Conformations of a Long Polymer in a Melt of Shorter Chains: Generalizations of the Flory Theorem
Identifieur interne : 000120 ( Pmc/Corpus ); précédent : 000119; suivant : 000121Conformations of a Long Polymer in a Melt of Shorter Chains: Generalizations of the Flory Theorem
Auteurs : Michael Lang ; Michael Rubinstein ; Jens-Uwe SommerSource :
- ACS Macro Letters [ 2161-1653 ] ; 2015.
Abstract
Large-scale simulations of the swelling
of a long
Url:
DOI: 10.1021/mz500777r
PubMed: 26543675
PubMed Central: 4621164
Links to Exploration step
PMC:4621164Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Conformations of a Long Polymer in a Melt of Shorter
Chains: Generalizations of the Flory Theorem</title><author><name sortKey="Lang, Michael" sort="Lang, Michael" uniqKey="Lang M" first="Michael" last="Lang">Michael Lang</name><affiliation><nlm:aff id="aff1"><institution>Leibniz Institute of Polymer Research Dresden</institution>, Hohe Straße 6, 01069 Dresden,<country>Germany</country></nlm:aff></affiliation></author><author><name sortKey="Rubinstein, Michael" sort="Rubinstein, Michael" uniqKey="Rubinstein M" first="Michael" last="Rubinstein">Michael Rubinstein</name><affiliation><nlm:aff id="aff2">Department of Chemistry,<institution>University of North Carolina</institution>, Chapel Hill, North Carolina 27599-3290,<country>United States</country></nlm:aff></affiliation></author><author><name sortKey="Sommer, Jens Uwe" sort="Sommer, Jens Uwe" uniqKey="Sommer J" first="Jens-Uwe" last="Sommer">Jens-Uwe Sommer</name><affiliation><nlm:aff id="aff1"><institution>Leibniz Institute of Polymer Research Dresden</institution>, Hohe Straße 6, 01069 Dresden,<country>Germany</country></nlm:aff></affiliation><affiliation><nlm:aff id="aff3">Institute of Theoretical Physics,<institution>Technische Universität Dresden</institution>, Zellescher Weg 17, 01062 Dresden,<country>Germany</country></nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">26543675</idno><idno type="pmc">4621164</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4621164</idno><idno type="RBID">PMC:4621164</idno><idno type="doi">10.1021/mz500777r</idno><date when="2015">2015</date><idno type="wicri:Area/Pmc/Corpus">000120</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000120</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Conformations of a Long Polymer in a Melt of Shorter
Chains: Generalizations of the Flory Theorem</title><author><name sortKey="Lang, Michael" sort="Lang, Michael" uniqKey="Lang M" first="Michael" last="Lang">Michael Lang</name><affiliation><nlm:aff id="aff1"><institution>Leibniz Institute of Polymer Research Dresden</institution>, Hohe Straße 6, 01069 Dresden,<country>Germany</country></nlm:aff></affiliation></author><author><name sortKey="Rubinstein, Michael" sort="Rubinstein, Michael" uniqKey="Rubinstein M" first="Michael" last="Rubinstein">Michael Rubinstein</name><affiliation><nlm:aff id="aff2">Department of Chemistry,<institution>University of North Carolina</institution>, Chapel Hill, North Carolina 27599-3290,<country>United States</country></nlm:aff></affiliation></author><author><name sortKey="Sommer, Jens Uwe" sort="Sommer, Jens Uwe" uniqKey="Sommer J" first="Jens-Uwe" last="Sommer">Jens-Uwe Sommer</name><affiliation><nlm:aff id="aff1"><institution>Leibniz Institute of Polymer Research Dresden</institution>, Hohe Straße 6, 01069 Dresden,<country>Germany</country></nlm:aff></affiliation><affiliation><nlm:aff id="aff3">Institute of Theoretical Physics,<institution>Technische Universität Dresden</institution>, Zellescher Weg 17, 01062 Dresden,<country>Germany</country></nlm:aff></affiliation></author></analytic><series><title level="j">ACS Macro Letters</title><idno type="eISSN">2161-1653</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 content-type="toc-graphic"><graphic xlink:href="mz-2014-00777r_0007" id="ab-tgr1"></graphic></p><p>Large-scale simulations of the swelling
of a long <italic>N</italic>-mer in a melt of chemically identical <italic>P</italic>-mers are
used to investigate a discrepancy between theory and experiments.
Classical theory predicts an increase of probe chain size <italic>R</italic> ∼ <italic>P</italic><sup>–0.18</sup> with
decreasing degree of polymerization <italic>P</italic> of melt chains
in the range of 1 < <italic>P</italic> < <italic>N</italic><sup>1/2</sup>. However, both experiment and simulation data are more
consistent with an apparently slower swelling <italic>R</italic> ∼ <italic>P</italic><sup>–0.1</sup> over a wider range of melt degrees
of polymerization. This anomaly is explained by taking into account
the recently discovered long-range bond correlations in polymer melts
and corrections to excluded volume. We generalize the Flory theorem
and demonstrate that it is in excellent agreement with experiments
and simulations.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Flory, P J" uniqKey="Flory P">P. J. Flory</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Flory, P J" uniqKey="Flory P">P. J. Flory</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="De Gennes, P G" uniqKey="De Gennes P">P. G. De Gennes</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Joanny, J F" uniqKey="Joanny J">J. F. Joanny</name></author><author><name sortKey="Grant, P" uniqKey="Grant P">P. Grant</name></author><author><name sortKey="Turkevich, L A" uniqKey="Turkevich L">L. A. Turkevich</name></author><author><name sortKey="Pincus, P" uniqKey="Pincus P">P. Pincus</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Raphael, E" uniqKey="Raphael E">E. Raphael</name></author><author><name sortKey="Fredrickson, G H" uniqKey="Fredrickson G">G. H. Fredrickson</name></author><author><name sortKey="Pincus, P" uniqKey="Pincus P">P. Pincus</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="De X0a Gennes, P G" uniqKey="De X0a Gennes P">P. G. De
