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At least 469 records · Page 26

Method of Calibrating a Force Balance

A calibration system and method utilizes acceleration of a mass to generate a force on the mass. An expected value of the force is calculated based on the magnitude and acceleration of the mass. A fixture is utilized to mount the mass to a force balance, and the force balance is calibrated to provide a reading consistent with the expected force determined for a given acceleration. The acceleration can be varied to provide different expected forces, and the force balance can be calibrated for different applied forces. The acceleration may result from linear acceleration of the mass or rotational movement of the mass.

Parker, Peter A.↗

Plasmonic Force Propulsion Revolutionizes Nano/Picosatellite Capability

This project investigated a new type of small spacecraft propulsion for attitude control, specifically proximity and precision pointing control. Plasmonic force propulsion uses solar light focused on deep-subwavelength nanostructures to excite strong optical forces that accelerate and expel nanoparticle propellant. The goal of the project was to assess the feasibility of plasmonic force propulsion for nano/pico-satellite applications by evaluating key mission parameters for a nano/pico-satellite using plasmonic force propulsion in a NASA-relevant mission context. We achieved this goal and objective by evaluating plasmonic force propulsion within a NASA mission that required attitude control and precision pointing of a small satellite. We numerically simulated plasmonic force fields with asymmetric/gradient geometry and relevant solar light constraints, predicted nanoparticle velocity, mass flow rate, and resulting propulsion performance (thrust, specific impulse), and evaluated spacecraft position control resolution and pointing precision. Additionally we compared the precision pointing capabilities of plasmonic propulsion, as well as the mass, volume, and power requirements, with other state-of-the-art control techniques, such as reaction wheels and colloid/electrospray electric propulsion. The results are very exciting. Plasmonic force propulsion can significantly enhance the state-of-the-art in small spacecraft position and attitude control by 1-2 orders of magnitude. This is most succinctly shown in the figure below, which compares proximity and attitude control of plasmonic force propulsion (PFP) with other state-of-the-art thruster systems (μCAT, VAT, electrospray). Additionally this figure also shows the proximity and attitude control required for different existing (James Webb Space Telescope, Hubble) and future (LISA and Stellar Imager) NASA missions. While some of these NASA missions are not small spacecraft missions, the requirements serve to illustrate the fact that more precise proximity and attitude control will be required for future NASA science missions. Stellar imager is a proposed NASA missions that requires an extremely high pointing precision of 0.1 milliarcseconds (2.7x10(exp -7) deg.) for an ultraviolet telescope that has over 200× the resolution of the Hubble Space Telescope, is able to take images showing details on the surfaces of other stars, consists of 20-30 small "mirror sats" flying in formation to produce a giant mirror, and requires each mirror-sat to be placed with nanometer precision and control its attitude with milliarcsecond precision.

Rovey, Joshua L.↗

TURBOMAT-FR: Turbomachinery Aeroelastic Forced Response Analysis Automation Using MATLAB® Part 2

The primary aim of this effort is to develop a software tool named “Turbomachinery Aeroelastic Analysis Tool for Forced Response (TURBOMAT-FR)” using MATLAB® (The MathWorks, Inc.) scripts. The tool will aid in performing routine forced response analysis of turbomachinery fan blades with the aid of the modal summation method (MSUM). The MSUM approach requires a modal solution consisting of modal displacements, frequencies, and stresses. The MATLAB® scripts facilitate easy extraction of the required quantities from the computational fluid dynamics (CFD) and MSC Nastran (Hexagon AB) solution files. These quantities are subsequently utilized by the MSUM approach to perform a complete forced response analysis using the MATLAB® scripts. The MATLAB® scripts in TURBOMAT-FR are utilized in postprocessing of the MSC Nastran finite element analysis outputs in three types of solutions or steps, namely (1) stresses and displacements from static or steady response solution, (2) unsteady stresses and displacements from direct forced response solution, and (3) modal solution containing modal displacements and modal stresses. In step (4), separate MATLAB® scripts are also developed to read the processed outputs in step (3) and perform forced response analysis using MSUM. It should be noted that the direct forced response solution from step (2) will provide for a comparison with those obtained from the MSUM approach in steps (3) and (4). The TURBOMAT-FR MATLAB® scripts mentioned previously are presented in two reports designated Parts 1 and 2. Part 1 presents the scripts needed to accomplish steps (1) and (2). The current report, Part 2, presents the scripts needed to accomplish steps (3) and (4), noted previously. The TURBOMAT-FR MATLAB® scripts provided here in Part 2 facilitate the calculation of unsteady stresses using MSUM and provide an automated procedure associated with aeroelastic forced response analysis with different pressure files. The unsteady stresses calculated using MSUM can be compared with those presented in Part 1. A cantilevered blade-alone model subjected to a realistic pressure loading to a boundary was used for static and unsteady direct forced response analysis to demonstrate the effectiveness of the TURBOMATFR MATLAB® scripts.

