Atomic-scale friction of a tungsten tip on a graphite surface

Molecular dynamics simulations on the frictional behavior of a perfluoropolyether film sandwiched between diamond-like-carbon coatings

We perform molecular dynamics simulations to investigate the nanoscale frictional behavior of a perfluoropolyether (PFPE) film sandwiched between two diamond-like-carbon (DLC) coatings. We show that the PFPE films behave like a solid and can perform either a motion-station movement or a continuous motion with fluctuating velocities. The former movement is caused by the alternating stick and slip at the two individual interfaces, while the latter is due to the dynamic sliding motions simultaneously occurring at both interfaces. We reveal that these motion characteristics are governed by the competition between the two interfacial adhesion energies, which are strongly affected by the thermal vibrations and interface roughness fluctuations. We also find that the Amonton's law modified by incorporating the adhesion effect can be used to describe the mean friction traction vs normal pressure relation, but large fluctuations are present at low contact pressures. The magnitude of atomic level friction forces at the interface is found to be highly nonuniform. The directions of atomic level friction forces can even be opposite. With increasing the normal pressure, the nonuniformity of atomic level friction forces decreases first and then increases again. This change can be explained by the concurrent effects from the large difference in material stiffness and the changes in surface roughness under normal pressure. The present work reveals interesting insights into the sliding mechanisms in sandwiched structures and provides useful guidelines for the design of nanoscale lubricant systems.

Friction coefficient dependence on electrostatic tribocharging

Friction between dielectric surfaces produces patterns of fixed, stable electric charges that in turn contribute electrostatic components to surface interactions between the contacting solids. The literature presents a wealth of information on the electronic contributions to friction in metals and semiconductors but the effect of triboelectricity on friction coefficients of dielectrics is as yet poorly defined and understood. In this work, friction coefficients were measured on tribocharged polytetrafluoroethylene (PTFE), using three different techniques. As a result, friction coefficients at the macro- and nanoscales increase many-fold when PTFE surfaces are tribocharged, but this effect is eliminated by silanization of glass spheres rolling on PTFE. In conclusion, tribocharging may supersede all other contributions to macro- and nanoscale friction coefficients in PTFE and probably in other insulating polymers.

Triboelectricity in insulating polymers: evidence for a mechanochemical mechanism

Transfer of reaction products formed on the surfaces of two mutually rubbed dielectric solids makes an important if not dominating contribution to triboelectricity. New evidence in support of this statement is presented in this report, based on analytical electron microscopy coupled to electrostatic potential mapping techniques. Mechanical action on contacting surface asperities transforms them into hot-spots for free-radical formation, followed by electron transfer producing cationic and anionic polymer fragments, according to their electronegativity. Polymer ions accumulate creating domains with excess charge because they are formed at fracture surfaces of pulled-out asperities. Another factor for charge segregation is the low polymer mixing entropy, following Flory and Huggins. The formation of fractal charge patterns that was previously described is thus the result of polymer fragment fractal scatter on both contacting surfaces. The present results contribute to the explanation of the centuries-old difficulties for understanding the “triboelectric series” and triboelectricity in general, as well as the dissipative nature of friction, and they may lead to better control of friction and its consequences.

Internal friction characteristic and analysis of in-plane natural frequency of trilayer complexes formed from graphenes and boron nitride nanosheets

The internal friction and in-plane natural frequency of a trilayer complex formed by a monolayer graphene sandwiched in the bilayer of boron nitride nanosheets (BN/G/BN) and graphenes (G/G/G) are studied by using molecular dynamics.

Friction, tribochemistry and triboelectricity: recent progress and perspectives

Mechanochemical reactions during polymer friction or contact produce ionic fragments distributed on positive and negative domains at both surfaces.

Artificial intelligence in nanotechnology

During the last decade there has been increasing use of artificial intelligence tools in nanotechnology research. In this paper we review some of these efforts in the context of interpreting scanning probe microscopy, the study of biological nanosystems, the classification of material properties at the nanoscale, theoretical approaches and simulations in nanoscience, and generally in the design of nanodevices. Current trends and future perspectives in the development of nanocomputing hardware that can boost artificial-intelligence-based applications are also discussed. Convergence between artificial intelligence and nanotechnology can shape the path for many technological developments in the field of information sciences that will rely on new computer architectures and data representations, hybrid technologies that use biological entities and nanotechnological devices, bioengineering, neuroscience and a large variety of related disciplines.