Gennes</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Landry, M" uniqKey="Landry M">M. Landry</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Kirste, R G" uniqKey="Kirste R">R. G. Kirste</name></author><author><name sortKey="Lehnen, B R" uniqKey="Lehnen B">B. R. Lehnen</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Tangari, C" uniqKey="Tangari C">C. Tangari</name></author><author><name sortKey="Ullman, R" uniqKey="Ullman R">R. Ullman</name></author><author><name sortKey="King, J S" uniqKey="King J">J. S. King</name></author><author><name sortKey="Wignall, G D" uniqKey="Wignall G">G. D. Wignall</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Torikai, N" uniqKey="Torikai N">N. Torikai</name></author><author><name sortKey="Takabayashi, N" uniqKey="Takabayashi N">N. Takabayashi</name></author><author><name sortKey="Noda, I" uniqKey="Noda I">I. Noda</name></author><author><name sortKey="Suzuki, J" uniqKey="Suzuki J">J. Suzuki</name></author><author><name sortKey="Matsushita, Y" uniqKey="Matsushita Y">Y. Matsushita</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Torikai, N" uniqKey="Torikai N">N. Torikai</name></author><author><name sortKey="Takabayashi, N" uniqKey="Takabayashi N">N. Takabayashi</name></author><author><name sortKey="Suzuki, J" uniqKey="Suzuki J">J. Suzuki</name></author><author><name sortKey="Noda, I" uniqKey="Noda I">I. Noda</name></author><author><name sortKey="Matsushita, Y" uniqKey="Matsushita Y">Y. Matsushita</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Wittmer, J P" uniqKey="Wittmer J">J. P. Wittmer</name></author><author><name sortKey="Beckrich, P" uniqKey="Beckrich P">P. Beckrich</name></author><author><name sortKey="Meyer, H" uniqKey="Meyer H">H. Meyer</name></author><author><name sortKey="Cavallo, A" uniqKey="Cavallo A">A. Cavallo</name></author><author><name sortKey="Johner, A" uniqKey="Johner A">A. Johner</name></author><author><name sortKey="Baschnagel, J" uniqKey="Baschnagel J">J. Baschnagel</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Semenov, A N" uniqKey="Semenov A">A. N. Semenov</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Rubinstein, M" uniqKey="Rubinstein M">M. Rubinstein</name></author><author><name sortKey="Colby, R" uniqKey="Colby R">R. Colby</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Le X0a Guillou, J C" uniqKey="Le X0a Guillou J">J. C. Le
Guillou</name></author><author><name sortKey="Zinn Justin, J" uniqKey="Zinn Justin J">J. Zinn Justin</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Carmesin, I" uniqKey="Carmesin I">I. Carmesin</name></author><author><name sortKey="Kremer, K" uniqKey="Kremer K">K. Kremer</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Lang, M" uniqKey="Lang M">M. Lang</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Paul, W" uniqKey="Paul W">W. Paul</name></author><author><name sortKey="Binder, K" uniqKey="Binder K">K. Binder</name></author><author><name sortKey="Heermann, D W" uniqKey="Heermann D">D. W. Heermann</name></author><author><name sortKey="Kremer, K" uniqKey="Kremer K">K. Kremer</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct><analytic><author><name sortKey="Wu, D T" uniqKey="Wu D">D. T. Wu</name></author><author><name sortKey="Fredrickson, G H" uniqKey="Fredrickson G">G. H. Fredrickson</name></author><author><name sortKey="Carton, J P" uniqKey="Carton J">J.-P. Carton</name></author><author><name sortKey="Adjari, A" uniqKey="Adjari A">A. Adjari</name></author><author><name sortKey="Leibler, L" uniqKey="Leibler L">L. Leibler</name></author></analytic></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct></listBibl></div1></back></TEI><pmc article-type="rapid-communication" xml:lang="EN"><pmc-dir>properties open_access</pmc-dir>
<front><journal-meta><journal-id journal-id-type="nlm-ta">ACS Macro Lett</journal-id><journal-id journal-id-type="iso-abbrev">ACS Macro Lett</journal-id><journal-id journal-id-type="publisher-id">mz</journal-id><journal-id journal-id-type="coden">amlccd</journal-id><journal-title-group><journal-title>ACS Macro Letters</journal-title></journal-title-group><issn pub-type="epub">2161-1653</issn><publisher><publisher-name>American Chemical Society</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">26543675</article-id><article-id pub-id-type="pmc">4621164</article-id><article-id pub-id-type="doi">10.1021/mz500777r</article-id><article-categories><subj-group><subject>Letter</subject></subj-group></article-categories><title-group><article-title>Conformations of a Long Polymer in a Melt of Shorter
Chains: Generalizations of the Flory Theorem</article-title></title-group><contrib-group><contrib contrib-type="author" corresp="yes" id="ath1"><name><surname>Lang</surname><given-names>Michael</given-names></name><xref rid="cor1" ref-type="other">*</xref><xref rid="aff1" ref-type="aff">†</xref></contrib><contrib contrib-type="author" id="ath2"><name><surname>Rubinstein</surname><given-names>Michael</given-names></name><xref rid="aff2" ref-type="aff">‡</xref></contrib><contrib contrib-type="author" id="ath3"><name><surname>Sommer</surname><given-names>Jens-Uwe</given-names></name><xref rid="aff1" ref-type="aff">†</xref><xref rid="aff3" ref-type="aff">¶</xref></contrib><aff id="aff1"><label>†</label><institution>Leibniz Institute of Polymer Research Dresden</institution>, Hohe Straße 6, 01069 Dresden,<country>Germany</country></aff><aff id="aff2"><label>‡</label>Department of Chemistry,<institution>University of North Carolina</institution>, Chapel Hill, North Carolina 27599-3290,<country>United States</country></aff><aff id="aff3"><label>¶</label>Institute of Theoretical Physics,<institution>Technische Universität Dresden</institution>, Zellescher Weg 17, 01062 Dresden,<country>Germany</country></aff></contrib-group><author-notes><corresp id="cor1"><label>*</label>E-mail: <email>lang@ipfdd.de</email>.</corresp></author-notes><pub-date pub-type="epub"><day>14</day><month>01</month><year>2015</year></pub-date><pub-date pub-type="ppub"><day>17</day><month>02</month><year>2015</year></pub-date><pub-date pub-type="pmc-release"><day>14</day><month>01</month><year>2016</year></pub-date><volume>4</volume><issue>2</issue><fpage>177</fpage><lpage>181</lpage><history><date date-type="received"><day>09</day><month>12</month><year>2014</year></date><date date-type="accepted"><day>05</day><month>01</month><year>2015</year></date></history><permissions><copyright-statement>Copyright © 2015 American Chemical
Society</copyright-statement><copyright-year>2015</copyright-year><copyright-holder>American Chemical
Society</copyright-holder><license><license-p>This is an open access article published under an ACS AuthorChoice <ext-link ext-link-type="uri" xlink:href="http://pubs.acs.org/page/policy/authorchoice_termsofuse.html">License</ext-link>, which permits copying and redistribution of the article or any adaptations for non-commercial purposes.</license-p></license></permissions><abstract><p content-type="toc-graphic"><graphic xlink:href="mz-2014-00777r_0007" id="ab-tgr1"></graphic></p><p>Large-scale simulations of the swelling
of a long <italic>N</italic>-mer in a melt of chemically identical <italic>P</italic>-mers are
used to investigate a discrepancy between theory and experiments.
