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Tada T, Persson BNJ. On the origin of the breakloose friction force. J Chem Phys 2025; 162:174709. [PMID: 40326598 DOI: 10.1063/5.0266065] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2025] [Accepted: 03/28/2025] [Indexed: 05/07/2025] Open
Abstract
We discuss the origin of the static or breakloose friction force and describe a kinetic effect that can manifest itself as an (effective) breakloose friction force. The kinetic effect is important if the kinetic friction force Fk(v) has a maximum at low sliding speed, say for v = vc. If the driving speed v is higher than vc, the friction force at the onset of slip will first increase rapidly to approximately Fk(vc) and then rapidly decrease to Fk(v), resulting in an effective breakloose friction force larger than the kinetic friction force. We present experimental results for sliding friction of a racer rubber tread compound and a passenger car tread compound, which exhibit very different breakloose friction as a result of this kinetic effect. We show that pre-slip in the contact makes the breakloose friction be less than the product of the true contact area and static shear stress. For elastically soft materials, in the absence of the kinetic effect described above, this may result in a breakloose friction force equal to the kinetic friction force. We present experimental results for a passenger car tread compound, which illustrate this effect.
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Affiliation(s)
- T Tada
- Sumitomo Rubber Industries, Research and Development, HQ. 2-1-1, Tsutsui-cho, Chuo-ku, Kobe 651-0071, Japan
| | - B N J Persson
- Peter Grünberg Institute (PGI-1), Forschungszentrum Jülich, 52425 Jülich, Germany
- State Key Laboratory of Solid Lubrication, Lanzhou Institute of Chemical Physics, Chinese Academy of Sciences, 730000 Lanzhou, China
- MultiscaleConsulting, Wolfshovener Str. 2, 52428 Jülich, Germany
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2
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Miyashita N, Persson BNJ. Dry and lubricated sliding friction for rubber on concrete: the role of surface energies. SOFT MATTER 2024; 20:7843-7853. [PMID: 39311896 DOI: 10.1039/d4sm00843j] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/10/2024]
Abstract
We study the influence of lubricant fluids (water-glycerol mixtures) on rubber sliding friction for two different rubber tread compounds on a concrete surface. We find that for the lubricated contacts the sliding friction below a critical velocity vc is similar to that of the dry contact, but for v > vc the friction drops fast with increasing sliding speed. We discuss the origin of this effect and show that it is not a "normal" mixed lubrication effect but depends on surface (or interfacial) energies.
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Affiliation(s)
- N Miyashita
- The Yokohama Rubber Company, 2-1 Oiwake, Hiratsuka, Kanagawa 254-8601, Japan
| | - B N J Persson
- Peter Grünberg Institute (PGI-1), Forschungszentrum Jülich, 52425, Jülich, Germany.
- State Key Laboratory of Solid Lubrication, Lanzhou Institute of Chemical Physics, Chinese Academy of Sciences, 730000 Lanzhou, China
- Multiscale Consulting, Wolfshovener str. 2, 52428 Jülich, Germany
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Miyashita N, Persson BNJ. Role of strain softening and viscoelastic memory for the rolling friction of two tire tread compounds. J Chem Phys 2024; 161:074702. [PMID: 39145561 DOI: 10.1063/5.0223550] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2024] [Accepted: 07/30/2024] [Indexed: 08/16/2024] Open
Abstract
Rolling friction is of great importance for many applications, such as tires and conveyor belts. We study the rolling friction for hard cylinders rolling on flat rubber sheets. The rolling friction depends on the number of rolling cycles, the rolling speed, and the temperature. We show that when the rubber is cooled down below the glass transition temperature, the deformations of the rubber surface are frozen-in, resulting in a non-flat rolling track where uphill and downhill rolling movements strongly affect the rolling force. The experimental data are analyzed using the Persson rolling friction theory; good agreement with the experiments is obtained when the non-linear (strain-softening) properties of the viscoelastic modulus are taken into account.