T S R Reddy↗

Development of a new method in force limited vibration testing

The force limited vibration test approaches discussed in NASA-HDBK-7004C were developed to reduce overtesting associated with base shake vibration tests of aerospace hardware. This handbook outlines several different methods of specifying force limiting specifications. The basis for deriving force limiting specifications is related to the differences between mechanical impedances of typical aerospace hardware interfacing the mounting structures in flight configurations and the large impedances inherent in shaker vibration testing. Among these approaches, the semi-empirical method is presently the most widely used method to derive the force limiting specifications. This method is developed based on the assumption that the hardware interface responses are coherently excited. The inclusion of the incoherent excitation of the aerospace structures at mounting interfaces provides the basis for developing a more realistic force limiting specification that may be used to qualify flight hardware for launch environments using shaker testing. In this paper the semi-empirical method for defining force limiting specifications discussed in NASA handbook are reviewed using data recently obtained from a series of acoustic and vibration tests. A mathematical formation of a new force limiting approach based on considering the incoherent excitation of the hardware at their mounting interfaces is developed. Results from the analysis, which are correlated to the data obtained from a series of well instrumented acoustic and vibration tests, are discussed. The new approach provides much more realistic force limits that may further remove conservatism inherent in shaker vibration testing not accounted for by methods discussed in the NASA handbook.

Kolaini, Ali R.↗

Nano-Newton Electrostatic Force Actuators for Femto-Newton-Sensitive Measurements: System Performance Test in the LISA Pathfinder Mission

Electrostatic force actuation is a key component of the system of geodesic reference test masses (TM) for the LISA orbiting gravitational wave observatory and in particular for performance at low frequencies, below 1 mHz, where the observatory sensitivity is limited by stray force noise. The system needs to apply forces of order 10 −9 N while limiting fluctuations in the measurement band to levels approaching 10 −15 N=Hz 1/2 . We present here the LISA actuation system design, based on audio-frequency voltage carrier signals, and results of its in-flight performance test with the LISA Pathfinder test mission. In LISA, TM force actuation is used to align the otherwise free-falling TM to the spacecraft-mounted optical metrology system, without any forcing along the critical gravitational wave-sensitive interferometry axes. In LISA Pathfinder, on the other hand, the actuation was used also to stabilize the TM along the critical x axis joining the two TM, with the commanded actuation force entering directly into the mission’s main differential acceleration science observable. The mission allowed demonstration of the full compatibility of the electrostatic actuation system with the LISA observatory requirements, including dedicated measurement campaigns to amplify, isolate, and quantify the two main force noise contributions from the actuation system, from actuator gain noise and from low frequency “in band” voltage fluctuations. These campaigns have shown actuation force noise to be a relevant, but not dominant, noise source in LISA Pathfinder and have allowed performance projections for the conditions expected in the LISA mission.

M. Armanos↗

Influence of 21st century atmospheric and sea surface temperature forcing on West African climate

he persistence of extended drought events throughout West Africa during the 20th century has motivated a substantial effort to understand the mechanisms driving African climate variability, as well as the possible response to elevated greenhouse gas (GHG) forcing. We use an ensemble of global climate model experiments to examine the relative roles of future direct atmospheric radiative forcing and SST forcing in shaping potential future changes in boreal summer precipitation over West Africa. We find that projected increases in precipitation throughout the Western Sahel result primarily from direct atmospheric radiative forcing. The changes in atmospheric forcing generate a slight northward displacement and weakening of the African easterly jet (AEJ), a strengthening of westward monsoon flow onto West Africa and an intensification of the tropical easterly jet (TEJ). Alternatively, we find that the projected decreases in precipitation over much of the Guinea Coast region are caused by SST changes that are induced by the atmospheric radiative forcing. The changes in SSTs generate a weakening of the monsoon westerlies and the TEJ, as well as a decrease in low-level convergence and resultant rising air throughout the mid levels of the troposphere. Our experiments suggest a potential shift in the regional moisture balance of West Africa should global radiative forcing continue to increase, highlighting the importance of climate system feedbacks in shaping the response of regional-scale climate to global-scale changes in radiative forcing.

