A new class of tunable hypersonic phononic crystals based on polymer-tethered colloids
Identifieur interne : 000023 ( Pmc/Curation ); précédent : 000022; suivant : 000024A new class of tunable hypersonic phononic crystals based on polymer-tethered colloids
Auteurs : E. Alonso-Redondo [Allemagne] ; M. Schmitt [États-Unis] ; Z. Urbach [Allemagne] ; C. M. Hui [États-Unis] ; R. Sainidou [France] ; P. Rembert [France] ; K. Matyjaszewski [États-Unis] ; M. R. Bockstaller [États-Unis] ; G. Fytas [Allemagne, Grèce]Source :
- Nature Communications [ 2041-1723 ] ; 2015.
Abstract
The design and engineering of hybrid materials exhibiting tailored phononic band gaps are fundamentally relevant to innovative material technologies in areas ranging from acoustics to thermo-optic devices. Phononic hybridization gaps, originating from the anti-crossing between local resonant and propagating modes, have attracted particular interest because of their relative robustness to structural disorder and the associated benefit to ‘manufacturability'. Although hybridization gap materials are well known, their economic fabrication and efficient control of the gap frequency have remained elusive because of the limited property variability and expensive fabrication methodologies. Here we report a new strategy to realize hybridization gap materials by harnessing the ‘anisotropic elasticity' across the particle–polymer interface in densely polymer-tethered colloidal particles. Theoretical and Brillouin scattering analysis confirm both the robustness to disorder and the tunability of the resulting hybridization gap and provide guidelines for the economic synthesis of new materials with deliberately controlled gap position and width frequencies.
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DOI: 10.1038/ncomms9309
PubMed: 26390851
PubMed Central: 4595630
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Alonso-Redondo</name><affiliation wicri:level="1"><nlm:aff id="a1"><institution>Max Planck Institute for Polymer Research</institution>, Ackermannweg 10, 55128 Mainz,<country>Germany</country></nlm:aff><country xml:lang="fr">Allemagne</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="Schmitt, M" sort="Schmitt, M" uniqKey="Schmitt M" first="M." last="Schmitt">M. Schmitt</name><affiliation wicri:level="1"><nlm:aff id="a2"><institution>Department of Materials Science and Engineering, Carnegie Mellon University</institution>, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213, USA</nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213</wicri:regionArea></affiliation></author><author><name sortKey="Urbach, Z" sort="Urbach, Z" uniqKey="Urbach Z" first="Z." last="Urbach">Z. Urbach</name><affiliation wicri:level="1"><nlm:aff id="a1"><institution>Max Planck Institute for Polymer Research</institution>, Ackermannweg 10, 55128 Mainz,<country>Germany</country></nlm:aff><country xml:lang="fr">Allemagne</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="Hui, C M" sort="Hui, C M" uniqKey="Hui C" first="C. M." last="Hui">C. M. Hui</name><affiliation wicri:level="1"><nlm:aff id="a3"><institution>Department of Chemistry, Carnegie Mellon University</institution>, 4400 Fifth Avenue, Pittsburgh, Pennsylvania 15213, USA</nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>, 4400 Fifth Avenue, Pittsburgh, Pennsylvania 15213</wicri:regionArea></affiliation></author><author><name sortKey="Sainidou, R" sort="Sainidou, R" uniqKey="Sainidou R" first="R." last="Sainidou">R. Sainidou</name><affiliation wicri:level="1"><nlm:aff id="a4"><institution>Laboratoire Ondes et Milieux Complexes, UMR CNRS 6294, University of Le Havre</institution>, 75 Rue Bellot, 76600 Le Havre,<country>France</country></nlm:aff><country xml:lang="fr">France</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="Rembert, P" sort="Rembert, P" uniqKey="Rembert P" first="P." last="Rembert">P. Rembert</name><affiliation wicri:level="1"><nlm:aff id="a4"><institution>Laboratoire Ondes et Milieux Complexes, UMR CNRS 6294, University of Le Havre</institution>, 75 Rue Bellot, 76600 Le Havre,<country>France</country></nlm:aff><country