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Affiliation(s)
- N Miyashita
- The Yokohama Rubber Company, 2-1 Oiwake, Hiratsuka, Kanagawa 254-8601, Japan
| | - B N J Persson
- State Key Laboratory of Solid Lubrication, Lanzhou Institute of Chemical Physics, Chinese Academy of Sciences, 730000 Lanzhou, China
- Peter Grünberg Institute (PGI-1), Forschungszentrum Jülich, 52425 Jülich, Germany
- MultiscaleConsulting, Wolfshovener Str. 2, 52428 Jülich, Germany
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Meneses A, Naya S, Francisco-Fernández M, López-Beceiro J, Gracia-Fernández C, Tarrío-Saavedra J. TTS package: Computational tools for the application of the Time Temperature Superposition principle. Heliyon 2023; 9:e15816. [PMID: 37215836 PMCID: PMC10192834 DOI: 10.1016/j.heliyon.2023.e15816] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2022] [Revised: 04/20/2023] [Accepted: 04/21/2023] [Indexed: 05/24/2023] Open
Abstract
The TTS package has been developed in R software to predict the mechanical properties of viscoelastic materials, at short and long observation times/frequencies by applying the Time Temperature Superposition (TTS) principle. TTS is a physical principle used in material science to estimate mechanical properties beyond the experimental range of observed times/frequencies by shifting data curves obtained at other temperatures relative to a reference temperature in the dataset. It is a methodology related to accelerated life-tests and reliability, whereas the TTS library is one of the first open source computational tool to apply the TTS principle. This R package provides free computational tools to obtain master curves that characterize materials from a thermal-mechanical approach. The TTS package also proposes, implements and explains our own method to obtain the shift factors and the master curve in a TTS analysis, based on horizontal shifting of the first derivative function of viscoelastic properties. This procedure provides shift factors estimates and smooth master curve estimates using B-spline fitting, in a fully automatic way, without assuming any parametric expression. Williams-Landel-Ferry (WLF) and Arrhenius TTS parametric models are also implemented in the TTS package. They can be fitted from shifts obtained by the our first derivative based method.
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Affiliation(s)
| | - Salvador Naya
- CITIC, Grupo MODES, Departamento de Matemáticas, Escola Politécnica de Enxeñaría de Ferrol, Universidade da Coruña, Ferrol, Spain
| | - Mario Francisco-Fernández
- CITIC, Grupo MODES, Departamento de Matemáticas, Facultade de Informática, Universidade da Coruña, A Coruña, Spain
| | - Jorge López-Beceiro
- Campus Industrial de Ferrol, CITENI, Grupo PROTERM, Departamento de Ingeniería Industrial y Naval, Escola Politécnica de Enxeñaría de Ferrol, Universidade da Coruña, Ferrol, Spain
| | | | - Javier Tarrío-Saavedra
- CITIC, Grupo MODES, Departamento de Matemáticas, Escola Politécnica de Enxeñaría de Ferrol, Universidade da Coruña, Ferrol, Spain
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La S, Wang C. Experimental and Numerical Calculation of the Friction Performance of a Concrete Surface. MATERIALS (BASEL, SWITZERLAND) 2023; 16:2989. [PMID: 37109825 PMCID: PMC10145778 DOI: 10.3390/ma16082989] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/14/2023] [Revised: 02/21/2023] [Accepted: 03/07/2023] [Indexed: 06/19/2023]
Abstract
The presented self-developed high-precision contact friction test device conducts experimental research on the friction characteristics of concrete pavement. First, the error analysis of the test device is carried out. The structure shows that the test device meets the test requirements. Subsequently, the device was used to carry out experimental research on the friction performance of concrete pavement under different roughness and temperature changes. The results showed that the friction performance of concrete pavement increased with the increase in surface roughness, and decreased with the increase in temperature. It has a small volume and significant stick-slip properties. Finally, the spring slider model is used to simulate the friction characteristics of the concrete pavement, then the shear modulus and viscous force of the concrete material are adjusted to achieve the calculation of the friction force over time under temperature changes, which is consistent with the experimental structure.