Classical theory predicts an increase of probe chain size <italic>R</italic> ∼ <italic>P</italic><sup>–0.18</sup> with
decreasing degree of polymerization <italic>P</italic> of melt chains
in the range of 1 < <italic>P</italic> < <italic>N</italic><sup>1/2</sup>. However, both experiment and simulation data are more
consistent with an apparently slower swelling <italic>R</italic> ∼ <italic>P</italic><sup>–0.1</sup> over a wider range of melt degrees
of polymerization. This anomaly is explained by taking into account
the recently discovered long-range bond correlations in polymer melts
and corrections to excluded volume. We generalize the Flory theorem
and demonstrate that it is in excellent agreement with experiments
and simulations.</p></abstract><funding-group><funding-statement><funding-source>National Institutes of Health, United States</funding-source></funding-statement></funding-group><custom-meta-group><custom-meta><meta-name>document-id-old-9</meta-name><meta-value>mz500777r</meta-value></custom-meta><custom-meta><meta-name>document-id-new-14</meta-name><meta-value>mz-2014-00777r</meta-value></custom-meta><custom-meta><meta-name>ccc-price</meta-name><meta-value></meta-value></custom-meta></custom-meta-group></article-meta></front><body><p id="sec3">The description of macromolecular
conformations in various environments is one of the cornerstones of
polymer physics research. A fundamental test of standard models of
polymer conformations is the problem of an <italic>N</italic>-mer
immersed in a melt of chemically identical <italic>P</italic>-mers.
This question has been addressed by numerous works in the past<sup><xref ref-type="bibr" rid="ref1">1</xref>−<xref ref-type="bibr" rid="ref6">6</xref></sup> and was considered to be well understood from the theoretical point
of view. In all of these works it was argued that a scaling variable
proportional to <italic>N</italic>/<italic>P</italic><sup>2</sup> describes
the crossover between ideal and swollen chain conformations.<sup><xref ref-type="bibr" rid="ref5">5</xref></sup> However, the available experimental data<sup><xref ref-type="bibr" rid="ref7">7</xref>−<xref ref-type="bibr" rid="ref11">11</xref></sup> turned out to be in rather poor agreement with the theoretical predictions.
Some studies even propose a significantly different scaling of the
chain size<sup><xref ref-type="bibr" rid="ref7">7</xref>,<xref ref-type="bibr" rid="ref10">10</xref>,<xref ref-type="bibr" rid="ref11">11</xref></sup> in its crossover
between swollen and ideal chain conformations with increasing molecular
weight of the matrix polymers. In this letter, we generalize the Flory
theory by taking into account the recently proposed long-range bond
correlations<sup><xref ref-type="bibr" rid="ref12">12</xref>,<xref ref-type="bibr" rid="ref13">13</xref></sup> and corrections to excluded volume
in dense polymer systems. We demonstrate that the generalized Flory
theorem is in excellent agreement with experimental and simulation
data.</p><p>In the following, we use <italic>P</italic> and <italic>N</italic> to denote the number of Kuhn segments per matrix and guest
chains,
respectively. The Flory theorem is based upon the concept that the
monomeric excluded volume parameter in a melt of <italic>P</italic>-mers is screened by a factor of 1/<italic>P</italic> and, thus,
very small for large <italic>P</italic>. The root-mean-square radius
of gyration <italic>R</italic> of dilute <italic>N</italic>-mers in
a melt of <italic>P</italic>-mers can be estimated by minimizing free
energy Δ<italic>F</italic> (dropping all coefficients on the
order of unity)<disp-formula id="eq1"><graphic xlink:href="mz-2014-00777r_m001" position="anchor"></graphic><label>1</label></disp-formula>where <italic>k</italic> is the Boltzmann
constant; <italic>T</italic> is the absolute temperature; and <italic>v</italic> is the excluded volume in a liquid of Kuhn monomers. The
first term of the free energy describes the entropic penalty for swelling
a chain with <italic>N</italic> segments from its ideal size <italic>R</italic><sub>0</sub> ≈ <italic>b</italic>(<italic>N</italic>/6)<sup>1/2</sup> to <italic>R</italic>, whereby <italic>b</italic> is the length of a Kuhn segment. The second term is the mean field
estimate of the interaction of <italic>N</italic> monomers with excluded
volume parameter <italic>v</italic>/<italic>P</italic> randomly distributed
over the chain volume <italic>R</italic><sup>3</sup>. The third term
is the confinement free energy. Denoting the swelling ratio by α
= <italic>R</italic>/<italic>R</italic><sub>0</sub>, eq <xref rid="eq1" ref-type="disp-formula">1</xref> can be rewritten as<disp-formula id="eq2"><graphic xlink:href="mz-2014-00777r_m002" position="anchor"></graphic><label>2</label></disp-formula>where the interaction parameter<disp-formula id="eq3"><graphic xlink:href="mz-2014-00777r_m003" position="anchor"></graphic><label>3</label></disp-formula>determines the strength of the
excluded volume
interactions (<italic>Z</italic><sup>2</sup> is proportional to the
number of “thermal blobs”<sup><xref ref-type="bibr" rid="ref14">14</xref></sup> per <italic>N</italic>-mer). Minimization of free energy Δ<italic>F</italic> (eq <xref rid="eq2" ref-type="disp-formula">2</xref>) leads to a swelling ratio
as a function of the interaction parameter<disp-formula id="eq4"><graphic xlink:href="mz-2014-00777r_m004" position="anchor"></graphic><label>4</label></disp-formula></p><p>For
melt chains with <italic>P</italic> ≫ <italic>N</italic><sup>1/2</sup> this interaction parameter is small, <italic>Z</italic> ≪ 1, and the size of an <italic>N</italic>-mer in the melt
of relatively long <italic>P</italic>-mers (determined by balancing
the first and third terms of eq <xref rid="eq2" ref-type="disp-formula">2</xref>) is almost
ideal with swelling ratio α ≈ <italic>f</italic>(0) =
1.</p><p>For shorter melt chains <italic>P</italic> ≪ <italic>N</italic><sup>1/2</sup>, the interaction parameter is large, <italic>Z</italic> ≫ 1, and the size of the polymer is determined
by balancing
the first two terms of the free energy eq <xref rid="eq2" ref-type="disp-formula">2</xref> (since