African monsoon↗

Adhesion and aerodynamic forces for the resuspension of non-spherical particles in outdoor environments

Particles deposited on an outdoor surface can be resuspended by wind gusts, become airborne, and be inhaled if small enough. If toxic or infectious, these particles may be dangerous for the populace health. It is therefore important to determine under which weather conditions a deposit of particle could be resuspended to implement the best response actions and plan clean-up. To this scope, one needs to consider the competing forces acting on the particle keeping it attached to the surface (gravity and adhesion) or trying to remove it (aerodynamic forces, i.e., lift and drag). Here, this article reviews the current understanding of the aforementioned forces for colloidal spherical particles and extends the existing theories to rod-shaped particles, representative for instance of Bacillus spores. In particular, for the adhesion force, the Derjaguin approximation was used and the adhesion force was computed from the radii of curvature of the particle and the surface at the point of closest approach. For the aerodynamic forces, we re-derived the equations for the drag and lift forces accounting for the shape of the particle. Both smooth and rough surfaces will be discussed, the former as idealized cases, the latter as more representative of real outdoor surfaces.

54 ENVIRONMENTAL SCIENCES↗

Prototype cantilevers for quantitative lateral force microscopy

Prototype cantilevers are presented that enable quantitative surface force measurements using contact-mode atomic force microscopy (AFM). The ''hammerhead'' cantilevers facilitate precise optical lever system calibrations for cantilever flexure and torsion, enabling quantifiable adhesion measurements and friction measurements by lateral force microscopy (LFM). Critically, a single hammerhead cantilever of known flexural stiffness and probe length dimension can be used to perform both a system calibration as well as surface force measurements in situ, which greatly increases force measurement precision and accuracy. During LFM calibration mode, a hammerhead cantilever allows an optical lever ''torque sensitivity'' to be generated for the quantification of LFM friction forces. Precise calibrations were performed on two different AFM instruments, in which torque sensitivity values were specified with sub-percent relative uncertainty. To examine the potential for accurate lateral force measurements using the prototype cantilevers, finite element analysis predicted measurement errors of a few percent or less, which could be reduced via refinement of calibration methodology or cantilever design. The cantilevers are compatible with commercial AFM instrumentation and can be used for other AFM techniques such as contact imaging and dynamic mode measurements.

42 ENGINEERING↗

Local nonlinear rf forces in inhomogeneous magnetized plasmas

The local nonlinear forces induced by radio frequency (rf) waves are derived in inhomogeneous magnetized plasmas, where the inhomogeneity exists in the rf fields, in the static magnetic field as well as in the equilibrium density and temperature. The local parallel force is completely resonant, but a novel component dependent on those inhomogeneities is obtained as the result of the inhomogeneous transport of parallel resonant-absorbed momentum by the nonlinear perpendicular drift flux. In the local poloidal force, the component induced by the inhomogeneity of rf power absorption is also confirmed and it can be recognized as the residual effect from the incomplete cancellation between the rate of the diamagnetic poloidal momentum gain and the Lorentz force due to the radial diffusion-like flux. The compact expression for radial force is also obtained for the first time, whose nonresonant component is expressed as the sum of the ponderomotive force on particles and the gradients of the nonresonant perpendicular pressure and of the nonresonant momentum flux due to the finite temperature effect. Numerical calculations in a 1-D slab model show that the resonant component dependent on the inhomogeneities may be significant when the ion absorption dominates the resonant wave-particle interaction. A quantitative estimation shows that the novel component in the parallel force is important to understand the experiments of the ion-cyclotron-frequency mode-conversion flow drive.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Perspective: Ab initio force field methods derived from quantum mechanics

