Rows Of Particles Research Articles

Correlation between meso-structures and magnetoelectric properties in 0-3 magnetoelectric composites

The quantitative characterization of macroscopic properties of 0-3 polymer-based magnetoelectric (ME) composites is the basis of their application, and the meso-structure is the source that affects their macroscopic properties. In this study, theoretical models consistent with the meso-structures of the composites are developed to address scenarios involving random, chain, and agglomeration particle distributions. The focus is on investigating the nonlinear mechanical behavior of multi-field coupling. The validity and rationality of the theoretical model are confirmed by comparing the theoretical results with existing experimental data. Based on this, the distribution characteristics of elastic field, electric field and magnetic field of 0-3 polymer-based ME composites with different particle arrangements are analyzed. Additionally, the influence of several factors on the ME properties, such as magnetic field strength and direction, particle proportion and frequency, is discussed. The results indicate that the ME performance is optimized when the direction of the densest chain row of particles aligns with the direction of the applied magnetic field. Furthermore, the phenomenon of current leakage is examined when the direction of the chain row of ferromagnetic particles coincides with the direction of polarization. To mitigate this leakage effect, a solution is proposed by adding thin layers of piezoelectric polymer to the surface of the composite, resulting in a 62.6% increase in ME performance. The research findings reveal the response mechanisms and essence of 0-3 polymer-based ME composites under applied magnetic fields, establishing a connection between meso-structures and macro properties. This provides guidance for the design and optimization of mesomorphic structures and performance, and serves as a foundation for the design, fabrication, and development of a new generation of ME devices.

A Microshutter for the Nanofabrication of Plasmonic Metal Alloys with Single Nanoparticle Composition Control.

Alloying offers an increasingly important handle in nanomaterials design in addition to the already widely explored size and geometry of nanostructures of interest. As the key trait, the mixing of elements at the atomic level enables nanomaterials with physical or chemical properties that cannot be obtained by a single element alone, and subtle compositional variations can significantly impact these properties. Alongside the great potential of alloying, the experimental scrutiny of its impact on nanomaterial function is a challenge because the parameter space that encompasses nanostructure size, geometry, chemical composition, and structural atomic-level differences among individuals is vast and requires unrealistically large sample sets if statistically relevant and systematic data are to be obtained. To address this challenge, we have developed a microshutter device for spatially highly resolved physical vapor deposition in the lithography-based fabrication of nanostructured surfaces. As we demonstrate, it enables establishing compositional gradients across a surface with single nanostructure resolution in terms of alloy composition, which subsequently can be probed in a single experiment. As a showcase, we have nanofabricated arrays of AuAg, AuPd, and AgPd alloy nanoparticles with compositions systematically controlled at the level of single particle rows, as verified by energy dispersive X-ray and single particle plasmonic nanospectroscopy measurements, which we also compared to finite-difference time-domain simulations. Finally, motivated by their application in state-of-the-art plasmonic hydrogen sensors, we investigated PdAu alloy gradient arrays for their hydrogen sorption properties. We found distinctly composition-dependent kinetics and hysteresis and revealed a composition-dependent contribution of a single nanoparticle response to the ensemble average, which highlights the importance of alloy composition screening in single experiments with single nanoparticle resolution, as offered by the microshutter nanofabrication approach.

Influence of ion structure and solvent electric dipole on ultrananoporous supercapacitor: a lattice model study

A five-state modified BEG model is proposed for describing electrolyte in polar solvent in cylindrical electrode pore in which only one row of particles can be stationed. Two properties the closest to the actual situation of the aqueous electrolyte are integrated into the model: cation is dimer comprised of two tangentially tethered hard spheres of different diameters: one is charged, while the other is neutral, and polar solvent is modeled as neutral hard sphere with an electric dipole. The system partition function is obtained by numerically solving maximum real root of characteristic equation of Kramers-Wannier transfer matrix. The model is used to investigate effects of ion structure and electric valence, solvent electric dipole, and cylindrical pore radius on specific differential capacitance and specific energy storage of the resulting supercapacitor. Main findings are briefly described below. (i) The dimer counter-ion electric valence does not exert obvious influence on the saturation value of the but reduces the threshold electrode potential strength beyond which the value is just reached. (ii) Whether the dimer counter-ion is monovalent or bivalent, the peak height and the value tend to rise with the dimer neutral site size decreasing, but influence of the value on the value is very small. Only when the value is small enough, the value rises a little with decrease of the value. (iii) Increase of solvent electric dipole moment (whether by increasing the dipole charge or dipole length) always raises the value, and enlarges width of the pure solvent domain or zero and zero domain around the zero electrode potential, but does not change the value obviously. (iv) With the pore size increasing, the value reduces and the value increases. So, using small size pore is favorable to improve the supercapacitor performance from the perspective of energy storage and electrode potential applied. (v) The three parameters: dipole charge, and dipole length, are cooperative in influencing the value.