Atomic roughness enhanced friction on hydrogenated graphene

Atomic friction on hydrogenated graphene is investigated using molecular dynamics simulations. Hydrogenation is found to increase friction significantly, and the atomic-level information provided by the simulations reveals that atomic roughness induced by hydrogenation is the primary cause of the friction enhancement. Other proposed mechanisms, specifically adhesion and rigidity, are excluded based on the simulation results and analyses performed using the Prandtl-Tomlinson model. In addition, it is found that friction does not monotonically increase with hydrogen coverage on the graphene surface; instead, a maximum friction is observed at a hydrogen coverage between 5 and 10%.

Atomic structure affects the directional dependence of friction

Friction between two objects can be understood by the making, stretching, and breaking of thousands of atomic-scale asperities. We have probed single atoms in a nonisotropic surface [the H-terminated Si(100) surface] with a lateral force microscope operating in noncontact mode. We show that these forces are measurably different, depending upon the direction. Experimentally, these differences are observable in both the line profiles and the maximum stiffnesses. Density functional theory calculations show a concerted motion of the whole Si dimer during the tip-sample interaction. These results demonstrate that on an asperity-by-asperity basis, the surface atomic structure plays a strong role in the directional dependence of friction.

A single-molecule stretching method for lateral and normal AFM lever calibration

A novel method for quantitative lateral force measurement (LFM) calibration has been developed. Using a single-molecule spectroscopy approach it is possible to calibrate the AFM levers for both lateral and normal spring constants with a single image scan. Moreover, our method does not involve tip modifications. Dextran molecules were chosen for testing our calibration procedure due to their characteristic plateau feature in the force-elongation curve which enables an easy identification of single-molecule stretching events. Using a non-standard (tilted) geometry of AFM scanning, it is possible to observe different components of the stretching force on both normal and lateral force signals. These signals can be further compared to the values obtained by standard (normal) spectroscopic measurements. The values of the normal spring constant obtained with our method are in good agreement with the results obtained from the method exploiting the energy equipartition theorem. The statistical analysis shows that the approach proposed in our paper gives reproducible results of the lateral sensitivity with a relative standard deviation less than 15%.

Puckering stick-slip friction induced by a sliding nanoscale contact

An atomic force microscope reveals that the sliding of a nanotip on a graphite surface occurs through a nanoscale stick-slip mechanism. The angle between the sliding direction and a stiff crystallographic axis determines the periodicity of the slip events defining domains of various friction properties. The experimental data are interpreted using the reaction rate theory, with the energy barrier driven by a local deformation of the surface and a thermally activated relaxation.

Optimal geometry for a quartz multipurpose SPM sensor

We propose a geometry for a piezoelectric SPM sensor that can be used for combined AFM/LFM/STM. The sensor utilises symmetry to provide a lateral mode without the need to excite torsional modes. The symmetry allows normal and lateral motion to be completely isolated, even when introducing large tips to tune the dynamic properties to optimal values.

Measurement of the friction between single polystyrene nanospheres and silicon surface using atomic force microscopy

In the present work, the individual nanoparticles have been manipulated on a silicon surface, using atomic force microscope (AFM) techniques. As a model system, near-spherical polystyrene nanoparticles with radii from 28.85 nm to 228.2 nm were deposited on a nanosmooth silicon wafer. Experiments demonstrated that when the normal force is above a threshold load, nanoparticles could steadily be pushed by the tip of the AFM along the defined pathway. The tests allow us to quantitatively study the interfacial friction between the nanoparticle and the surface. It was found that the friction could be affected by various factors such as the load, the particle size, and the surface treatment. The results showed that the friction between particles and substrate is proportional to the two-third power of the radius, which is in agreement with the Hertzian theory. It can also be seen that the ratio between the kinetic and the static friction was slightly changed from 0.3 to 0.6, depending on the size of the particles. However, the value of the ratio was little affected by other factors such as the particles' location, the tip normal force and the surface modification. The results provided new insights into the intriguing friction phenomenon on the nanoscale.