It is often desirable to accurately and efficiently model the behavior of large molecular systems in the condensed phase (thousands to tens of thousands of atoms) over long time scales (from nanoseconds to milliseconds). In these cases, ab initio methods are difficult due to the increasing computational cost with the number of electrons. A more computationally attractive alternative is to perform the simulations at the atomic level using a parameterized function to model the electronic energy. Many empirical force fields have been developed for this purpose. However, the functions that are used to model interatomic and intermolecular interactions contain many fitted parameters obtained from selected model systems, and such classical force fields cannot properly simulate important electronic effects. Furthermore, while such force fields are computationally affordable, they are not reliable when applied to systems that differ significantly from those used in their parameterization. They also cannot provide the information necessary to analyze the interactions that occur in the system, making the systematic improvement of the functional forms that are used difficult. Ab initio force field methods aim to combine the merits of both types of methods. The ideal ab initio force fields are built on first principles and require no fitted parameters. Ab initio force field methods surveyed in this perspective are based on fragmentation approaches and intermolecular perturbation theory. This perspective summarizes their theoretical foundation, key components in their formulation, and discusses key aspects of these methods such as accuracy and formal computational cost. The ab initio force fields considered here were developed for different targets, and this perspective also aims to provide a balanced presentation of their strengths and shortcomings. Finally, this perspective suggests some future directions for this actively developing area.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Acoustic force spectroscopy reveals subtle differences in cellulose unbinding behavior of carbohydrate-binding modules

Protein adsorption to solid carbohydrate interfaces is critical to many biological processes, particularly in biomass deconstruction. To engineer more-efficient enzymes for biomass deconstruction into sugars, it is necessary to characterize the complex protein–carbohydrate interfacial interactions. A carbohydrate-binding module (CBM) is often associated with microbial surface-tethered cellulosomes or secreted cellulase enzymes to enhance substrate accessibility. However, it is not well known how CBMs recognize, bind, and dissociate from polysaccharides to facilitate efficient cellulolytic activity, due to the lack of mechanistic understanding and a suitable toolkit to study CBM–substrate interactions. Our work outlines a general approach to study the unbinding behavior of CBMs from polysaccharide surfaces using a highly multiplexed single-molecule force spectroscopy assay. Here, we apply acoustic force spectroscopy (AFS) to probe a Clostridium thermocellum cellulosomal scaffoldin protein (CBM3a) and measure its dissociation from nanocellulose surfaces at physiologically relevant, low force loading rates. An automated microfluidic setup and method for uniform deposition of insoluble polysaccharides on the AFS chip surfaces are demonstrated. The rupture forces of wild-type CBM3a, and its Y67A mutant, unbinding from nanocellulose surfaces suggests distinct multimodal CBM binding conformations, with structural mechanisms further explored using molecular dynamics simulations. Applying classical dynamic force spectroscopy theory, the single-molecule unbinding rate at zero force is extrapolated and found to agree with bulk equilibrium unbinding rates estimated independently using quartz crystal microbalance with dissipation monitoring. However, our results also highlight critical limitations of applying classical theory to explain the highly multivalent binding interactions for cellulose–CBM bond rupture forces exceeding 15 pN.

09 BIOMASS FUELS↗

Data-model files associated with the manuscript "The Effects of Spatial and Temporal Resolution of Gridded Meteorological Forcing on Watershed Hydrological Responses" (Shuai et al., 2022 HESS)