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Affiliation(s)
- Shiren La
- School of Civil and Traffic Engineering, Qinghai Nationalities University, Xining 810007, China
| | - Chong Wang
- School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China
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Pabst F, Kraus J, Reynolds M, Mattsson J, Blochowicz T. Preserving fast ion dynamics while introducing mechanical rigidity in gelatin-based ionogels. SOFT MATTER 2023; 19:1418-1428. [PMID: 36723269 DOI: 10.1039/d2sm01143c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
Ionogels are gels containing ions, often an ionic liquid (IL), and a gelling agent. They are promising candidates for applications including batteries, photovoltaics or fuel cells due to their chemical stability and high ionic conductivity. In this work we report on a thermo-irreversible ionic gel prepared from a mixture of the ionic liquid 1-butyl-3-methylimidazolium ([BMIM]) dicyanamide ([DCA]), water and gelatin, which combines the advantages of an ionic liquid with the low cost of gelatin. We use (i) dielectric spectroscopy to monitor the ion transport, (ii) dynamic light scattering techniques to access the reorientational motions of the ions, as well as fluctuations of the gel matrix, and (iii) rheology to determine the shear response from above room temperature down to the glass transition. In this way, we are able to connect the microscopic ion dynamics with the meso- and macroscopic behavior of the gelatin matrix. We show, by comparing our results to those for a IL-water mixture from a previous study, that although some weak additional slow relaxation modes are present in the gel, the overall ion dynamics is hardly changed by the presence of gelatin. The macroscopic mechanical response, as probed by rheology, is however dominated by the gel matrix. This behaviour can be highly useful e.g. in battery gel electrolytes which prevent electrolyte leakage and combine mechanical rigidity and flexibility.
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Affiliation(s)
- Florian Pabst
- TU Darmstadt, Institute for Condensed Matter Physics, 64289 Darmstadt, Germany.
| | - Jennifer Kraus
- TU Darmstadt, Institute for Condensed Matter Physics, 64289 Darmstadt, Germany.
| | - Matthew Reynolds
- School of Physics and Astronomy, University of Leeds, LS2 9JT Leeds, UK
| | - Johan Mattsson
- School of Physics and Astronomy, University of Leeds, LS2 9JT Leeds, UK
| | - Thomas Blochowicz
- TU Darmstadt, Institute for Condensed Matter Physics, 64289 Darmstadt, Germany.
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Yang J, Melton M, Sun R, Yang W, Cheng S. Decoupling the Polymer Dynamics and the Nanoparticle Network Dynamics of Polymer Nanocomposites through Dielectric Spectroscopy and Rheology. Macromolecules 2019. [DOI: 10.1021/acs.macromol.9b01584] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Affiliation(s)
- Jie Yang
- College of Polymer Science and Engineering, State Key Laboratory of Polymer Materials Engineering, Sichuan University, Chengdu, Sichuan 610065, People’s Republic of China
- Department of Chemical Engineering and Materials Science, Michigan State University, East Lansing, Michigan 48824, United States
| | - Matthew Melton
- Department of Chemical Engineering and Materials Science, Michigan State University, East Lansing, Michigan 48824, United States
| | - Ruikun Sun
- Department of Chemical Engineering and Materials Science, Michigan State University, East Lansing, Michigan 48824, United States
| | - Wei Yang
- College of Polymer Science and Engineering, State Key Laboratory of Polymer Materials Engineering, Sichuan University, Chengdu, Sichuan 610065, People’s Republic of China
| | - Shiwang Cheng
- Department of Chemical Engineering and Materials Science, Michigan State University, East Lansing, Michigan 48824, United States
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Tiwari A, Miyashita N, Persson BNJ. Rolling friction of elastomers: role of strain softening. SOFT MATTER 2019; 15:9233-9243. [PMID: 31651922 DOI: 10.1039/c9sm01764j] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
We study the temperature and velocity dependency of rolling friction. Steel and PMMA cylinders are rolled on sheets of nitrile butadiene rubber (NBR), with and without filler, and fluoroelastomer (FKM) with filler. Measurements of the rolling friction are performed for temperatures between -40 °C and 20 °C, and for velocities between 5 μm s-1 and 0.5 cm s-1. For the unfilled NBR, a smooth rolling friction master curve is obtained using the bulk viscoelastic frequency-temperature shift factor aT. For the filled rubber compounds, a small deviation from the bulk viscoelastic shift factor is observed at low temperatures. The experimental data are analyzed using an analytical theory of rolling friction. For the filled compounds, good agreement with theory is obtained when strain softening is included, which increases the rolling friction by a factor ∼2 for the filled FKM and ∼3 for the filled NBR compounds. For the unfilled NBR, the maximum of the rolling friction occurs at higher sliding speeds than predicted by the theory. We discuss the role of the adhesive (crack-opening) contribution to the rolling friction, and the role of frozen-in elastic deformations as the rubber is cooled down below the rubber glass transition temperature.