the confinement term is not important for swollen chains) with <italic>f</italic>(<italic>Z</italic>) ∼ <italic>Z</italic><sup>1/5</sup>. In this case of <italic>P</italic> ≪ <italic>N</italic><sup>1/2</sup>, one obtains<sup><xref ref-type="bibr" rid="ref1">1</xref>,<xref ref-type="bibr" rid="ref14">14</xref></sup> the size of a swollen <italic>N</italic>-mer<disp-formula id="eq5"><graphic xlink:href="mz-2014-00777r_m005" position="anchor"></graphic><label>5</label></disp-formula></p><p>For monomeric solvent, <italic>P</italic> = 1, this leads to the
well-known <italic>R</italic> ≈ <italic>bN</italic><sup>1/2</sup>(<italic>N</italic><sup>1/2</sup><italic>v</italic><italic>b</italic><sup>3</sup>)<sup>2ν–1</sup> with scaling exponent ν
= 3/5 close to the exponent ν = 0.588 obtained by more accurate
numerical methods.<sup><xref ref-type="bibr" rid="ref15">15</xref></sup></p><p>In order to
test the above predictions and in order to explain
the discrepancy between theoretical prediction and experimental data
we simulated bidisperse melts of linear chains using the bond fluctuation
model.<sup><xref ref-type="bibr" rid="ref16">16</xref></sup> This simulation method was frequently
used to study polymer melts and networks (see refs (<xref ref-type="bibr" rid="ref12">12</xref> and <xref ref-type="bibr" rid="ref17">17</xref>) and the references therein).
To distinguish between Kuhn segments and degrees of polymerization
of the simulated chains, we denote the latter by small letters <italic>p</italic> and <italic>n</italic> for melt and guest chains, respectively.
We relate the degree of polymerization to the number of Kuhn segments
via <italic>p</italic> = <italic>C</italic><sub>∞</sub><italic>P</italic> and <italic>n</italic> = <italic>C</italic><sub>∞</sub><italic>N</italic>.<sup><xref ref-type="bibr" rid="ref18">18</xref></sup> All samples of the
present study contain 2<sup>17</sup> monomers at a “lattice
occupation density” of 50% of the maximum possible monomer
occupation density, which refers to a monomer number density of 1/16
that is considered<sup><xref ref-type="bibr" rid="ref19">19</xref></sup> as a concentrated
solution with melt-like properties. As a starting point we used a
well-equilibrated monodisperse melt of chains containing 512 monomers
each. The degree of polymerization <italic>p</italic> of the bulk
material and test chain degree of polymerization <italic>n</italic> were chosen as <italic>p</italic> = 2<sup><italic>i</italic></sup> with <italic>i</italic> = 0, ..., 9 and <italic>n</italic> = 2<sup><italic>j</italic></sup> with <italic>j</italic> = 3, ..., 9. A randomly
selected fraction of 1/32 of the 512-mers was cut down to degree of
polymerization <italic>n</italic>, while the 31/32 of all 512-mers
were cut down to <italic>p</italic>. This volume fraction is sufficiently
low that the <italic>n</italic>-mers in all samples are below their
overlap volume fraction. Thereafter, the samples were relaxed for
at least one more relaxation time of the longest chains in the sample.
Furthermore, it was checked that the melt chains reached conformations
with corrections to ideal behavior as described in ref (<xref ref-type="bibr" rid="ref12">12</xref>). Afterward, conformations
were sampled for a duration of 10<sup>9</sup> simulation steps for <italic>n</italic> = 512, which is roughly five relaxation times of the longest
chains in a monodisperse melt. For shorter chains, the conformations
were sampled for at least 20 relaxation times, as defined by the end-to-end
vector autocorrelation time. Error bars for the root-mean-square radius
of gyration <italic>R</italic> were computed from the mean fluctuations
of the ensemble average of the <italic>R</italic> data as a function
of time divided by the square root of the number of relaxation times
of the chains in order to provide an accurate estimate of the statistical
significance of each data point.</p><p>The classical works based on
the Flory theorem<sup><xref ref-type="bibr" rid="ref1">1</xref>−<xref ref-type="bibr" rid="ref6">6</xref></sup> predict a universal plot for chain swelling ratio α = <italic>R</italic>/<italic>R</italic><sub>0</sub> as a function of <italic>Z</italic> ∝ <italic>N</italic><sup>1/2</sup>/<italic>P</italic>. Figure <xref rid="fig1" ref-type="fig">1</xref>, which is a naive plot of α<sup>2</sup> as a function of <italic>N</italic>/<italic>P</italic><sup>2</sup> ∝ <italic>Z</italic><sup>2</sup> using <italic>N</italic>/<italic>P</italic><sup>2</sup> = <italic>nC</italic><sub>∞</sub>/<italic>p</italic><sup>2</sup> with Flory’s characteristic
ratio <italic>C</italic><sub>∞</sub> = 1.52 from ref (<xref ref-type="bibr" rid="ref12">12</xref>), shows that this procedure
does not lead to a collapse of the data. Thus, neither the experimental
data<sup><xref ref-type="bibr" rid="ref8">8</xref>−<xref ref-type="bibr" rid="ref11">11</xref></sup> nor our simulation data agree with the classical prediction that
the swelling ratio α = <italic>R</italic>/<italic>R</italic><sub>0</sub> is a universal function of the interaction parameter <italic>Z</italic> ∝ <italic>N</italic><sup>1/2</sup>/<italic>P</italic>.</p><fig id="fig1" position="float"><label>Figure 1</label><caption><p>Lack of overlap of the simulation data of the ratios of the mean
square radii of gyration <italic>R</italic><sup>2</sup> of chains
with <italic>N</italic> Kuhn segments in a melt of chains with <italic>P</italic> Kuhn segments to their ideal mean square radii of gyration <italic>R</italic><sub>0</sub><sup arrange="stack">2</sup> = <italic>b</italic><sup>2</sup><italic>N</italic>/6 using the classical
scaling variable <italic>N</italic>/<italic>P</italic><sup>2</sup>.</p></caption><graphic xlink:href="mz-2014-00777r_0001" id="gr1" position="float"></graphic></fig><p>Several previous experimental
studies proposed a significantly
different scaling of the chain size<sup><xref ref-type="bibr" rid="ref7">7</xref>,<xref ref-type="bibr" rid="ref10">10</xref>,<xref ref-type="bibr" rid="ref11">11</xref></sup> as a function of <italic>N</italic>/<italic>P</italic> instead of <italic>N</italic><sup>1/2</sup>/<italic>P</italic>.