Effect of wall slip on the viscoelastic particle ordering in a microfluidic channel.

The formation of a line of equally spaced particles at the centerline of a microchannel, referred as "particle ordering," is desired in several microfluidic applications. Recent experiments and simulations highlighted the capability of viscoelastic fluids to form a row of particles characterized by a preferential spacing. When dealing with non-Newtonian fluids in microfluidics, the adherence condition of the liquid at the channel wall may be violated and the liquid can slip over the surface, possibly affecting the orderingefficiency. In this work, we investigate the effect of wall slip on the ordering of particles suspended in a viscoelastic liquid by numerical simulations. The dynamics of a triplet of particles in an infinite cylindrical channel is first addressed by solving the fluid and particle governing equations. The relative velocities computed for the three-particle system are used to predict the dynamics of a train of particles flowing in a long microchannel. The distributions of the interparticle spacing evaluated at different slip coefficients, linear particle concentrations, and distances from the channel inlet show that wall slip slows down the self-assembly mechanism. For strong slipping surfaces, no significant change of the initial microstructure is observed at low particle concentrations, whereas strings of particles in contact form at higher concentrations. The detrimental effect of wall slip on viscoelastic ordering suggests care when designing microdevices, especially in case of hydrophobic surfaces that may enhance the slippingphenomenon.

Ultrananoporous supercapacitor with ionic liquid comprised of two-site cation: an Ising model study (II)

A four-state one-dimension discrete spin model with the nearest neighbor and partial next nearest neighbor lattice interactions is proposed for describing electrical double layer formed by ionic liquid in cylindrical pore that can accommodate only one row of particles. The model, in which cation is modeled as a dimer formed by two hard spheres (HSs): one is positively charged with diameter , while the other is neutral with diameter , and anion is modeled by charged HS with diameter , is designed to investigate effects of pore diameter , ion transfer free energy, ion size and electrical valence on behaviors of specific differential capacitance and specific energy storage and its saturation value . Relevant Kramers–Wannier transfer matrix is , which gives system partition function by numerically solving maximum real root of the relevant characteristic equation. Main findings are briefly described below. (i) The ion valence has little effect on the value, but does reduce the threshold voltage strength , beyond which the energy storage reaches . However, the ion valence (whether co- or counter-ion) raises both the and values as the relevant ion size decreases enough. (ii) Decreasing of the value weakly reduces the value, but raises the peak height and the value obviously; correlation between the value and value becomes stronger with the ions getting more pore hostile. (iii) With the counter-ion or co-ion charged site size decreasing, both the value and value rise. However, the effect of the co-ion charged site size is weakened as the co-ion gets more pore hostile, and the cation neutral site size influences on the value less than the cation charged site size does. (iv) Pore friendly co-ion or pore hostile counter-ion always helps in raising the value; the two have significant synergistic effect with the co-ion pore friendliness playing more important role than the counter-ion pore hostility.

3D Arrangement of Magnetic Particles in Thin Polymer Films Assisted by Magnetically Patterned Exchange Bias Layer Systems

AbstractThe possibility to arrange and embed magnetic micro‐ and nanoparticles in thin polymer film systems using flat magnetically patterned substrate templates is investigated. In contrast to self‐organized particle rows forming by applying a homogeneous magnetic field, particles adapt to the magnetic field landscape of the substrate's magnetic pattern prior to polymer crosslinking. Crosslinking then fixes the particle positions in the polymer. The process is tested for composites of hydrophobic polydimethylsiloxane (PDMS) and maghemite nanoparticles as well as for hydrophilic polyvinyl alcohol (PVOH) and hydrophilic functionalized, superparamagnetic core–shell microspheres. The substrate template is an exchange bias layer system magnetically patterned into parallel‐stripe domains with in‐plane magnetizations and head‐to‐head/tail‐to‐tail remanent magnetization orientation in adjacent magnetic domains. A high occupancy percentage of magnetic beads on a domain wall as well as anisotropic actuation of the composite is achieved.