Energy loss triggered by atomic-scale lateral force

We perform bimodal atomic force microscopy measurements on a Br-doped NaCl (001) surface to investigate the mechanisms behind frequency shift and energy dissipation contrasts. The peculiar pattern of the dissipated energy in the torsional channel, related to frictional processes, is increased at the positions of Br impurities, otherwise indistinguishable from Cl ions in the other measured channels. Our simulations reveal how the energy dissipates by the rearrangement of the tip apex and how the process is ultimately governed by lateral forces. Even the slightest change in lateral forces, induced by the presence of a Br impurity, is enough to trigger the apex reconstruction more often, thus increasing the dissipation contrast; the predicted dissipation pattern and magnitude are in good quantitative agreement with the measurements.

Superlubric sliding of graphene nanoflakes on graphene

The lubricating properties of graphite and graphene have been intensely studied by sliding a frictional force microscope tip against them to understand the origin of the observed low friction. In contrast, the relative motion of free graphene layers remains poorly understood. Here we report a study of the sliding behavior of graphene nanoflakes (GNFs) on a graphene surface. Using scanning tunneling microscopy, we found that the GNFs show facile translational and rotational motions between commensurate initial and final states at temperatures as low as 5 K. The motion is initiated by a tip-induced transition of the flakes from a commensurate to an incommensurate registry with the underlying graphene layer (the superlubric state), followed by rapid sliding until another commensurate position is reached. Counterintuitively, the average sliding distance of the flakes is larger at 5 K than at 77 K, indicating that thermal fluctuations are likely to trigger their transitions from superlubric back to commensurate ground states.

Scanning tunneling microscope-quartz crystal microbalance study of temperature gradients at an asperity contact

Investigations of atomic-scale friction frequently involve setups where a tip and substrate are initially at different temperatures. The temperature of the sliding interface upon contact has thus become a topic of interest. A method for detecting initial tip-sample temperature differences at an asperity contact is described, which consists of a scanning tunneling microscope (STM) tip in contact with the surface electrode of a quartz crystal microbalance (QCM). The technique makes use of the fact that a QCM is extremely sensitive to abrupt changes in temperature. In order to demonstrate the technique's capabilities, QCM frequency shifts were recorded for varying initial tip-substrate temperature differences as an STM tip was brought into and out of contact. The results are interpreted within the context of a recent model for thermal heat conduction at an asperity contact, and it is concluded that the transient frequency response is attributable to small changes in temperature close to the region of contact rather than a change in the overall temperature of the QCM itself. For the assumed model parameters, the results moreover reveal substantial temperature discontinuities at the boundary between the tip and the sample, for example, on the order of 10-15 °C for initial temperature differences of 20 °C.

Friction measurement on free standing plates using atomic force microscopy

A method is introduced to measure friction on small, free standing objects, specifically microfabricated silicon plates, based on atomic force microscopy (AFM). An AFM tip is brought into contact with the plate resting on a substrate. The substrate is displaced laterally and, provided the AFM tip does not slide over the plate, the twisting of the AFM cantilever is used to measure the friction of the underlying plate-substrate interface. The method can measure nano-Newton to micro-Newton forces (both friction and applied load) and provides a means to measure friction of macroscopic structures at low load.

Real-time transmission electron microscope observation of nanofriction at a single Ag asperity

The observation of nanoscale deformation in real time is a significant step toward understanding the mechanisms of friction and lubrication. Our experimental setup of a micromachine combined with a transmission electron microscope allowed us to measure the deformation, force and cross-sectional area of a single Ag asperity during shear testing. The experimental results provided insight into one of the parameters that determines the frictional coefficient. Furthermore, we demonstrated that the energy loss associated with a shear fracture event is strongly correlated with the increase in total surface energy of the two surfaces formed here after the fracture of the nanocontact.

Control of nanoscale friction on gold in an ionic liquid by a potential-dependent ionic lubricant layer

The lubricating properties of an ionic liquid on gold surfaces can be controlled through application of an electric potential to the sliding contact. A nanotribology approach has been used to study the frictional behavior of 1-butyl-1-methylpyrrolidinium tris(pentafluoroethyl) trifluorophosphate ([Py(1,4)]FAP) confined between silica colloid probes or sharp silica tips and a Au(111) substrate using atomic force microscopy. Friction forces vary with potential because the composition of a confined ion layer between the two surfaces changes from cation-enriched (at negative potentials) to anion-enriched (at positive potentials). This offers a new approach to tuning frictional forces reversibly at the molecular level without changing the substrates, employing a self-replenishing boundary lubricant of low vapor pressure.