For this phenomenologial scaling, we obtain a better but yet not satisfactory
overlap of the simulation data in Figure <xref rid="fig2" ref-type="fig">2</xref>.
Notably, large deviations are found for <italic>N</italic>/<italic>P</italic> < 10 in Figure <xref rid="fig2" ref-type="fig">2</xref>. This figure
also contains a comparison<sup><xref ref-type="bibr" rid="ref20">20</xref></sup> with the data
of Landry.<sup><xref ref-type="bibr" rid="ref7">7</xref></sup> Both Figures <xref rid="fig1" ref-type="fig">1</xref> and <xref rid="fig2" ref-type="fig">2</xref> are suggesting that significant
corrections for the unswollen regime are necessary.</p><fig id="fig2" position="float"><label>Figure 2</label><caption><p>Partial overlap of the
ratios of mean square radii of gyration <italic>R</italic><sup>2</sup> of chains with <italic>N</italic> Kuhn segments
in a melt of chains with <italic>P</italic> Kuhn segments to their
ideal mean square radii of gyration <italic>R</italic><sub>0</sub><sup arrange="stack">2</sup> = <italic>b</italic><sup>2</sup><italic>N</italic>/6 using the nonclassical scaling variable <italic>N</italic>/<italic>P</italic> of Landry.<sup><xref ref-type="bibr" rid="ref7">7</xref></sup> Same symbols for simulation data as in Figure <xref rid="fig1" ref-type="fig">1</xref>; experimental data of ref (<xref ref-type="bibr" rid="ref7">7</xref>) are depicted by black stars.</p></caption><graphic xlink:href="mz-2014-00777r_0002" id="gr2" position="float"></graphic></fig><p>Recently, it was emphasized<sup><xref ref-type="bibr" rid="ref12">12</xref>,<xref ref-type="bibr" rid="ref13">13</xref></sup> that the intramolecular
bond correlation function in dense melts decays as a power law in
contrast to an exponential decay for chains without long-range correlations.
This leads to a partial swelling of polymer chains even in monodisperse
melts with the mean square radius of gyration of a chain with <italic>n</italic> monomers approximated<sup><xref ref-type="bibr" rid="ref12">12</xref></sup> by<disp-formula id="eq6"><graphic xlink:href="mz-2014-00777r_m006" position="anchor"></graphic><label>6</label></disp-formula>with characteristic ratio<disp-formula id="eq7"><graphic xlink:href="mz-2014-00777r_m007" position="anchor"></graphic><label>7</label></disp-formula>The coefficient <italic>c</italic> = 0.656,
root-mean-square bond length <italic>l</italic> = 2.636, and <italic>C</italic><sub>∞</sub> = 1.52 were determined in ref (<xref ref-type="bibr" rid="ref12">12</xref>) at simulation conditions
identical to the present study. We expect that long <italic>n</italic>-mers dissolved in small <italic>p</italic>-mers start swelling from
this new reference chain size <italic>R̅</italic><sub>0</sub>. Thus, we define the expansion factor α̅ ≡ <italic>R</italic>/<italic>R̅</italic><sub>0</sub> with this new reference
size for the discussion below.</p><p>To test this hypothesis quantitatively,
we compute the new swelling
ratio α̅ for all data and solve eq <xref rid="eq2" ref-type="disp-formula">2</xref> numerically. Note that we return here to the original interaction
parameter, eq <xref rid="eq3" ref-type="disp-formula">3</xref>, with the scaling variable ∝ <italic>N</italic>/<italic>P</italic><sup>2</sup> where <italic>N</italic> and <italic>P</italic> are numbers of Kuhn segments in test and
melt chains, respectively. The excluded volume <italic>v</italic> is
the only adjustable parameter. We obtain <italic>v</italic>/<italic>b</italic><sup>3</sup> = 0.17 from a best fit to all simulation data.
Figure <xref rid="fig3" ref-type="fig">3</xref> shows that the above correction improves
the overlap of the data at small <italic>N</italic>/<italic>P</italic><sup>2</sup> < 1, but no unique crossover function of the data
is obtained. Instead, the data show a systematic shift for small <italic>p</italic>.</p><p>For a lattice model with lattice constant smaller
than monomer
size or for off-lattice models, the excluded volume of a single monomer
is larger than the bare volume of a monomer in contrast to regular
lattice models including Flory–Huggins theory. Furthermore,
denser packing with submonomer-size spacing between monomers becomes
possible in models with a grid finer than monomer size. Thus, the
gain in the net conformational entropy when placing chain ends in
nearest positions next to inner monomers leads to a denser packing
of chain ends next to other monomers similar to the enrichment of
chain ends near a solid wall.<sup><xref ref-type="bibr" rid="ref21">21</xref></sup> This is
in accord with the fact that we detected a clearly larger fraction
of <italic>p</italic>-mer chain ends as compared to inner monomers
in nearest-neighbor positions of the monomers of <italic>n</italic>-mers.</p><fig id="fig3" position="float"><label>Figure 3</label><caption><p>Ratios of mean square radii of gyration <italic>R</italic><sup>2</sup> of chains with <italic>N</italic> Kuhn segments in a melt
of chains with <italic>P</italic> Kuhn segments to their mean square
radii of gyration <italic>R̅</italic><sub>0</sub><sup arrange="stack">2</sup> in monodisperse melt corrected for long-range
bond correlations (eq <xref rid="eq6" ref-type="disp-formula">6</xref>). The line is the best
fit of all data to the numerical solution of eq <xref rid="eq2" ref-type="disp-formula">2</xref> for swelling ratio α̅ = <italic>R</italic>/<italic>R̅</italic><sub>0</sub> with <italic>v</italic>/<italic>b</italic><sup>3</sup> = 0.17.</p></caption><graphic xlink:href="mz-2014-00777r_0003" id="gr3" position="float"></graphic></fig><p>An <italic>n</italic>-mer in a
melt of <italic>p</italic>-mers
is in contact with ∝ <italic>n</italic>/<italic>p</italic> ends
of surrounding chains. Since the ends of <italic>p</italic>-mers pack
closer to <italic>n</italic>-mers, the effective volume fraction ϕ̅
excluded by <italic>n</italic>-mers decreases with increasing concentration
of ends of <italic>p</italic>-mers (with decreasing <italic>p</italic>). Similarly, the ends of <italic>n</italic>-mers contribute less
to the total excluded volume of <italic>n</italic>-mers by a closer
packing to surrounding monomers. In consequence, the inner monomers
of the <italic>n</italic>-mers experience a different packing of ends
of the surrounding <italic>p</italic>-mers. Such corrections to ϕ
proportional to the concentration of chain ends of <italic>p</italic>-mers are expected to be<disp-formula id="eq8"><graphic xlink:href="mz-2014-00777r_m008" position="anchor"></graphic><label>8</label></disp-formula>with numerical
constants <italic>y</italic> and an <italic>n</italic>-dependent <italic>z</italic><sub><italic>n</italic></sub> that can be determined directly
from simulation
data.</p><p>To detect whether there are indeed such corrections to
the effective
excluded volume fraction ϕ̅<sub><italic>n</italic></sub> (eq <xref rid="eq8" ref-type="disp-formula">8</xref>), we performed the following analysis:
First, we calculated the number fraction of lattice sites that are
accessible for inserting an additional monomer in all bidisperse blends.