A three-dimensional beam formulation for large deformation and an accurate implementation of the free boundary

This paper presents a meshless model for quasi-static and dynamic analysis of a three-dimensional Timoshenko beam with geometric nonlinearity. A general mathematical formulation is constructed based on the corrective smoothed particle method (CSPM), which can correct the low precision and completeness deficiency of the standard smoothed particle hydrodynamics(SPH) method. The discrete governing equations as well as the boundary conditions in strong form for the three-dimensional beam are then derived by using the conservation conditions and the CSPM interpolation function. The developed model enables one to expediently discretize the geometric nonlinear beam with only a row of particles at the central axis and to automatically satisfy the free boundary condition without any additional treatment. Moreover, Lagrangian kernel function and stress points are adopted to eliminate tensile instability and instability induced by the rank deficiency within the particle methods. Finally, comparisons with several results obtained from the existing literature are provided to demonstrate the validity and potential of the present procedure.

Persistent PnV (Perina nuda virus) infection in a heterologous Lymantria xylina cell line, NTU-LY

ABSTRACT The NTU-LY-1 cell line is a Lymantria xylina cell line; however, during cell passage, obvious cytopathic effects (CPEs) were observed under light microscopy. Round inclusion body (IB)-like particles and a few irregular larger IB-like particles were found in the cytoplasm of NTU-LY-1 cells. To further investigate the CPEs in NTU-LY-1 cells, reverse transcription–PCR screening of cytoplasmic polyhedrosis virus, infectious flacherie virus and Perina nuda virus (PnV) was performed. The results indicated that NTU-LY-1 cells were persistently infected with PnV from days 1 to 15 of culture. NTU-LY-1 cells were further subjected to Western blotting and indirect immunofluorescence assays to confirm PnV infection. Based on transmission electronic microscopy observations, the CPEs in NTU-LY-1 cells were different from those in the PnV-homologous NTU-Pn-HF cell line (a cell line of P. nuda). Membrane-bound IBs were found in newly seeded NTU-LY-1 cells. Lysed NTU-LY-1 cells formed many microsomes after 15 days of culture, and three different types of IBs could be distinguished among the microsomes, including membrane-bound, fibril-bound and sheath-bound IBs. After 15 days of culture, the cytoplasm of un-lysed NTU-LY-1 cells contained zipper-like arrays of non-membrane-bound virogenic stroma and parallel rows of viral particles that were restricted by two membranes. These viral particles were purified and showed a PnV morphology. This is the first study to show persistent PnV infection in a heterologous cell line. The findings suggest a new strategy for microbial control; moreover, the findings could support the development of a model for membrane-associated replication of positive-sense single-stranded RNA viruses.

Simulation of flow and mixing for highly viscous fluid in a twin screw extruder with a conveying element using parallelized smoothed particle hydrodynamics

For simulation of the flow in twin screw extrusion, there are several challenges for mesh-based methods, such as complex geometry, moving boundary and partially filled regions. Smooth particle hydrodynamics (SPH) is a kind of meshless particle algorithm that is very suitable in the simulation of these problems. However, due to the high viscosity of polymer limits the time step, the simulation efficiency of SPH is very low. In this paper, we used a parallelized SPH algorithm based on graphics processing units (GPU) to simulate the full filled and partially filled flows of highly viscous fluid in a conveying element. Firstly, in order to verify the results of the SPH simulation, the results for the full filled case simulated by SPH, FEM and FVM were compared and it showed that they are consistent with each other. Consequently, we investigated the mixing process of two kinds of color particles in the conveying element respectively for full filled and half filled cases. We found a better mixing occurs in some regions with the increase of revolution. We also tracked the motion trajectory of two rows of particles and analyze the mixing behavior of these particles in depth. Our results show that by using this parallelized SPH, the computational efficiency can be greatly improved while the hardware cost is low. This work laid a foundation for further study of real extrusion process.

Wave transmission across surface interfaces in lattice structures

Within the lattice dynamics formulation, we present an exact solution for anti-plane surface waves in a square lattice strip with a surface row of material particles of two types separated by a linear interface. The considered problem is a discrete analog of an elastic half-space with surface stresses modelled through the simplified Gurtin–Murdoch model, where we have an interfacial line separating areas with different surface elastic properties. The main attention is paid to the transmittance and the reflectance of a wave across the interface. The presented results shed a light on the influence on surface waves of surface inhomogeneity in surface elastic properties such as grain and subgrain boundaries.