This data package contains the model inputs and outputs used in "The Effects of Spatial and Temporal Resolution of Gridded Meteorological Forcing on Watershed Hydrological Responses" (Shuai et al., 2022 HESS). The data.zip file contains the data used to drive the model simulations. The model.zip file contains the XML input file for ATS. The notebook.zip file contains the Jupyter notebooks for pre- and post- processing model results. The figures.zip file contains the raw figures associated with the manuscript.Meteorological forcing plays a critical role in accurately simulating the watershed hydrological cycle. With the advancement of high-performance computing and the development of integrated watershed models, simulating the watershed hydrological cycle at high temporal (hourly to daily) and spatial resolution (10s of meters) has become efficient and computationally affordable. These hyperresolution watershed models require high resolution of meteorological forcing as model input to ensure the fidelity and accuracy of simulated responses. In this study, we utilized the Advanced Terrestrial Simulator (ATS), an integrated watershed model, to simulate surface and subsurface flow and land surface processes using unstructured meshes at the Coal Creek Watershed near Crested Butte (Colorado). We compared simulated watershed hydrologic responses including streamflow, and distributed variables such as evapotranspiration, snow water equivalent (SWE), and groundwater table driven by three publicly available, gridded meteorological forcings (GMFs) -- Daily Surface Weather and Climatological Summaries (Daymet), Parameter-elevation Regressions on Independent Slopes Model (PRISM), and North American Land Data Assimilation System (NLDAS). By comparing various spatial resolutions (ranging from 400 m to 4 km) of PRISM, the simulated streamflow only becomes marginally worse when spatial resolution of meteorological forcing is coarsened to 4 km (or 30% of the watershed area). However, the 4 km resolution has much worse performance than finer resolution in spatially distributed variables such as SWE. Using temporally disaggregated PRISM, we compared models forced by different temporal resolutions (hourly to daily), sub-daily resolution preserves the dynamic watershed responses (e.g., diurnal fluctuation of streamflow) that are absent in results forced by daily resolution. Conversely, the simulated streamflow shows better performance using daily resolution compared to that using sub-daily resolution. Our findings suggest that the choice of GMF and its spatiotemporal resolution depends on the quantity of interest and its spatial and temporal scale, which may have important implications on model calibration and watershed management decisions.

54 ENVIRONMENTAL SCIENCES↗

Fabrication of oriented crystals as force measurement tips via focused ion beam and microlithography methods

Detailed knowledge of the forces between nanocrystals is very crucial for understanding many generic (e.g., random aggregation/assembly and rheology) and specific (e.g., oriented attachment) phenomena at macroscopic length scales, especially considering the additional complexities involved in nanocrystals such as crystal orientation and corresponding orientation-dependent physicochemical properties. Because there are a limited number of methods to directly measure the forces, little is known about the forces that drive the various emergent phenomena. Here we report on two methods of preparing crystals as force measurement tips used in an atomic force microscope (AFM): the focused ion beam method and microlithography method. The desired crystals are fabricated using these two methods and are fixed to the AFM probe using platinum deposition, ultraviolet epoxy, or resin, which allows for the orientation-dependent force measurements. These two methods can be used to attach virtually any solid particles (from the size of a few hundreds of nanometers to millimeters). We demonstrate the force measurements between aqueous media under different conditions such as pH.

Environmental Molecular Sciences Laboratory↗

Seasonal delay of Sahelian rainfall driven by an east–west contrast in radiative forcing in idealized CESM experiments

Recent studies suggest the observed seasonal delay of rainfall over the Sahel is mainly driven by anthropogenic aerosol forcing, which features a robust east–west contrasting changes in recent decades, with negative and positive top-of-the-atmosphere shortwave radiative forcing in South and East Asia (SA&EA) and in North America and Europe (NA&EU), respectively. Their individual effects on the Sahel rainfall annual cycle remain unclear. Here, by designing idealized sensitivity experiments based on Community Earth System Model (CESM1.2), we show that both negative radiative forcing over SA&EA and positive radiative forcing over NA&EU contribute to the delayed phase and enhanced amplitude of Sahel rainfall annual cycle. To understand the underlying physical processes, both the convective quasi-equilibrium (CQE) and atmospheric energetic frameworks are utilized. Both frameworks can well explain the seasonal delay in the rainfall annual cycle over the Sahel under the regional radiative forcings, with the CQE framework showing stronger explanatory power. Based on the insights from both frameworks, we conducted a moisture budget analysis and found that the negative radiative forcing over SA&EA causes an anomalous low-level anticyclonic circulation, which transports more moisture to the Sahel. As the anomalous anticyclone is stronger in summer and fall compared to spring, rainfall is delayed in the Sahel. Further, under the influence of the positive radiative forcing over NA&EU, more warming in the northern hemisphere causes northward transport of moisture to the northern tropics associated with the cross-equatorial transport of energy. The moisture transport increases relativehumidity over the Sahel and enhances the effective atmospheric heat capacity that delays the seasonal rainfall.