xml:lang="fr">France</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="Matyjaszewski, K" sort="Matyjaszewski, K" uniqKey="Matyjaszewski K" first="K." last="Matyjaszewski">K. Matyjaszewski</name><affiliation wicri:level="1"><nlm:aff id="a3"><institution>Department of Chemistry, Carnegie Mellon University</institution>, 4400 Fifth Avenue, Pittsburgh, Pennsylvania 15213, USA</nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>, 4400 Fifth Avenue, Pittsburgh, Pennsylvania 15213</wicri:regionArea></affiliation></author><author><name sortKey="Bockstaller, M R" sort="Bockstaller, M R" uniqKey="Bockstaller M" first="M. R." last="Bockstaller">M. R. Bockstaller</name><affiliation wicri:level="1"><nlm:aff id="a2"><institution>Department of Materials Science and Engineering, Carnegie Mellon University</institution>, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213, USA</nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea>, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213</wicri:regionArea></affiliation></author><author><name sortKey="Fytas, G" sort="Fytas, G" uniqKey="Fytas G" first="G." last="Fytas">G. Fytas</name><affiliation wicri:level="1"><nlm:aff id="a1"><institution>Max Planck Institute for Polymer Research</institution>, Ackermannweg 10, 55128 Mainz,<country>Germany</country></nlm:aff><country xml:lang="fr">Allemagne</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation><affiliation wicri:level="1"><nlm:aff id="a5"><institution>Department of Materials Science, FORTH-IESL,</institution> PO Box 1527, 71110 Heraklion,<country>Greece</country></nlm:aff><country xml:lang="fr">Grèce</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author></analytic><series><title level="j">Nature Communications</title><idno type="eISSN">2041-1723</idno><imprint><date when="2015">2015</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>The design and engineering of hybrid materials exhibiting tailored phononic band gaps are fundamentally relevant to innovative material technologies in areas ranging from acoustics to thermo-optic devices. Phononic hybridization gaps, originating from the anti-crossing between local resonant and propagating modes, have attracted particular interest because of their relative robustness to structural disorder and the associated benefit to ‘manufacturability'. Although hybridization gap materials are well known, their economic fabrication and efficient control of the gap frequency have remained elusive because of the limited property variability and expensive fabrication methodologies. Here we report a new strategy to realize hybridization gap materials by harnessing the ‘anisotropic elasticity' across the particle–polymer interface in densely polymer-tethered colloidal particles. Theoretical and Brillouin scattering analysis confirm both the robustness to disorder and the tunability of the resulting hybridization gap and provide guidelines for the economic synthesis of new materials with deliberately controlled gap position and width frequencies.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Maldovan, M" uniqKey="Maldovan M">M. Maldovan</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Han, T" uniqKey="Han T">T. Han</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Xu, H" uniqKey="Xu H">H. Xu</name></author><author><name sortKey="Shi, X" uniqKey="Shi X">X. Shi</name></author><author><name sortKey="Gao, F" uniqKey="Gao F">F. Gao</name></author><author><name sortKey="Sun, H" uniqKey="Sun H">H. Sun</name></author><author><name sortKey="Zhang, B" uniqKey="Zhang B">B. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Nat Commun</journal-id><journal-id journal-id-type="iso-abbrev">Nat Commun</journal-id><journal-title-group><journal-title>Nature Communications</journal-title></journal-title-group><issn pub-type="epub">2041-1723</issn><publisher><publisher-name>Nature Pub. Group</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">26390851</article-id><article-id pub-id-type="pmc">4595630</article-id><article-id pub-id-type="pii">ncomms9309</article-id><article-id pub-id-type="doi">10.1038/ncomms9309</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title-group><article-title>A new class of tunable hypersonic phononic crystals based on polymer-tethered colloids</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Alonso-Redondo</surname><given-names>E.