Anti-plane surface waves in media with surface structure: Discrete vs. continuum model

We present a comparison of the dispersion relations derived for anti-plane surface waves using the two distinct approaches of the surface elasticity vis-a-vis the lattice dynamics. We consider an elastic half-space with surface stresses described within the Gurtin–Murdoch model, and present a formulation of its discrete counterpart that is a square lattice half-plane with surface row of particles having mass and elastic bonds different from the ones in the bulk. As both models possess anti-plane surface waves we discuss similarities between the continuum and discrete viewpoint. In particular, in the context of the behaviour of phase velocity, we discuss the possible characterization of the surface shear modulus through the parameters involved in lattice formulation.

Geometric model of sintering of unequal-sized particles

Sintering is a materials processing method generally applied to high-melting point materials. Performance of a sintered product is influenced by its microstructure. The effects of processing parameters, such as sintering time and temperature, on the microstructure are usually understood with the help of geometric models. Most geometric models assume equal-sized particles in the powders. This assumption neglects coarsening by inter-particle mass transport and grain boundary migration. Moreover, this assumption is seldom true in practice; because, coarsening and grain boundary migration always occur during sintering, especially during sintering of nanosized and ultrafine particles. In this paper, we present a model to quantitatively describe the sintering process of powders with unequal-sized particles. The present model relies on calculations of diffusional flux for neck growth and coarsening during sintering. In addition, we derive an expression to describe the grain boundary migration. We apply the model to predict the sintering behaviour of two equal-sized particles, two unequal-sized particles and a row of unequal-sized particles. We validate the model by comparing with Kingery and Berg's model, phase field simulations, and sintering experiments. The comparison showed that the model can be used to predict the sintering process with a reasonable accuracy.

Two classes of short intraflagellar transport train with different 3D structures are present in Chlamydomonas flagella.

Intraflagellar transport (IFT) is responsible for the bidirectional trafficking of molecular components required for the elongation and maintenance of eukaryotic cilia and flagella. Cargo is transported by IFT 'trains', linear rows of multiprotein particles moved by molecular motors along the axonemal doublets. We have previously described two structurally distinct categories of 'long' and 'short' trains. Here, we analyse the relative number of these trains throughout flagellar regeneration and show that long trains are most abundant at the beginning of flagellar growth whereas short trains gradually increase in number as flagella elongate. These observations are incompatible with the previous hypothesis that short trains are derived solely from the reorganization of long trains at the flagellar tip. We demonstrate with electron tomography the existence of two distinct ultrastructural organizations for the short trains, we name these 'narrow' and 'wide', and provide the first 3D model of the narrow short trains. These trains are characterized by tri-lobed units, which repeat longitudinally every 16 nm and contact protofilament 7 of the B-tubule. Functional implications of the new structural evidence are discussed.

Collective ratchet effects and reversals for active matter particles on quasi-one-dimensional asymmetric substrates

Using computer simulations, we study a two-dimensional system of sterically interacting self-mobile run-and-tumble disk-shaped particles with an underlying periodic quasi-one-dimensional asymmetric substrate, and show that a rich variety of collective active ratchet behaviors arise as a function of particle density, activity, substrate period, and the maximum force exerted by the substrate. The net dc drift, or ratchet transport flux, is nonmonotonic since it increases with increased activity but is diminished by the onset of self-clustering of the active particles. Increasing the particle density decreases the ratchet transport flux for shallow substrates but increases the ratchet transport flux for deep substrates due to collective hopping events. At the highest particle densities, the ratchet motion is destroyed by a self-jamming effect. We show that it is possible to realize reversals of the direction of the net dc drift in the deep substrate limit when multiple rows of active particles can be confined in each substrate minimum, permitting emergent particle-like excitations to appear that experience an inverted effective substrate potential. We map out a phase diagram of the forward and reverse ratchet effects as a function of the particle density, activity, and substrate properties.

