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Unformatted text preview: ﬂ‘ httpzﬂhrennanmse.ufl.eduldileuplema616SlReferences1Chain_£xpasion_Faclor_PE_J_Phys_Chem_88_64  Windows Internet Explorer _.... lC;i v ‘g‘ http:ﬂbrennanmse.ufl.edu!dileupfema6lBSIReferences!Chain_Expasion_Factor_PE_J_Phys_Chem_88_6192_Mattice(1934).de V ‘7 X l p '
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E Commerls Expansion of the EndtoEnd Dlstance and Radlus of Gyratlon In Perturbed Polymethylene Chalns Wayne L. Mattice Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 708031804 (Received: April 6, 1984) Two approaches have been employed for evaluation of the expansion of realistic rotational isomericstate models of ﬁnite
polymcthylcne chains. Simulations were used for chains of 100750 bonds in which atoms participating in longrange interactions
behave as hard spheres. An approximate generator matrix method permitted extension of the study to longer chains. The
ratio of expansion factors for the mean square endtoend distance and mean square radius of gyration approaches a limit
that is signiﬁcantly smaller than that estimated from several earlier studies of lattice and offlattice chains with hardsphere
interactions. The present limit for (r2) [(5’) is closer to Debye‘s limit for (Rio/(52).; than limits estimated previously by using lattice chains. Introduction The dimensions of a ﬂexible chain molecule comprised of :1
bonds of length i are often characterized by the mean square
endtoend distance, (r1), or mean square radius of gyration, (s2).
Debye demonstrated that (9)0] (6)0 approaches six as it becomes
infinite.l Zero as a subscript denotes the average for an ensemble
unperturbed by longrange interactions. A much more elusive
quantity is the precise limiting value of (rib/(:2) when chains
are expanded as a consequence of the excluded volume effect.
Alternatively stated, there is some ambiguity as to the high n limit
for inf/a}, where expansion factors are deﬁned as at, = (r1)/ (r1)o
and a}  (sh/(5%. Many investigators have reported vii/at} > 1 for expanded
chains with ﬁnite n. The approaches used include discrete enu "'[zlll . ‘nu  UII‘  ll I ‘h. 1'32} and then checked for the absence of longrange hardsphere in
teractions. Surviving chains provide ensembles of chains expanded by the intramolccular excluded volume effect.10 The values of a} and a} are obtained for chains having )2 as large as 750 by
evaluating (3)0 and (52)!) for the initial ensembles and {r2} and
(52) for those chains that survive hardsphere longrange inter~
actions. The second method makes use of a generator matrix
approximation to the behavior of perturbed polymethylene chains.'3
The usual12 generator matrices are used to calculate (r2)o and
(3)0 for unperturbed polymethylene chains of speciﬁed n, and
the modiﬁed" generator matrices are employed to calculate (r2)
and (:2). The greater computational efﬁciency of generator matrix
calculations permits examination of (1,2 and (2,2 for much longer
chains than those accessible in the simulations. Both the simu
lations and generator matrix calculations predict limiting values in  : 1°f3 ‘i o ﬂ‘ http:,llil:urennan.rnse...t I g LectureS.ppt[Read... 9 Unknown Zone ‘2'. PowerPoint Slide Sho. .. I t, .‘ul, 9:33 AM ﬂ‘ httpzﬂhrennanmse.ufl.eduldileuplema616SlReferences1Chain_£xpasion_Faclor_PE_J_Phys_Chem_88_64  Windows Internet Explorer _.... lC;i v ‘g‘ http:ﬂbrennanmse.ufl.edu!dileupfema6lBSIReferences!Chain_Expasion_Factor_PE_J_Phys_Chem_88_6192_Mattice(1934).de V ‘7 X l p '
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6492 J. Phys. Chem. 1984, 88, 64926494 ‘ m
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E Commerls Expansion of the EndtoEnd Dlstance and Radlus of Gyratlon In Perturbed Polymethylene Chalns Wayne L. Mattice Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 708031804 (Received: April 6, 1984) Two approaches have been employed for evaluation of the expansion of realistic rotational isomericstate models of ﬁnite
polymcthylcne chains. Simulations were used for chains of 100750 bonds in which atoms participating in longrange interactions
behave as hard spheres. An approximate generator matrix method permitted extension of the study to longer chains. The
ratio of expansion factors for the mean square endtoend distance and mean square radius of gyration approaches a limit
that is signiﬁcantly smaller than that estimated from several earlier studies of lattice and offlattice chains with hardsphere
interactions. The present limit for (r2) [(5’) is closer to Debye‘s limit for (Rio/(52).; than limits estimated previously by using lattice chains. Introduction The dimensions of a ﬂexible chain molecule comprised of :1
bonds of length i are often characterized by the mean square
endtoend distance, (r1), or mean square radius of gyration, (s2).