Next, we removed all <italic>n</italic>-mers (a number fraction of
1/32 of all monomers) from all snapshots of the bidisperse samples
and repeated this analysis. The difference in the number fraction
of accessible sites for bidisperse samples with <italic>n</italic>-mers removed and the original bidisperse samples measures the effective
volume fraction ϕ̅<sub><italic>n</italic></sub> that is
blocked by <italic>n</italic>-mers. The results of this analysis display
a reasonable agreement with the correction proposed in eq <xref rid="eq8" ref-type="disp-formula">8</xref> as shown in Figure <xref rid="fig4" ref-type="fig">4</xref>. As
we see from the fit, the effective excluded volume fraction for long
matrix chains <italic>p</italic> ≫ 1 increases with degree
of polymerization of test chains <italic>n</italic> as ϕ̅
∝ ϕ(1 – <italic>y</italic>/<italic>n</italic>),
due to the smaller excluded volume of chain ends. The asymptotic value
ϕ = 0.0194 for <italic>n</italic>, <italic>p</italic> ≫
1, corresponds to an average of 9.93 lattice sites excluded per test
chain monomer. The effect of the ends of <italic>p</italic>-mers described
by parameter <italic>z</italic><sub><italic>n</italic></sub> ≈
0.26(1 – 1.2/<italic>n</italic>) (see Figure <xref rid="fig4" ref-type="fig">4</xref> inset) increases roughly with the fraction of inner monomer
sections 1 – 1/<italic>n</italic> at which the ends of <italic>p</italic>-mers prefer to pack. In this work we consider relatively
long<sup><xref ref-type="bibr" rid="ref22">22</xref></sup> test chains; therefore, we neglect
the <italic>y</italic>/<italic>n</italic> correction to ϕ̅
in eq <xref rid="eq8" ref-type="disp-formula">8</xref>. For similar reasons, we use the limiting
value of <italic>z</italic><sub>∞</sub> in our analysis below.<sup><xref ref-type="bibr" rid="ref23">23</xref></sup> This reduces the expression for the effective
volume fraction to<disp-formula id="eq9"><graphic xlink:href="mz-2014-00777r_m009" position="anchor"></graphic><label>9</label></disp-formula>with ϕ = 0.0194 and <italic>z</italic><sub>∞</sub> = 0.26 ± 0.01.</p><fig id="fig4" position="float"><label>Figure 4</label><caption><p>Corrections to the volume
fraction of <italic>n</italic>-mers in
binary blends with <italic>p</italic>-mers (eq <xref rid="eq8" ref-type="disp-formula">8</xref>), with ϕ = 0.0194 ± 0.001, <italic>y</italic> = 1.3 ±
0.1, and <italic>z</italic><sub><italic>n</italic></sub> ≈
0.26(1–1.2/<italic>n</italic>) approaching <italic>z</italic><sub>∞</sub> = 0.26 ± 0.01 for large <italic>n</italic> (inset).</p></caption><graphic xlink:href="mz-2014-00777r_0004" id="gr4" position="float"></graphic></fig><p>In the framework of the Flory–Huggins
model, the excluded
volume interaction is<sup><xref ref-type="bibr" rid="ref14">14</xref></sup> ∝ (v/<italic>P</italic>)ϕ̅<sup>2</sup>, and the <italic>p</italic>-dependence of the volume available for <italic>n</italic>-mers is
considered by replacing <italic>P</italic> by <italic>P</italic>/(1
– <italic>z</italic><sub>∞</sub>/(<italic>PC</italic><sub>∞</sub>))<sup>2</sup> with <italic>z</italic><sub>∞</sub> = 0.26 in the expression for the interaction parameter.</p><p>This
leads to a modified interaction parameter<disp-formula id="eq10"><graphic xlink:href="mz-2014-00777r_m010" position="anchor"></graphic><label>10</label></disp-formula></p><p>In Figure <xref rid="fig5" ref-type="fig">5</xref>, we plot the normalized
mean
square radius of gyration of chains with <italic>N</italic> Kuhn segments,
α̅<sup>2</sup> = <italic>R</italic><sup>2</sup>/<italic>R̅</italic><sub>0</sub><sup arrange="stack">2</sup>, as a function of <italic>N</italic>(1 – <italic>z</italic><sub>∞</sub>/(<italic>PC</italic><sub>∞</sub>))<sup>4</sup>/<italic>P</italic><sup>2</sup> ∝ <italic>Ẑ</italic><sup>2</sup>. The parameter <italic>c</italic> in the expression
for mean square reference size of chains, <italic>R̅</italic><sub>0</sub><sup arrange="stack">2</sup>, (eq <xref rid="eq6" ref-type="disp-formula">6</xref> and <xref rid="eq7" ref-type="disp-formula">7</xref>) is varied to optimize
the collapse of the data along the <italic>y</italic>-axis. Optimum
overlap is obtained for <italic>c</italic> = 0.86 ± 0.03, if
data for <italic>N</italic> < <italic>P</italic> are ignored (see
Figure <xref rid="fig5" ref-type="fig">5</xref>). Next, we fit the minimum of<disp-formula id="eq11"><graphic xlink:href="mz-2014-00777r_m011" position="anchor"></graphic><label>11</label></disp-formula>to the data in Figure <xref rid="fig5" ref-type="fig">5</xref> resulting in a best fit for the excluded volume
parameter <italic>v</italic>/<italic>b</italic><sup>3</sup> = 0.30
± 0.01 (see
red solid line in Figure <xref rid="fig5" ref-type="fig">5</xref>). Our optimum value