</given-names></name><xref ref-type="aff" rid="a1">1</xref></contrib><contrib contrib-type="author"><name><surname>Schmitt</surname><given-names>M.</given-names></name><xref ref-type="aff" rid="a2">2</xref></contrib><contrib contrib-type="author"><name><surname>Urbach</surname><given-names>Z.</given-names></name><xref ref-type="aff" rid="a1">1</xref></contrib><contrib contrib-type="author"><name><surname>Hui</surname><given-names>C. M.</given-names></name><xref ref-type="aff" rid="a3">3</xref></contrib><contrib contrib-type="author"><name><surname>Sainidou</surname><given-names>R.</given-names></name><xref ref-type="aff" rid="a4">4</xref></contrib><contrib contrib-type="author"><name><surname>Rembert</surname><given-names>P.</given-names></name><xref ref-type="aff" rid="a4">4</xref></contrib><contrib contrib-type="author"><name><surname>Matyjaszewski</surname><given-names>K.</given-names></name><xref ref-type="aff" rid="a3">3</xref></contrib><contrib contrib-type="author"><name><surname>Bockstaller</surname><given-names>M. R.</given-names></name><xref ref-type="aff" rid="a2">2</xref></contrib><contrib contrib-type="author"><name><surname>Fytas</surname><given-names>G.</given-names></name><xref ref-type="corresp" rid="c1">a</xref><xref ref-type="aff" rid="a1">1</xref><xref ref-type="aff" rid="a5">5</xref></contrib><aff id="a1"><label>1</label><institution>Max Planck Institute for Polymer Research</institution>, Ackermannweg 10, 55128 Mainz,<country>Germany</country></aff><aff id="a2"><label>2</label><institution>Department of Materials Science and Engineering, Carnegie Mellon University</institution>, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213, USA</aff><aff id="a3"><label>3</label><institution>Department of Chemistry, Carnegie Mellon University</institution>, 4400 Fifth Avenue, Pittsburgh, Pennsylvania 15213, USA</aff><aff id="a4"><label>4</label><institution>Laboratoire Ondes et Milieux Complexes, UMR CNRS 6294, University of Le Havre</institution>, 75 Rue Bellot, 76600 Le Havre,<country>France</country></aff><aff id="a5"><label>5</label><institution>Department of Materials Science, FORTH-IESL,</institution> PO Box 1527, 71110 Heraklion,<country>Greece</country></aff></contrib-group><author-notes><corresp id="c1"><label>a</label><email>fytas@mpip-mainz.mpg.de</email></corresp></author-notes><pub-date pub-type="epub"><day>22</day><month>09</month><year>2015</year></pub-date><pub-date pub-type="collection"><year>2015</year></pub-date><volume>6</volume><elocation-id>8309</elocation-id><history><date date-type="received"><day>29</day><month>04</month><year>2015</year></date><date date-type="accepted"><day>11</day><month>08</month><year>2015</year></date></history><permissions><copyright-statement>Copyright © 2015, Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.</copyright-statement><copyright-year>2015</copyright-year><copyright-holder>Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.</copyright-holder><license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by/4.0/"><pmc-comment>author-paid</pmc-comment>
<license-p>This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/4.0/">http://creativecommons.org/licenses/by/4.0/</ext-link></license-p></license></permissions><abstract><p>The design and engineering of hybrid materials exhibiting tailored phononic band gaps are fundamentally relevant to innovative material technologies in areas ranging from acoustics to thermo-optic devices. Phononic hybridization gaps, originating from the anti-crossing between local resonant and propagating modes, have attracted particular interest because of their relative robustness to structural disorder and the associated benefit to ‘manufacturability'. Although hybridization gap materials are well known, their economic fabrication and efficient control of the gap frequency have remained elusive because of the limited property variability and expensive fabrication methodologies. Here we report a new strategy to realize hybridization gap materials by harnessing the ‘anisotropic elasticity' across the particle–polymer interface in densely polymer-tethered colloidal particles. Theoretical and Brillouin scattering analysis confirm both the robustness to disorder and the tunability of the resulting hybridization gap and provide guidelines for the economic synthesis of new materials with deliberately controlled gap position and width frequencies.