Exploring atomic-scale lateral forces in the attractive regime: a case study on graphite (0001)

A non-contact atomic force microscopy-based method has been used to map the static lateral forces exerted on an atomically sharp Pt/Ir probe tip by a graphite surface. With measurements carried out at low temperatures and in the attractive regime, where the atomic sharpness of the tip can be maintained over extended time periods, the method allows the quantification and directional analysis of lateral forces with piconewton and picometer resolution as a function of both the in-plane tip position and the vertical tip-sample distance, without limitations due to a finite contact area or to stick-slip-related sudden jumps of tip apex atoms. After reviewing the measurement principle, the data obtained in this case study are utilized to illustrate the unique insight that the method offers. In particular, the local lateral forces that are expected to determine frictional resistance in the attractive regime are found to depend linearly on the normal force for small tip-sample distances.

Accurate and precise calibration of AFM cantilever spring constants using laser Doppler vibrometry

Accurate cantilever spring constants are important in atomic force microscopy both in control of sensitive imaging and to provide correct nanomechanical property measurements. Conventional atomic force microscope (AFM) spring constant calibration techniques are usually performed in an AFM. They rely on significant handling and often require touching the cantilever probe tip to a surface to calibrate the optical lever sensitivity of the configuration. This can damage the tip. The thermal calibration technique developed for laser Doppler vibrometry (LDV) can be used to calibrate cantilevers without handling or touching the tip to a surface. Both flexural and torsional spring constants can be measured. Using both Euler-Bernoulli modeling and an SI traceable electrostatic force balance technique as a comparison we demonstrate that the LDV thermal technique is capable of providing rapid calibrations with a combination of ease, accuracy and precision beyond anything previously available.

Stochastic noise in atomic force microscopy

Having reached the quantum and thermodynamic limits of detection, atomic force microscopy (AFM) experiments are routinely being performed at the fundamental limit of signal to noise. A critical understanding of the statistical properties of noise leads to more accurate interpretation of data, optimization of experimental protocols, advancements in instrumentation, and new measurement techniques. Furthermore, accurate simulation of cantilever dynamics requires knowledge of stochastic behavior of the system, as stochastic noise may exceed the deterministic signals of interest, and even dominate the outcome of an experiment. In this article, the power spectral density (PSD), used to quantify stationary stochastic processes, is introduced in the context of a thorough noise analysis of the light source used to detect cantilever deflections. The statistical properties of PSDs are then outlined for various stationary, nonstationary, and deterministic noise sources in the context of AFM experiments. Following these developments, a method for integrating PSDs to provide an accurate standard deviation of linear measurements is described. Lastly, a method for simulating stochastic Gaussian noise from any arbitrary power spectral density is presented. The result demonstrates that mechanical vibrations of the AFM can cause a logarithmic velocity dependence of friction and induce multiple slip events in the atomic stick-slip process, as well as predicts an artifactual temperature dependence of friction measured by AFM.

Instrumentation for dual-probe scanning near-field optical microscopy

To investigate local carrier motions, we developed a dual-probe scanning near-field optical microscope (SNOM) with two fiber probes where one is for photoexcitation and the other is for light collection. This instrumentation is based on two important techniques: the design of probe structures and distance control between the sample surface and the two probes. A finite-difference time-domain method numerically analyzed and optimized the design for high efficiency photoexcitation and light collection, while a dual band modulation realized distance control. Real time detection of the oscillations of the probe tips using different frequencies independently controls the distance between the probe tip and the sample surface as well as the distance between the two probes. Thus, the collection probe can be scanned around an illumination probe without destroying the probe tips. To demonstrate our SNOM, we performed photoluminescence spectroscopy under the dual-probe configuration and observed carrier motions in an InGaN quantum well.

Molecular dynamics simulation of atomic-scale frictional behavior of corrugated nano-structured surfaces

Surface morphology is one of the critical parameters that affect the frictional behavior of two contacting bodies in relative motion. It is important because the real contact area as well as the contact stiffness is dictated by the micro- and nano-scale geometry of the surface. In this regard, the frictional behavior may be controlled by varying the surface morphology through nano-structuring. In this study, molecular dynamics simulations were conducted to investigate the effects of contact area and structural stiffness of corrugated nano-structures on the fundamental frictional behavior at the atomic-scale. The nano-structured surface was modeled as an array of corrugated carbon atoms with a given periodicity. It was found that the friction coefficient of the nano-structured surface was lower than that of a smooth surface under specific contact conditions. The effect of applied load on the friction coefficient was dependent on the size of the corrugation. Furthermore, stiffness of the nano-structure was identified to be an important variable in dictating the frictional behavior.