Debye demonstrated that (9)0] (6)0 approaches six as it becomes
infinite.l Zero as a subscript denotes the average for an ensemble
unperturbed by longrange interactions. A much more elusive
quantity is the precise limiting value of (rib/(:2) when chains
are expanded as a consequence of the excluded volume effect.
Alternatively stated, there is some ambiguity as to the high n limit
for inf/a}, where expansion factors are deﬁned as at, = (r1)/ (r1)o
andurh’t'szll (52% Many investigators have reported vii/at} > 1 for expanded
chains with ﬁnite n. The approaches used include discrete enu "'[zlll . ‘nu  UII‘  ll I ‘h. 1'32} and then checked for the absence of longrange hardsphere in
teractions. Surviving chains provide ensembles of chains expanded by the intramolccular excluded volume effect.10 The values of a} and a} are obtained for chains having )2 as large as 750 by
evaluating (3)0 and (52)!) for the initial ensembles and {r2} and
(52) for those chains that survive hardsphere longrange inter~
actions. The second method makes use of a generator matrix
approximation to the behavior of perturbed polymethylene chains.'3
The usual12 generator matrices are used to calculate (r2)o and
(3)0 for unperturbed polymethylene chains of speciﬁed n, and
the modiﬁed" generator matrices are employed to calculate (r2)
and (:2). The greater computational efﬁciency of generator matrix
calculations permits examination of (1,2 and (2,2 for much longer
chains than those accessible in the simulations. Both the simu
lations and generator matrix calculations predict limiting values in  : 1°f3 ‘i o ﬂ‘ http:,llil:urennan.rnse...t I g LectureS.ppt[Read... 9 Unknown Zone ‘2'. PowerPoint Slide Sho. .. I t, .‘ul, 9:4? AM C httpzﬂhrennanmse.ufl.eduldileuplemao16SlReferences1Chain_Expasion_Factor_PE_J_Phys_Chem_88_64  Windows Internet Explorer File
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Introduction The dimensions of a ﬂexible chain molecule comprised of to
bonds of length i are often characterized by the mean square endtoend distance, (r2), or mean square radius of gyration, (:2).
Debye demonstrated that (r2>o/ (52h, approaches six as it becomes
inﬁnite.l Zero as a subscript denotes the average for an ensemble
unperturbed by long—range interactions. A much more elusive
quantity is the precise limiting value of (r’)/(sz) when chains
are expanded as a consequence of the excluded volume effect.