</p></abstract><abstract abstract-type="web-summary"><p><inline-graphic id="i1" xlink:href="ncomms9309-i1.jpg"></inline-graphic>Hybridization-type band gaps are known to persist in phononic crystals, but their fabrication remains challenging for all-solid hypersonic composites. Here, the authors utilize the elastic anisotropy at the interface of polymer-tethered colloidal particles to control phonon propagation in GHz regime.</p></abstract></article-meta></front><floats-group><fig id="f1"><label>Figure 1</label><caption><title>Schematic representation of Bragg and hybridization gaps.</title><p>(<bold>a</bold>) Structure-directed, ‘Bragg' (BG) gap occurring at a frequency <italic>f</italic>∼<italic>q</italic><sub>BZ</sub><italic>c</italic>/2<italic>π</italic> at the edge of the Brillouin zone (BZ), <italic>q</italic><sub>BZ</sub>=<italic>π</italic>/<italic>a</italic>, where <italic>c</italic> is the sound velocity in the composite medium and <italic>a</italic> the lattice constant. (<bold>b</bold>) Hybridization (HG) gap is originating from an anti-crossing opening up at <italic>q</italic>*<<italic>q</italic><sub>BZ</sub> and involving a local resonant mode that occurs at a frequency <italic>f</italic><sub>HG</sub> related to the particle resonance characteristics (see the text).</p></caption><graphic xlink:href="ncomms9309-f1"></graphic></fig><fig id="f2"><label>Figure 2</label><caption><title>Structure of gradient-interface particle brush colloidal crystal.</title><p>(<bold>a</bold>) Bright-field transmission electron microscopy image of particle brush film revealing projection of a four particle-stack. In the micrograph, the sixfold intersection of contrast regions is indicative of fcc packing. The inset shows a magnified image of individual brush particles enlightening the ‘inner shell' of extended polymer segments (faint ring in the image). The scheme illustrates the conformational transition and interdigitation of tethered polymer chains on silica spheres. (<bold>b</bold>) Schematic of fcc packing structure of brush particles with interparticle distance <italic>d</italic> along with an illustration of the projection image that is expected for fcc structure along [111] direction (black arrow in left panel).</p></caption><graphic xlink:href="ncomms9309-f2"></graphic></fig><fig id="f3"><label>Figure 3</label><caption><title>Recording the phononic band diagram.</title><p>(<bold>a</bold>,<bold>b</bold>) Brillouin light scattering geometries probing phonon propagation along the wave vector <bold>q=k</bold><sub>s</sub>–<bold>k</bold><sub>i</sub>, with <bold>k</bold><sub>s</sub>, <bold>k</bold><sub>i</sub> being, respectively, the incident laser and scattered light wave vector. The direction of <bold>q</bold> is selected either in-plane ((<bold>a</bold>) transmission geometry) or normal to the plane ((<bold>b</bold>) reflection geometry). In the transmission geometry, the magnitude <italic>q</italic> is tuned by varying the scattering angle <italic>θ</italic> and is independent of the refractive index of the medium. (<bold>c</bold>) Experimental dispersion relation (frequency versus wavenumber <italic>q</italic>) for DP100, DP400 and DP1000 samples, obtained from the corresponding deducted BLS spectra (insets) recorded at a given <italic>q</italic> (vertical arrows) and fitted as a sum of Lorentzian shapes (red lines). The deducted isotropic (grey) spectra is the difference between the intensities recorded in vv (black) and vh (blue) polarizations, <italic>I</italic><sub>vv</sub><italic>-xI</italic><sub>vh</sub>, with <italic>x</italic> being a variable factor between 0.7 and 4/3. In each plot, a clear band gap region (patterned area) and a localized mode (open circles) are observed. The red lines in the low <italic>q</italic> regime represent the effective medium acoustic mode; the dashed grey lines in the high-frequency branch are guides to the eye, connecting the data acquired with <bold>q</bold> perpendicular to the substrate plane (blue shaded area) using the reflection geometry in <bold>a</bold>. The frequency of the flat mode is indicated by a grey dashed arrow, and wavenumber number of the gap opening is marked with a black solid arrow, with an error of ∼0.002 nm<sup>−1</sup>.</p></caption><graphic xlink:href="ncomms9309-f3"></graphic></fig><fig id="f4"><label>Figure 4</label><caption><title>Theoretical versus experimental band diagrams.</title><p>(<bold>a</bold>) Reciprocal fcc lattice showing a [112] plane (orange hexagon) and the ΓM direction probed in the experiment. (<bold>b</bold>) Schematic presentation of PBC and IBC models applied at the SiO<sub>2</sub>-PS interface. In the IBC model, the springs of respective stiffness <italic>k</italic><sub>L,T</sub> represent the discontinuity of the displacement field across this interface. The theoretical band diagram of the DP400 is shown for PBC along [111] with the bulk PS sound velocities in <bold>c</bold> and for IBC with PS velocities (<italic>c</italic><sub>L</sub>=2,560 m s<sup>−1</sup>, <italic>c</italic><sub>T</sub>=1,320 m s<sup>−1</sup>) about 9% higher than bulk PS along [111] (<bold>d</bold>) and [112] (<bold>e</bold>). Along ΓL dark/light solid and dotted blue lines denote non-degenerate (longitudinal), double degenerate (transverse) and deaf computed bands, respectively. In <bold>e</bold>, all bands are non-degenerate of mixed character; the flat band is highlighted (dotted line). Solid and open circles indicate the experimental points (see <xref ref-type="fig" rid="f3">Fig. 3</xref>). Hatched regions denote hybridization gaps (LHG for longitudinal modes; HG for all modes).</p></caption><graphic xlink:href="ncomms9309-f4"></graphic></fig><fig id="f5"><label>Figure 5</label><caption><title>Physical origin of the dispersion characteristics and their evolutive formation with dimensionality.</title><p>Calculated DOS for (<bold>a</bold>) one SiO<sub>2</sub> particle in PS (solid/broken lines spheroidal/torsional <italic>l</italic>=1 modes) and (<bold>b</bold>) for one (111) plane of particles of the DP600 film at slightly off-normal incidence (dark/light blue lines: non-degenerate/double degenerate modes). The mode corresponding to the very sharp peak in DOS plot of (111) plane, is associated to the flat band of the crystal (see <bold>c</bold>). Its field intensity representation within the unit cell passing at the centre of the sphere (<italic>z</italic>=0), at <italic>f</italic>=5.24 GHz, is shown in the inset. White arrows represent the quasi-pure rotational character of the field everywhere in the unit cell. (<bold>c</bold>) The band structure of the corresponding crystal along [111], with non-degenerate (left) and double degenerate (right) calculated bands together with the experimental points (symbols). The notation used is that of <xref ref-type="fig" rid="f3">Fig. 3</xref>. (<bold>d</bold>) Schematic representation of the evolution of these modes when passing from a single sphere to the whole crystal. Parenthesized numbers denote the number of states per level. In the case of three-dimensional (3D) crystal, the levels indicating the hybridized modes correspond to HG centres.</p></caption><graphic xlink:href="ncomms9309-f5"></graphic></fig><fig id="f6"><label>Figure 6</label><caption><title>Controlling the core-brush assemblies behaviour.