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Affiliation(s)
| | - N Miyashita
- The Yokohama Rubber Company, 2-1 Oiwake, Hiratsuka, Kanagawa 254-8601, Japan
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Morishita Y, Tsunoda K, Urayama K. Universal relation between crack-growth dynamics and viscoelasticity in glass-rubber transition for filled elastomers. POLYMER 2019. [DOI: 10.1016/j.polymer.2019.121651] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Abstract
We study the linear and nonlinear viscoelastic properties of two tire tread compounds. We discuss the difference in nonlinear response between the oscillatory tensile and shear modes. We also analyze strain relaxation (creep) data for the same systems. We discuss what type of measurements are most suitable for obtaining the viscoelastic modulus used in rubber friction calculations.
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12
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Affiliation(s)
- Ari J. Tuononen
- Department of Mechanical Engineering, Aalto University, Espoo, Finland
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13
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Savadkoohi S, Kasapis S. High pressure effects on the structural functionality of condensed globular-protein matrices. Int J Biol Macromol 2016; 88:433-42. [PMID: 27060534 DOI: 10.1016/j.ijbiomac.2016.04.012] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/22/2015] [Revised: 04/05/2016] [Accepted: 04/05/2016] [Indexed: 11/26/2022]
Abstract
High pressure technology is the outcome of consumer demand for better quality control of processed foods. There is great potential to apply HPP to condensed systems of globular proteins for the generation of industry-relevant biomaterials with advanced techno- and biofunctionality. To this end, research demonstrates that application of high hydrostatic pressure generates a coherent structure and preserves the native conformation in condensed globular proteins, which is an entirely unexpected but interesting outcome on both scientific and technological grounds. In microbiological challenge tests, high pressure at conventional commercial conditions, demonstrated to effectively reduce the concentration of typical Gram negative or Gram positive foodborne pathogens, and proteolytic enzymes in high-solid protein samples. This may have industrial significance in relation to the formulation and stabilisation of "functional food" products as well as in protein ingredients and concentrates by replacing spray dried powders with condensed HPP-treated pastes that maintain structure and bioactivity. Fundamental concepts and structural functionality of condensed matrices of globular proteins are the primary interest in this mini-review, which may lead to opportunities for industrial exploitation, but earlier work on low-solid systems is also summarised presently to put recent developments in context of this rapidly growing field.
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Affiliation(s)
- Sobhan Savadkoohi
- School of Science, RMIT University, Bundoora West Campus, Plenty Road, Vic 3083, Australia
| | - Stefan Kasapis
- School of Science, RMIT University, Bundoora West Campus, Plenty Road, Vic 3083, Australia.
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Morishita Y, Tsunoda K, Urayama K. Velocity transition in the crack growth dynamics of filled elastomers: Contributions of nonlinear viscoelasticity. Phys Rev E 2016; 93:043001. [PMID: 27176379 DOI: 10.1103/physreve.93.043001] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2015] [Indexed: 06/05/2023]
Abstract
The crack growth dynamics of the carbon-black (CB) filled elastomers is studied experimentally and analyzed while focusing on both kinetics and crack tip profiles. The CB amounts are varied to change the mechanical properties of the elastomers. Static crack growth measurements simultaneously reveal the discontinuous-like transition of the crack growth rate v between the "slow mode" (v≈10^{-5}-10^{-3} m/s) and "fast mode" (v≈10^{-1}-10^{2} m/s) in a narrow range of the input tearing energy Γ and the accompanying changes in the crack tip profiles from blunt to sharp shapes. The crack tip profiles are characterized by two specific parameters, i.e., the deviation δ from the parabolic profile and the opening displacement a in the loading direction. The analysis based on the linear and weakly nonlinear elasticity theories of fracture dynamics demonstrates that the Γ dependence of δ and a is simply classified into three groups depending on the mode (slow or fast) and the magnitudes of δ, independent of CB volume fractions. The theories well explain the results in the slow and fast modes with small magnitudes of δ, while they fail to describe the data in the fast mode with large magnitudes of δ, where the contributions of the strong nonlinearity and/or energy dissipation become significant. The correlation between a power-law relationship Γ∼v^{α} observed in the fast mode and the linear viscoelasticity spectrum is also discussed. The correlation in elastomers with low CB volume fractions is quantitatively explained by the theory of Persson and Brener [Phys. Rev. E 71, 036123 (2005)PLEEE81539-375510.1103/PhysRevE.71.036123], whereas the deviation from the theory becomes appreciable for elastomers with higher CB volume fractions which exhibit strong nonlinear viscoelasticity.