Ultrastructural studies of Intraflagellar Transport trains in Chlamydomonas reinhardtii suggest a revision of the current model for IFT trafficking in the flagellar compartment

Intraflagellar Transport (IFT) is the molecular process responsible for the active bidirectional trafficking of structural and functional components that occurs in the flagellar compartment of eukaryotic cells. Flagellar components undergo a constant turnover at flagellar tip and multiple evidences indicate that flagellar elongation, maintenance and reabsorption depend on the correct balance between anterograde and retrograde trafficking. IFT particles are formed by >22 polypeptides assembled into two subcomplexes, A and B, and are moved bidirectionally along the outer surface of axonemal doublets as linear rows of IFT particles, for which we proposed the term train. Anterograde IFT trains are moved by kinesin II and carry to flagellar tip the retrograde motor cytoplasmic dynein 1b, responsible for retrograde IFT. In a previous study carried out on Chlamydomonas flagella we identified two types of IFT trains we named long and short trains, each characterized by a specific ultrastructure and a definite internal repeat, and proposed that long, less compact trains could represent anterograde IFT while the short, more compact trains could be retrograde. To challenge such model, we monitored by transmission electron microscopy the IFT trains expressed both in wt regenerating flagella and during flagellar reabsorption induced in the ts mutant pf1-fla10. We also progressed in our electron tomographic 3D modeling of short IFT trains. Our data suggest that long IFT trains are not the only anterograde IFT component. Rather, anterograde IFT is contributed also by a subclass of short trains that is expressed in a flagellar length-dependent fashion.

Morphological evidence for a permeability barrier in the testis and spermatic duct ofGymnotus carapo(Teleostei: Gymnotidae)

Cell-cell interactions play essential roles in the regulation of gametogenesis. The involvement of junctional complexes in permeability barriers, for example, provides structural and physiological support for male germ-cell development. This study describes morphological characteristics of the reproductive system of Gymnotus carapo, a neo-tropical freshwater fish widely distributed in South and Central America, focusing on the detection of permeability barriers using morphological and biochemical approaches. Ultrastructural analysis of testes treated with the lanthanum nitrate exclusion technique showed that the tracer penetrated the interstitial compartment of the testis, surrounding and appearing within cysts containing spermatogonia and spermatocytes in early stages of meiosis, but was not detected in the spermatid cysts or inside the lumen of spermatogenic tubules. These results suggest the presence of a permeability barrier that is stabilized after meiosis is completed and serves to protect the haploid cells from the vascular system. In the spermatic-duct region, the tracer was obstructed near the lumen of the duct. Junctional complexes and focal tight junctions between adjacent cells were observed in the testis and spermatic duct. Freeze-fracture methods indeed confirmed the presence of tight junctions, which were visualized as parallel rows of individual particles between adjacent cells. More evidence supporting the existence of a permeability barrier was gathered from differences observed in the electrophoretic protein profiles of testis and spermatic-duct fluids compared to blood plasma. Together, these observations demonstrate the existence of a permeability barrier formed by tight junctions in the testis and spermatic duct of G. carapo.

Theory of Sintering in Presence of Pressure and Torque

Experiments like the sintering of single crystal spheres on a single crystal plate, the sintering of particle rows or 2D arrangements of powder particles as well as recent experiments by in‐situ Synchrotron Computer Tomography show movements of entire particles beyond the particle center approach described by the well‐known two particle model. Several attempts exist to describe these cooperative material transport processes by an analytical model of the sintering of two wires by grain‐boundary diffusion. The described mechanisms are either rotations caused by asymmetric sintering necks or the application of a torque or changes of the rate of sintering by a normal stress in the sintering neck. To develop a more comprehensive model of sintering, volume diffusion was included in addition to grain‐boundary diffusion and all driving forces mentioned above are used. As a result, the influence of normal stress, torque, and ratio of the sintering neck curvatures on the stress along the contact grain boundary, the diffusion current and the rate of removal along the sintering neck are shown. The contribution of volume diffusion and grain‐boundary diffusion to sintering was also investigated, confirming that a large sintering neck and high sintering temperatures result in a dominating influence of volume diffusion. The shrinkage rate and the rate of rotation provided by this model were used in a subsequent numerical simulation. For several stages of sintering, the influence of particle radius and sintering temperature on the rolling angle of sintering particles was analyzed in 2D. Furthermore, the impact of external normal stresses on the rolling angle is shown. The simulation model gives a surprising result: The speed of wire rolling shows the same behavior as sintering kinetics—a high speed for small particles, small sintering neck diameter and high temperature, but the cumulated rolling angle for small particles is low compared to that for larger particles. Finally, a new approach for simulating particle rotation and rolling caused by torques due to grain‐boundary anisotropy in 3D is presented, which is based on an existing Discrete Element Method sintering model. This model is applied to a row of particles confirming previous observations from the 2D sintering simulations.