of parameter <italic>c</italic> = 0.86 is larger than <italic>c</italic> = 0.656 of refs (<xref ref-type="bibr" rid="ref12">12</xref> and <xref ref-type="bibr" rid="ref13">13</xref>) which was obtained for monodisperse melt data ignoring the contribution
of excluded volume interactions. Our slightly larger value of <italic>c</italic> corresponds to a smaller size of test chains in a melt
of infinitely long matrix chains in comparison to weakly swollen chains
in monodispersed melt. The increase of the excluded volume parameter
from <italic>v</italic>/<italic>b</italic><sup>3</sup> = 0.17 in Figure <xref rid="fig3" ref-type="fig">3</xref> to <italic>v</italic>/<italic>b</italic><sup>3</sup> = 0.30 in Figure <xref rid="fig5" ref-type="fig">5</xref> results from the corrections
to the interaction parameter (eq <xref rid="eq10" ref-type="disp-formula">10</xref>).</p><p>The experimental data of ref (<xref ref-type="bibr" rid="ref7">7</xref>) are added to Figure <xref rid="fig5" ref-type="fig">5</xref> using the
optimal value of parameter <italic>c</italic> ≈ 5 ± 3
to ensure <italic>R</italic><sup>2</sup>/<italic>R̅</italic><sub>0</sub><sup arrange="stack">2</sup> = 1 for
small <italic>N</italic>/<italic>P</italic><sup>2</sup>. The best
fit of experimental data to the solution of eq <xref rid="eq11" ref-type="disp-formula">11</xref> is shown by the dashed curve in Figure <xref rid="fig5" ref-type="fig">5</xref> with
excluded volume parameter <italic>v</italic>/<italic>b</italic><sup>3</sup> = 0.38 ± 0.03.<sup><xref ref-type="bibr" rid="ref24">24</xref></sup> We observe
a clearly better collapse of the experimental data as a function of
the corrected interaction parameter ∝ <italic>Ẑ</italic><sup>2</sup> with a chain expansion that follows the classical prediction
instead of a scaling ∝ <italic>N</italic>/<italic>P</italic> as suggested in ref (<xref ref-type="bibr" rid="ref7">7</xref>).</p><p>Interestingly, it is not possible to fully collapse the simulation
data in Figure <xref rid="fig5" ref-type="fig">5</xref> at small <italic>N</italic>(1 – <italic>z</italic><sub>∞</sub>/(<italic>PC</italic><sub>∞</sub>))<sup>4</sup>/<italic>P</italic><sup>2</sup>,
if <italic>N</italic> < <italic>P</italic>. The Flory approach
seems to break down for <italic>N</italic> < <italic>P</italic>, possibly because the surrounding <italic>p</italic>-mers no longer
fit into the pervaded volume of an <italic>n</italic>-mer. Instead,
only sections of ≈ <italic>n</italic> monomers of the larger <italic>p</italic>-mers are overlapping with the <italic>n</italic>-mer.
In consequence, the excluded volume contribution (but not the correction
due to packing of ends) becomes similar to the contribution in monodisperse
melts ∝ <italic>v</italic>/<italic>N</italic> instead of ∝ <italic>v</italic>/<italic>P</italic> for <italic>N</italic> < <italic>P</italic>. This can be taken into account by using a new interaction
parameter<disp-formula id="eq12"><graphic xlink:href="mz-2014-00777r_m012" position="anchor"></graphic><label>12</label></disp-formula></p><fig id="fig5" position="float"><label>Figure 5</label><caption><p>Ratio
of mean square radius of gyration <italic>R</italic><sup>2</sup> of
chains with <italic>N</italic> Kuhn segments in a melt
of chains with <italic>P</italic> Kuhn segments to their mean square
radius of gyration <italic>R̅</italic><sub>0</sub><sup arrange="stack">2</sup> corrected for long-range bond correlations
(eq <xref rid="eq6" ref-type="disp-formula">6</xref>). Abscissa is proportional to the square
of the interaction parameter <italic>Ẑ</italic><sup>2</sup> corrected for the effective volume fraction of test chains (eq <xref rid="eq10" ref-type="disp-formula">10</xref>). Red solid line is best fit of the numerical solution
of eq <xref rid="eq11" ref-type="disp-formula">11</xref> to simulation data, and dashed line
is the fit of solution of eq <xref rid="eq11" ref-type="disp-formula">11</xref> to experimental
data.</p></caption><graphic xlink:href="mz-2014-00777r_0005" id="gr5" position="float"></graphic></fig><p>The resulting plot in Figure <xref rid="fig6" ref-type="fig">6</xref> leads to
an overlap of all data<sup><xref ref-type="bibr" rid="ref25">25</xref></sup> confirming that
these three corrections are necessary to understand the conformations
of <italic>n</italic>-mers in a melt of <italic>p</italic>-mers: (a)
long-range correlations due to intramolecular contacts, (b) <italic>p</italic>-dependence of the volume fraction ϕ̅<sub><italic>n</italic></sub> occupied by <italic>n</italic>-mers, and (c) the
correction to the excluded volume for long matrix chains <italic>p</italic> > <italic>n</italic>.</p><p>Our analysis demonstrates that the
swelling of <italic>n</italic>-mers approaches the classical prediction
for sufficiently large <italic>p</italic> and <italic>n</italic>.