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Affiliation(s)
- Yoshihiro Morishita
- Research Department I, Central Research, Bridgestone Corporation, Tokyo 187-8531, Japan
- Department of Macromolecular Science and Engineering, Kyoto Institute of Technology, Kyoto 606-8585, Japan
| | - Katsuhiko Tsunoda
- Research Department I, Central Research, Bridgestone Corporation, Tokyo 187-8531, Japan
| | - Kenji Urayama
- Department of Macromolecular Science and Engineering, Kyoto Institute of Technology, Kyoto 606-8585, Japan
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15
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Panyoyai N, Kasapis S. A free-volume interpretation of the decoupling parameter in bioactive-compound diffusion from a glassy polymer. Food Hydrocoll 2016. [DOI: 10.1016/j.foodhyd.2015.10.019] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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16
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Lorenz B, Oh YR, Nam SK, Jeon SH, Persson BNJ. Rubber friction on road surfaces: Experiment and theory for low sliding speeds. J Chem Phys 2015; 142:194701. [DOI: 10.1063/1.4919221] [Citation(s) in RCA: 76] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- B. Lorenz
- PGI, FZ Jülich, 52425 Jülich, Germany
| | - Y. R. Oh
- Hankook Tire Co. LTD., 112 Gajeongbuk-ro, Yuseong-gu, Daejeon 305-725, South Korea
| | - S. K. Nam
- Hankook Tire Co. LTD., 112 Gajeongbuk-ro, Yuseong-gu, Daejeon 305-725, South Korea
| | - S. H. Jeon
- Hankook Tire Co. LTD., 112 Gajeongbuk-ro, Yuseong-gu, Daejeon 305-725, South Korea
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Scaraggi M, Persson BNJ. Friction and universal contact area law for randomly rough viscoelastic contacts. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:105102. [PMID: 25662338 DOI: 10.1088/0953-8984/27/10/105102] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We present accurate numerical results for the friction force and the contact area for a viscoelastic solid (rubber) in sliding contact with hard, randomly rough substrates. The rough surfaces are self-affine fractal with roughness over several decades in length scales. We calculate the contribution to the friction from the pulsating deformations induced by the substrate asperities. We also calculate how the area of real contact, A(v, p), depends on the sliding speed v and on the nominal contact pressure p, and we show how the contact area for any sliding speed can be obtained from a universal master curve A(p). The numerical results are found to be in good agreement with the predictions of an analytical contact mechanics theory.
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Affiliation(s)
- M Scaraggi
- DII, Universitá del Salento, 73100 Monteroni-Lecce, Italy. PGI, FZ-Jülich, 52425 Jülich, Germany
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Mujtaba A, Keller M, Ilisch S, Radusch HJ, Beiner M, Thurn-Albrecht T, Saalwächter K. Detection of Surface-Immobilized Components and Their Role in Viscoelastic Reinforcement of Rubber-Silica Nanocomposites. ACS Macro Lett 2014; 3:481-485. [PMID: 35590787 DOI: 10.1021/mz500192r] [Citation(s) in RCA: 97] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Immobilized polymer fractions have been claimed to be of pivotal importance for the large mechanical reinforcement observed in nanoparticle-filled elastomers but remained elusive in actual application-relevant materials. We here isolate the additive filler network contribution to the storage modulus of industrial styrene-butadiene rubber (SBR) nanocomposites filled with silica at different frequencies and temperatures and demonstrate that it is viscoelastic in nature. We further quantify the amount of immobilized polymer using solid-state NMR and establish a correlation with the mechanical reinforcement, identifying a direct, strongly nonlinear dependence on the immobilized polymer fraction. The observation of a temperature-independent filler percolation threshold suggests that immobilized polymer fractions may not necessarily form contiguous layers around the filler particles but could only reside in highly confined regions between closely packed filler particles, where they dominate the bending modulus of aggregated particles.
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Affiliation(s)
- A. Mujtaba
- Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
| | - M. Keller
- Zentrum für Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
| | - S. Ilisch
- Zentrum für Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
| | - H.-J. Radusch
- Zentrum für Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
| | - M. Beiner
- Fraunhofer Institut für
Werkstoffmechanik IWM, Walter-Hülse-Str. 1, 06120 Halle (Saale), Germany
| | - T. Thurn-Albrecht
- Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
| | - K. Saalwächter
- Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
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