</title><p>(<bold>a</bold>) Evolution of the frequency gap region (shaded area) for all modes with <italic>N</italic>, following the theoretical predictions (see the text) along ΓM. Inset: normalized gap width (<italic>f</italic><sub>c</sub> being the central frequency of the gap). Variation of the tangential stiffness values <italic>k</italic><sub>T</sub> with (<bold>b</bold>) the frequency of the flat band <italic>f</italic><sub>flat</sub> calculated in the middle of the band along ΓM (<italic>qd</italic><sub>ΓM</sub>=0.5<italic>π</italic>), and with (<bold>c</bold>) the degree of polymerization <italic>N</italic> of the assemblies considered, showing a power-law behaviour in both cases.</p></caption><graphic xlink:href="ncomms9309-f6"></graphic></fig><table-wrap position="float" id="t1"><label>Table 1</label><caption><title>Samples characteristics and longitudinal sound velocities in the assemblies.</title></caption><table frame="hsides" rules="groups" border="1"><colgroup><col align="left"></col><col align="char" char="."></col><col align="char" char="."></col><col align="char" char="."></col><col align="char" char="."></col><col align="char" char="."></col><col align="char" char="."></col></colgroup><thead valign="bottom"><tr><th align="left" valign="top" charoff="50"><bold>Sample</bold></th><th align="center" valign="top" char="." charoff="50"><italic><bold>σ</bold></italic><bold>(nm</bold><sup><bold>−2</bold></sup><bold>)</bold></th><th align="center" valign="top" char="." charoff="50"><italic><bold>N</bold></italic></th><th align="center" valign="top" char="." charoff="50"><italic><bold>d</bold></italic><bold>(nm)</bold></th><th align="center" valign="top" char="." charoff="50"><italic><bold>c</bold></italic><sub><bold>PS</bold></sub><bold>(m s<sup>−1</sup>)</bold></th><th align="center" valign="top" char="." charoff="50"><italic><bold>c</bold></italic><sub><bold>eff-th</bold></sub><bold>(m s<sup>−1</sup>)</bold></th><th align="center" valign="top" char="." charoff="50"><italic><bold>c</bold></italic><sub><bold>eff-exp</bold></sub><bold>(m s<sup>−1</sup>)</bold></th></tr></thead><tbody valign="top"><tr><td align="left" valign="top" charoff="50">DP100</td><td align="char" valign="top" char="." charoff="50">0.61</td><td align="char" valign="top" char="." charoff="50">130</td><td align="char" valign="top" char="." charoff="50">142</td><td align="char" valign="top" char="." charoff="50">2,820</td><td align="char" valign="top" char="." charoff="50">2,688</td><td align="char" valign="top" char="." charoff="50">2,710</td></tr><tr><td align="left" valign="top" charoff="50">DP400</td><td align="char" valign="top" char="." charoff="50">0.61</td><td align="char" valign="top" char="." charoff="50">400</td><td align="char" valign="top" char="." charoff="50">176</td><td align="char" valign="top" char="." charoff="50">2,562</td><td align="char" valign="top" char="." charoff="50">2,482</td><td align="char" valign="top" char="." charoff="50">2,480</td></tr><tr><td align="left" valign="top" charoff="50">DP600</td><td align="char" valign="top" char="." charoff="50">0.56</td><td align="char" valign="top" char="." charoff="50">630</td><td align="char" valign="top" char="." charoff="50">201</td><td align="char" valign="top" char="." charoff="50">2,526</td><td align="char" valign="top" char="." charoff="50">2,469</td><td align="char" valign="top" char="." charoff="50">2,420</td></tr><tr><td align="left" valign="top" charoff="50">DP1000</td><td align="char" valign="top" char="." charoff="50">0.48</td><td align="char" valign="top" char="." charoff="50">980</td><td align="char" valign="top" char="." charoff="50">214</td><td align="char" valign="top" char="." charoff="50">2,350</td><td align="char" valign="top" char="." charoff="50">2,309</td><td align="char" valign="top" char="." charoff="50">2,360</td></tr></tbody></table><table-wrap-foot><fn id="t1-fn1"><p><italic>σ</italic>, grafting density; <italic>N</italic>, degree of polymerization; <italic>d</italic>, distance core to core; <italic>c</italic><sub>PS</sub>, polystyrene sound velocity; <italic>c</italic><sub>eff,th</sub>, computed effective sound velocity; <italic>c</italic><sub>eff,exp</sub>, measured effective sound velocity.</p></fn></table-wrap-foot></table-wrap></floats-group></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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