Less than 10% shift from the limiting
case is obtained for our simulation data, if <italic>n</italic> >
100<italic>c</italic> ≈ 86 (shift along <italic>y</italic>-axis)
or <italic>p</italic> ≳ 38<italic>z</italic><sub>∞</sub> ≈ 10 (shift along <italic>x</italic>-axis), when plotting
the data as a function of <italic>Z̃</italic><sup>2</sup>. It
is also evident from our discussions that the effect of correlations
in bond orientations modify predominantly the unswollen regime <italic>N</italic>/<italic>P</italic><sup>2</sup> < 1, while the corrections
to excluded volume are most important for small <italic>p</italic> and, thus, predominantly for <italic>N</italic>/<italic>P</italic><sup>2</sup> > 1. Furthermore, data with <italic>P</italic> > <italic>N</italic> can be collapsed, if the excluded volume is computed as
a function of min(<italic>P, N</italic>). Previous observations<sup><xref ref-type="bibr" rid="ref7">7</xref>,<xref ref-type="bibr" rid="ref10">10</xref>,<xref ref-type="bibr" rid="ref11">11</xref></sup> of the data scaling with <italic>N</italic>/<italic>P</italic> rather than <italic>N</italic>/<italic>P</italic><sup>2</sup> can be understood from the combined effect
of these corrections.</p><fig id="fig6" position="float"><label>Figure 6</label><caption><p>Computer simulation data<sup><xref ref-type="bibr" rid="ref23">23</xref></sup> from
Figure <xref rid="fig5" ref-type="fig">5</xref> with overlap correction to excluded
volume (eq <xref rid="eq12" ref-type="disp-formula">12</xref>).</p></caption><graphic xlink:href="mz-2014-00777r_0006" id="gr6" position="float"></graphic></fig></body><back><notes notes-type="COI-statement" id="NOTES-d42e2261-autogenerated"><p>The authors declare no
competing financial interest.</p></notes><ack><title>Acknowledgments</title><p>The authors acknowledge a generous
grant of computing time
at the ZIH Dresden for the project BiBPoDiA. Financial support was
given by the DFG grants LA 2735/2-1 and SO 277/7-1. MR would like
to acknowledge financial support from the NSF under grants DMR-1309892,
DMR-1121107, DMR-1436201, and DMR-1122483, the NIH under 1-P01-HL108808-01A1,
and the Cystic Fibrosis Foundation.</p></ack><ref-list><title>References</title><ref id="ref1"><mixed-citation publication-type="journal" id="cit1"><name><surname>Flory</surname><given-names>P. J.</given-names></name><source>J. Chem. Phys.</source><year>1949</year>, <volume>17</volume>, <fpage>303</fpage>–<lpage>310</lpage>.</mixed-citation></ref><ref id="ref2"><mixed-citation publication-type="journal" id="cit2"><name><surname>Flory</surname><given-names>P. J.</given-names></name><source>J. Chem. Phys.</source><year>1942</year>, <volume>10</volume>, <fpage>51</fpage>–<lpage>61</lpage>.</mixed-citation></ref><ref id="ref3"><mixed-citation publication-type="book" id="cit3"><person-group person-group-type="allauthors"><name><surname>De Gennes</surname><given-names>P. G.</given-names></name></person-group><source>Scaling Concepts in
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Flory ratio <italic>C</italic><sub><italic>n</italic></sub> by <italic>C</italic><sub>∞</sub>.</p></note></ref><ref id="ref19"><mixed-citation publication-type="journal" id="cit19a"><name><surname>Paul</surname><given-names>W.</given-names></name>; <name><surname>Binder</surname><given-names>K.</given-names></name>; <name><surname>Heermann</surname><given-names>D. W.</given-names></name>; <name><surname>Kremer</surname><given-names>K.</given-names></name><source>J. Phys. (France)
II</source><year>1991</year>, <volume>1</volume>, <fpage>37</fpage>–<lpage>60</lpage>.</mixed-citation><mixed-citation publication-type="journal" id="cit19b"><name><surname>Paul</surname><given-names>W.</given-names></name>; <name><surname>Binder</surname><given-names>K.</given-names></name>; <name><surname>Heermann</surname><given-names>D. W.</given-names></name>; <name><surname>Kremer</surname><given-names>K.</given-names></name><source>J. Chem. Phys.</source><year>1991</year>, <volume>95</volume>, <fpage>7726</fpage>–<lpage>7740</lpage>.</mixed-citation></ref><ref id="ref20"><note><p>To convert molecular weights
to number of
Kuhn segments we used <italic>N = C</italic><sub>∞</sub><italic>M</italic>/(52g/mol) with <italic>C</italic><sub>∞</sub> ≃
10 as done in ref (<xref ref-type="bibr" rid="ref7">7</xref>).</p></note></ref><ref id="ref21"><mixed-citation publication-type="journal" id="cit21"><name><surname>Wu</surname><given-names>D. T.</given-names></name>; <name><surname>Fredrickson</surname><given-names>G. H.</given-names></name>; <name><surname>Carton</surname><given-names>J.-P.</given-names></name>; <name><surname>Adjari</surname><given-names>A.</given-names></name>; <name><surname>Leibler</surname><given-names>L.</given-names></name><source>J. Polym. Sci. B</source><year>1995</year>, <volume>33</volume>, <fpage>2373</fpage>–<lpage>2389</lpage>.</mixed-citation></ref><ref id="ref22"><note><p>Short chains of <italic>n</italic> ≈
10 hardly swell; a 10% correction to <italic>ϕ̅</italic> for <italic>n</italic> ≈ 10, for instance, leads to less
than 2% swelling of chain size, which is about the symbol size in
Figure <xref rid="fig5" ref-type="fig">5</xref>.</p></note></ref><ref id="ref23"><note><p>When using <italic>z</italic><sub>∞</sub> instead of <italic>z</italic><sub><italic>n</italic></sub> a maximum
error of about 1% is found for estimating chain size of <italic>n</italic> = 10.</p></note></ref><ref id="ref24"><note><p>The smallest <italic>p</italic> are ≥24
and thus too large for a reliable determination of <italic>z</italic><sub>∞</sub>. Therefore, we used <italic>z</italic><sub>∞</sub> = 0.26 from the simulation data as a first estimate. Additional
data at small <italic>N</italic> and <italic>P</italic> are required
for a better determination of <italic>c</italic> and <italic>z</italic><sub>∞</sub>.</p></note></ref><ref id="ref25"><note><p>Data
of Landry<sup><xref ref-type="bibr" rid="ref7">7</xref></sup> were not included because no
blends with test chains shorter than
matrix polymers, <italic>N</italic> < <italic>P</italic>, were
reported.</p></note></ref></ref-list></back></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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