silicone oil heat treatment showed excellent corrosion resistance. The mechanism for wettability change and anti-corrosion were analyzed. The influence of laser parameters and surface modification procedures on the wettability and corrosion resistance of Al surfaces was also analyzed systematically through a series of surface characterization techniques, potentiodynamic polarization tests, and electrochemical impedance spectroscopy tests. The corrosion protection efficiency of the fabricated superhydrophobic Al surface has reached up to 99.40 % as compared with an untreated flat surface. Furthermore, the fabricated superhydrophobic surfaces show good stability even after prolonged exposure to air, fresh water, seawater, and high temperature environments. The performance of the non-wetting surfaces is demonstrated through self-cleaning, water jetting, and water droplet bouncing phenomena. This research provides a novel and sustainable approach for superhydrophobic metal surfaces and improved corrosion resistance for potential practical applications.

Drop impact dynamics on the hydrophobic leaf surface of an aquatic plant: a case study of Pistia stratiotes.

Pistia stratiotes is an aquatic plant with a complex structure that allows it to stay afloat. It grows quickly, and in large numbers becomes an undesirable plant as an invasive species. Describing the dynamics of a water drop splash on P. stratiotes leaves can contribute to increasing knowledge of its behavior and finding alternative methods for eradicating it or using it for the benefit of the environment. The non-wettable surface of P. stratiotes presents a complex structure-simple uniseriate trichomes and also ridges and veins. We analyzed the drop impact on a leaf placed on the water surface and recorded it by high-speed cameras. Based on the recordings, quantitative and qualitative analyses were performed. After impacting the leaf, the water drop spread until it reached its maximum surface area accompanied by the ejection of early droplets in the initial stage. Thereafter, three scenarios of water behavior were observed: (i) drop receding and stabilization; (ii) drop receding and ejection of late droplets formed in the later stage as an effect of elastic deformation of the leaf; and (iii) drop breaking apart and ejection of late droplets. The results indicated that the increasing kinetic energy of the impacting drops expressed by the Weber number and the complex leaf surface have an effect on the course of the splash. The simple uniseriate trichomes of the P. stratiotes leaf and the high energy of the falling drops were responsible for the formation and characteristics of the early droplets. The presence of ridges and veins and the leaf's mechanical response had an impact on the occurrence of late droplets.

Capillary-Driven Water Transport by Contrast Wettability-Based Durable Surfaces.

Controlling water transport and management is crucial for continuous and reliable system operation in harsh weather conditions. Passive strategies based on nonwetting surfaces are desirable, but so far, the implementation of superhydrophobic coatings into real-world applications has been limited by durability issues and, in some cases, lack of compliance with environmental regulations. Inspired by surface patterning observed on living organisms, in this study we have developed durable surfaces based on contrast wettability for capillary-driven water transport and management. The surface fabrication process combines a hydrophobic coating with hard-anodized aluminum patterning, using a scalable femtosecond laser microtexturing technique. The concept targets heavy-duty engineering applications; particularly in aggressive weather conditions where corrosion is prevalent and typically the anodic aluminum oxide-based coating is used to protect the surface from corrosion, the concept has been validated on anodic aluminum oxide coated aluminum alloy substrates. Such substrates with contrast wettable characteristics show long-term durability in both natural and lab-based artificial UV and corrosion tests where superhydrophobic coatings tend to degrade.

When coalescing droplets jump: A unified energy conversion model incorporating droplet size and surface adhesion

Jumping-droplet condensation pushes the boundary of condensation heat transfer by enabling microdroplet shedding via coalescence-induced droplet jumping. The latter is empowered by surface-to-kinetic energy conversion. Regardless of extensive studies of droplet jumping on ideally non-wetting surfaces, a quantitative description of droplet jumping from realistic surfaces remains a challenge due to limited insight into the complex energy conversion process that is strongly coupled with droplet–droplet and droplet–substrate interactions. Here, we use a three-dimensional (3D) pseudopotential multiphase multiple-relaxation-time lattice Boltzmann method (MRT-LBM) to simulate binary-droplet coalescence with various droplet sizes and surface wettability. Then, we developed a comprehensive and unified energy conversion model, derived by rigorously analyzing the dynamic droplet–surface interaction and quantifying the roles of droplet size scale, droplet size mismatch, and surface wettability. Our simulations capture coalescence and jumping dynamics of arbitrary-sized droplets on surfaces having various wettability and reveal the effect of droplet size and surface wettability. Validated by experiments, the energy model is then used to define the jumping/non-jumping boundaries for coalescing droplets on nanostructured surfaces. Our work demonstrates the key physics and a universal criterion governing self-propelled droplet shedding, key to the design of surfaces for enhanced condensation heat transfer, anti-frosting/icing, self-cleaning, and water/energy harvesting.

A mechanistic model of critical heat flux for pool boiling based on supply failure mechanisms depending on the contact angle

Experimental and theoretical analyses are conducted to develop a mechanistic model of critical heat flux (CHF) for application to pool boiling of fluids with contact angles between 0° and 180°. To accurately predict the CHF over the entire range of the contact angle, it is postulated that an unquenchable dry patch occurs when liquid fails to be supplied to the dry patch; A supply failure mechanism is classified into two different mechanisms depending on the wettability of the boiling surface. The first is an evaporation mechanism which is dominant for non-wetting surfaces. This type occurs when the liquid supply rate in the liquid film due to the capillary force becomes lower than the evaporation rate. The second mechanism is a levitation mechanism which is dominant for wetting surfaces. This type occurs when the surface temperature under the dry patch exceeds the Leidenfrost point. To check the validity of the postulate, experiments involving the pool boiling of water and FC-72 are conducted on a titanium surface, which is a non-wetting surface for water but a wetting surface for FC-72. In the experiments, the temperature and heat flux distributions on the boiling surface are measured using infrared thermography for a quantitative characterization of boiling at the CHF. Experimental analyses confirm the postulate by showing that the CHF occurs due to the evaporation mechanism for the pool boiling of water, whereas the levitation mechanism for the pool boiling of FC-72. It is also observed that a large area of liquid film exists for the pool boiling of water, whereas this liquid film hardly exists for the pool boiling of FC-72. This observation indicates that the change in the dominant mechanism is due to the dependence of the liquid film area on the wettability of the boiling surface. After validating the postulate, a CHF correlation is developed that reflects the change of the supply failure mechanism depending on the contact angle. The developed correlation shows a good match to the extensive sets of CHF data previously obtained by various investigators to within 20%.