54 ENVIRONMENTAL SCIENCES↗

Impact of peptide clustering on unbinding forces in the context of fusion mimetics

Highlights: ► Coiled-coil peptides as SNARE mimetics for membrane fusion. ► Interaction forces assessed by colloidal probe microscopy. ► Lateral organization of lipopeptides visualized by atomic force microscopy. -- Abstract: Coiled-coil zipping and unzipping is a pivotal process in SNARE-regulated membrane fusion. In this study we examine this process mediated by a minimal model for coiled-coil formation employing force spectroscopy in the context of membrane-coated surfaces and probes. The interaction forces of several hundred pN are surprisingly low considering the proposed amount of molecular bonds in the contact zone. However, by means of high-resolution imaging employing atomic force microscopy and studying the lateral mobility of lipids and peptides as a function of coiled-coil formation, we are able to supply a detailed view on processes occurring on the membrane surfaces during force measurements. The interaction forces determined here are not only dependent on the peptide concentration on the surface, but also on the regional organization of lateral peptide clusters found prior to coiled-coil formation.

60 APPLIED LIFE SCIENCES↗

Understanding Rapid Adjustments to Diverse Forcing Agents

Rapid adjustments are responses to forcing agents that cause a perturbation to the top of atmosphere energy budget but are uncoupled to changes in surface warming. Different mechanisms are responsible for these adjustments for a variety of climate drivers. These remain to be quantified in detail. It is shown that rapid adjustments reduce the effective radiative forcing (ERF) of black carbon by half of the instantaneous forcing, but for CO 2 forcing, rapid adjustments increase ERF. Competing tropospheric adjustments for CO 2 forcing are individually significant but sum to zero, such that the ERF equals the stratospherically adjusted radiative forcing, but this is not true for other forcing agents. Additional experiments of increase in the solar constant and increase in CH 4 are used to show that a key factor of the rapid adjustment for an individual climate driver is changes in temperature in the upper troposphere and lower stratosphere.

58 GEOSCIENCES↗

Benchmark calculations of radiative forcing by greenhouse gases

Changes in concentrations of greenhouse gases lead to changes in radiative fluxes throughout the atmosphere. The value of this change, the instantaneous radiative forcing, varies across climate models, due partly to differences in the distribution of clouds, humidity, and temperature across models, and partly due to errors introduced by approximate treatments of radiative transfer. This paper describes an experiment within the Radiative Forcing Model Intercomparision Project that uses benchmark calculations made with line–by–line models to identify parameterization error in the representation of absorption and emission by greenhouse gases. Clear–sky instantaneous forcing by greenhouse gases is computed using a set of 100 profiles, selected from a re–analysis of present–day conditions, that represent the global annual mean forcing from pre–industrial times to the present–day with sampling errors of less than 0.01 W m –2 . Six contributing line–by–line models agree in their estimate of this forcing to within 0.025 W m –2 while even recently–developed parameterizations have typical errors four or more times larger, suggesting both that the samples reveal true differences among line–by–line models and that parameterization error will be readily identifiable. Agreement among line–by–line models is better in the longwave than in the shortwave where differing treatments of the water vapor vapor continuum affect estimates of forcing by carbon dioxide and methane. The impacts of clouds on instantaneous radiative forcing are estimated from climate model simulations. Here, the adjustment due to stratospheric temperature change by assuming fixed dynamical heating. Adjustments are large only for ozone and for carbon dioxide, for which stratospheric cooling introduces modest non–linearity.

54 ENVIRONMENTAL SCIENCES↗

Unfolding of globular polymers by external force

We examine the problem of a polymer chain, folded into a globule in poor solvent, subjected to a constant tensile force. Such a situation represents a Gibbs thermodynamic ensemble and is useful for analysing force-clamp atomic force microscopy measurements, now very common in molecular biophysics. Using a basic Flory mean-field theory, we account for surface interactions of monomers with solvent. Under an increasing tensile force, a first-order phase transition occurs from a compact globule to a fully extended chain, in an “all-or-nothing” unfolding event. This contrasts with the regime of imposed extension, first studied by Halperin and Zhulina [Europhys. Lett. 15, 417 (1991)], where there is a regime of coexistence of a partial globule with an extended chain segment. We relate the transition forces in this problem to the solvent quality and degree of polymerisation, and also find analytical expressions for the energy barriers present in the problem. Using these expressions, we analyse the kinetic problem of a force-ramp experiment and show that the force at which a globule ruptures depends on the rate of loading.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