Nonlinear transport of the Wigner crystal in symmetric and asymmetric FET-like structures

When floating on a two-dimensional surface of superfluid 4He, electrons arrange themselves in two-dimensional crystalline structure known as Wigner crystal. In channels, the boundaries interfere the crystalline order and in case of very narrow channels one observes a quasi-one-dimensional (quasi-1D) Wigner crystal formed by just a few rows of electrons and, ultimately, one row in the “quantum wire” regime. Recently, the “quantum wire” regime was accessed experimentally [D.G. Rees, H. Totsuji, K. Kono, Phys. Rev. Lett. 108, 176801 (2012)] resulting in unusual transport phenomena such as, e.g., oscillations in the electron conductance. Using molecular dynamics simulations, we study the nonlinear transport of electrons in channels with various types of constrictions: single and multiple symmetric and asymmetric geometrical constrictions with varying width and length, and saddle-point-type potentials with varying gate voltage. In particular, we analyze the average particle velocity of the particles and the corresponding electron current versus the driving force or the gate voltage. We have revealed a significant difference in the dynamics for long and short constrictions: The oscillations of the average velocity of the particles for the systems with short constrictions exhibit a clear correlation with the transitions between the states with different numbers of rows of particles; on the other hand, for the systems with longer constrictions these oscillations are suppressed. The obtained results qualitatively agree with the experimental observations. Next, we propose a FET-like structure that consists of a channel with asymmetric constrictions. We show that applying a transverse bias results either in increase of the average particle velocity or in its suppression thus allowing a flexible control tool over the electron transport. The advantage of the asymmetric FET is that it does not have a gate and it allows an easy control of relatively large electron flow. Furthermore, the asymmetric device can be used for rectification of an ac-driven electron flow. Our results bring important insights into the dynamics of electrons floating on the surface of superfluid 4He in channels with constrictions and allow the effective control over the electron transport.

Nonlinear Transport of the Wigner Crystal on Superfluid 4He in a Quasi-One-Dimensional Channel

We study transport properties of a Wigner crystal driven by an external force on the surface of superfluid 4 He , in the "quantum wire" regime, i.e. in the quasi-one-dimensional (quasi-1D) case when a typical width of the channel is comparable to the inter-electron separation. Utilizing molecular dynamics simulations, we investigate the electronic transport through the channels with different constrictions: (i) geometrical constrictions with varying shape and size, and (ii) a saddle-point potential with varying gate voltage. The average particle velocity has been calculated as a function of the driving force or gate voltage. We have revealed a significant difference in the dynamical behavior for long and short constrictions. In particular, we found that the oscillations of the average particle velocity in channels with short constrictions exhibit a clear correlation with the transitions between the states with different numbers of rows of particles in the constriction, while for channels with longer constrictions these oscillations are suppressed. The obtained results are in agreement with the recent experimental observations,1 and thus bring new important insights into the dynamics of electrons floating on the surface of superfluid 4 He in channels with constrictions. [Formula: see text] Special Issue Comments: This article presents results on the dynamics of electrons moving on the surface of liquid helium in narrow channels with constrictions, with a focus on the "quantum wire", i.e. single file, regime. This article is connected to the Special Issue articles about advanced statistical properties in single file dynamics39 and the experiments on liquid helium.40

Macroscopic Limits of Microscopic Models

Many physical systems are comprised of several discrete elements, the equations of motion of each element being known. If the system has a large number of degrees of freedom, it may be possible to treat it as a continuous system. In this event, one might wish to derive the equations of motion of the continuous (macroscopic) system by taking a suitable limit of the equations governing the discrete (microscopic) system. The classical example of this involves a row of particles with each particle connected to its nearest neighbor by a linear spring, its continuum counterpart being a linearly elastic bar; see Fig. 1. In a typical undergraduate engineering subject on, for example, Dynamics, the transition from a discrete system to a continuous system is usually carried out through a formal Taylor expansion of the terms of the discrete model about some reference configuration. The aim of this paper is to draw attention to the fact that a macroscopic model derived in this way should be examined critically in order to confirm that it provides a faithful representation of the underlying microscopic model. We use a specific (striking) example to make this point. In this example, a simple solution of the discrete model can be stable or unstable depending on the state of the system. However, the corresponding solution of the continuous system is always unstable! We go on to show how the dispersion relations of the two models can be used to identify the source of the discrepancy and to suggest how one might modify the continuous model.