Two-step controllable electrochemical etching of tungsten scanning probe microscopy tips

Dynamic electrochemical etching technique is optimized to produce tungsten tips with controllable shape and radius of curvature of less than 10 nm. Nascent features such as "dynamic electrochemical etching" and reverse biasing after "drop-off" are utilized, and "two-step dynamic electrochemical etching" is introduced to produce extremely sharp tips with controllable aspect ratio. Electronic current shut-off time for conventional dc "drop-off" technique is reduced to ∼36 ns using high speed analog electronics. Undesirable variability in tip shape, which is innate to static dc electrochemical etching, is mitigated with novel "dynamic electrochemical etching." Overall, we present a facile and robust approach, whereby using a novel etchant level adjustment mechanism, 30° variability in cone angle and 1.5 mm controllability in cone length were achieved, while routinely producing ultra-sharp probes.

Controlling nanoscale friction through the competition between capillary adsorption and thermally activated sliding

We demonstrate measurement and control of nanoscale single-asperity friction by using cantilever probes featuring an in situ solid-state heater in contact with silicon oxide substrates. The heater temperature was varied between 25 and 790 °C. By using a low thermal conductivity sample, silicon oxide, we are able to vary tip temperatures over a broad range from 25 ± 2 to 255 ± 25 °C. In ambient atmosphere with ∼30% relative humidity, the control of friction forces was achieved through the formation of a capillary bridge whose characteristics exhibit a strong dependence on temperature and sliding speed. The capillary condensation is observed to be a thermally activated process, such that heating in ambient air caused friction to increase due to the capillary bridge nucleating and growing. Above tip temperatures of ∼100 ± 10 °C, friction decreased drastically, which we attribute to controllably evaporating water from the contact at the nanoscale. In contrast, in a dry nitrogen atmosphere, friction was not affected appreciably by temperature changes. In the presence of a capillary, friction decreases at higher sliding speeds due to disruption of the capillary; otherwise, friction increases in accordance with the predictions of a thermally assisted sliding model. In ambient atmospheres, the rate of increase of friction with sliding speed at room temperature is sufficiently strong that the friction force changes from being smaller than the response at 76 ± 8 °C to being larger. Thus, an appropriate change in temperature can cause friction to increase at one sliding speed, while it decreases at another speed.

Frictional figures of merit for single layered nanostructures

We determine the frictional figures of merit for a pair of layered honeycomb nanostructures, such as graphane, fluorographene, MoS2 and WO2 moving over each other, by carrying out ab initio calculations of interlayer interaction under constant loading force. Using the Prandtl-Tomlinson model we derive the critical stiffness required to avoid stick-slip behavior. We show that these layered structures have low critical stiffness even under high loading forces due to their charged surfaces repelling each other. The intrinsic stiffness of these materials exceeds critical stiffness and thereby the materials avoid the stick-slip regime and attain nearly dissipationless continuous sliding. Remarkably, tungsten dioxide displays a much better performance relative to others and heralds a potential superlubricant. The absence of mechanical instabilities leading to conservative lateral forces is also confirmed directly by the simulations of sliding layers.

Anti-drift and auto-alignment mechanism for an astigmatic atomic force microscope system based on a digital versatile disk optical head

In this work, an anti-drift and auto-alignment mechanism is applied to an astigmatic detection system (ADS)-based atomic force microscope (AFM) for drift compensation and cantilever alignment. The optical path of the ADS adopts a commercial digital versatile disc (DVD) optical head using the astigmatic focus error signal. The ADS-based astigmatic AFM is lightweight, compact size, low priced, and easy to use. Furthermore, the optical head is capable of measuring sub-atomic displacements of high-frequency AFM probes with a sub-micron laser spot (~570 nm, FWHM) and a high-working bandwidth (80 MHz). Nevertheless, conventional DVD optical heads suffer from signal drift problems. In a previous setup, signal drifts of even thousands of nanometers had been measured. With the anti-drift and auto-alignment mechanism, the signal drift is compensated by actuating a voice coil motor of the DVD optical head. A nearly zero signal drift was achieved. Additional benefits of this mechanism are automatic cantilever alignment and simplified design.

Identifying the mechanisms of polymer friction through molecular dynamics simulation

Mechanisms governing the tribological behavior of polymer-on-polymer sliding were investigated by molecular dynamics simulations. Three main mechanisms governing frictional behavior were identified. Interfacial "brushing" of molecular chain ends over one another was observed as the key contribution to frictional forces. With an increase of the sliding speed, fluctuations in frictional forces reduced in both magnitude and periodicity, leading to dynamic frictional behavior. While "brushing" remained prevalent, two additional irreversible mechanisms, "combing" and "chain scission", of molecular chains were observed when the interfaces were significantly diffused.

Friction force spectroscopy as a tool to study the strength and structure of salivary films

In this work, we employ atomic force microscopy based friction force spectroscopy to study the strength and structure of salivary films. Specifically, films formed on model hydrophobic (methylated silica) and hydrophilic (clean silica) substrata have been studied in water at pHs in the range 3.3-7. Results reveal that films formed on both types of substrata can be described in terms of two different fractions, with only one of them being able to diffuse along the underlying substrata. We also show how the protective function of the films is reduced when the pH of the surrounding medium is lowered. Specifically, lowering of pH causes desorption of some components of the films formed on hydrophobic methylated surfaces, leading to weaker layers. In contrast, at low pHs, saliva no longer forms a layer on hydrophilic silica surfaces. Instead, an inhomogeneous distribution of amorphous aggregates is observed. Our data also suggest that hydrophobic materials in the oral cavity might be more easily cleaned from adsorbed salivary films. Finally, reproducible differences are observed in results from experiments on films from different individuals, validating the technique as a tool for clinical diagnosis of the resistance to erosion of salivary films.

Rotational positioning system adapted to atomic force microscope for measuring anisotropic surface properties

The diverse atomic configurations induce the anisotropic surface properties. For investigating anisotropic phenomena, we developed a rotational positioning system adapted to atomic force microscope (AFM). This rotational positioning system is applied to revolve the measured sample to defined angular direction, and it composed of an inertial rotational stepper and a visual angular measurement. The inertial rotational stepper with diameter 30 mm and height 7.6 mm can be easily attached to the AFM-system built in any general optical microscope. Based on a clearance less bearing and the inertial driving method, its bidirectional angular resolution reaches 0.005° per step. For realizing a close-loop controlled angular positioning function, the visual measurement method is utilized. Through the feedback control, the angular positioning error is less than 0.01°. For verifying the system performance, we used it to investigate the anisotropic surface properties of graphite. Through a modified cantilever tip, the atomic-scale stick-slip, and the anisotropic friction phenomena can be distinctly detected.

Atomic-scale nanoindentation: detection and identification of single glide events in three dimensions by force microscopy

Indentation experiments on the nanometre scale have been performed by means of atomic force microscopy in ultra-high vacuum on KBr(100) surfaces. The surfaces yield in the form of discrete surface displacements with a typical length scale of 1 Å. These surface displacements are detected in both normal and lateral directions. Measurement of the lateral tip displacement requires a load-dependent calibration due to the load dependence of the effective lateral compliance. Correlation of the lateral and normal displacements for each glide event allow identification of the activated slip system. The results are discussed in terms of the resolved shear stress in indentation experiments and of typical results in atomistic simulations of nanometre-scale indentation.

Theoretical and computational investigation of flagellin translocation and bacterial flagellum growth

The bacterial flagellum is a self-assembling filament, which bacteria use for swimming. It is built from tens of thousands of flagellin monomers in a self-assembly process that involves translocation of the monomers through the flagellar interior, a channel, to the growing tip. Flagellum monomers are pumped into the filament at the base, move unfolded along the channel and then bind to the tip of the filament, thereby extending the growing flagellum. The flagellin translocation process, due to the flagellum maximum length of 20 μm, is an extreme example of protein transport through channels. Here, we derive a model for flagellin transport through the long confining channel, testing the key assumptions of the model through molecular dynamics simulations that also furnish system parameters needed for quantitative description. Together, mathematical model and molecular dynamics simulations explain why the growth rate of flagellar filaments decays exponentially with filament length and why flagellum growth ceases at a certain maximum length.