Alternatively stated, there is some ambiguity as to the high it limit
for cal/a}, where expansion factors are defined as at, = (r2)/ (3),, and or,z  (fund)?
any Investigators ave reported uni/c151 > 1 for expanded chains with ﬁnite n. The approaches used include discrete enu
meration of all short selfavoiding lattice chains of speciﬁed n”3
and Monte Carlo simulations of longer selfavoiding freely jointed“
or lattice” chains. lattice chains with nonzero energies assigned
to nonbonded units situated at neighboring lattice sites,3 and
offlattice chains with realistic shortrange interactions and
hardsphere longrange interactions?»10 In some studies in which
chains of various n were examined, little or no trend could be
discerned for the longer chains studied. The average of the results
for the longer chains typically gives uni/a,” of 1.05—1.015’69 A
suitable extrapolation must be devised when the ratio is dependent
on n in the range covered. In the case of the discrete enumeration
of all short selfavoiding lattice chains} extrapolation to l /n n:
0 produced a limiting value of offer} = 1.075 :I: 0.007. Similar
extrapolation for the results obtained in a Monte Carlo study of
freely jointed chains with hardsphere interactions would produce
limiting values in the range 1.0461060.‘ While these investi
gations have not produced exactly the same result, they do agree
that a} is larger than a} for an inﬁnitely long perturbed chain.
They imply 01,2 exceeds a": by 595—895. The behavior of cal/a} is examined here by two methods. The
common starting point is a realistic rotational isomericstate model
for the unperturbed polymethylene chain.“ In the first method, u: ﬂ‘ http:,llil:irennan.rnse.... I ‘2'] Lecture5.ppt [Read... ILSeIect Qv Lisle 150% ‘v 5) l3"
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and then chec ed for the aegsence of longrange hardsphere in teractiOns. Surviving chains provide ensembles of chains expanded
by the intramolecular excluded volume effecr.” The values of a} and or,2 are obtained for chains having it as large as 750 by
evaluating (r2)o and (52),, for the initial ensembles and (r?) and
(52) for those chains that survive hardsphere longrange inter
actions. The second method makes use of a generator matrix
approximation to the behavior of perturbed polymethylene chains.”
The usual12 generator matrices are used to calculate (3)0 and
(52),, for unperturbed polymethylene chains of speciﬁed :3, and
the modiﬁed13 generator matrices are employed to calculate (r2)
and (52). The greater computational efﬁciency of generator matrix
calculations permits examination of a} and a,’ for much longer
chains than those accessible in the simulations. Both the simu
lations and generator matrix calculations predict limiting values
for aE/aﬁ that are closer to unity than previous estimates of
1.05—1.08. The limiting value for (r2)/(s’) is very close to the Db 'l"f (1} 2).
eye tmtt or r n/(s n <5z>/$2>
< o .— 2..
Meatdds D“ s 1 The unperturbed chain was a realistic rotational isomericstate
model for linear polymethylene.“ Bond angles were 112°, dihedral
angles for gauche states were displaced =l:120° from the dihedral
angle for a trans state, and first and second—order interaction
energies were 500 and 2000 cal mol", respectively. The tem
perature was taken to be 300 K. The conﬁguration partition
function was formulated as the serial product of a row vector, n
— 2 statistical weight matrices, and a column vector.” Representative unperturbed chains were generated with a
pseudorandom number generator in conjunction with a priori and
conditional probabilities deduced from the conﬁguration partition
function in the usual manner.12 Chain atoms participating in
longrange interactions were those separated by eight or more
bonds. These atoms behave as hard spheres of diameter r" = 0.81,
1.0!, or 1.2!. The method has been used previously to evaluate ' . . . . ' . . I ‘ . . . . . . . in il ' o t s 9 Unknown Zone ‘2'] Lecture4.ppt [Read. .. r2) zFu—Xﬁ/w tcied ii I 1, .‘ul, 9:49 AM ﬂ‘ httpzﬂhrennanmse.uft.eduldileuplema616SlReterences1Chain_£xpasion_Factor_PE_J_Phys_Chem_88_64  Windows Internet Explorer _..... t\': j v kg http:Hbrennanmse.ufl.edu.fdileupfema6lBSIReferences!Chain_Expasion_Factor_PE_J_Phys_Chem_88_6492_Mattice(1934)de ‘7 X File Edit Go To Favorites Help
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chains 0 various n were examined. little or no trend could be
discerned for the longer chains studied. The average of the results
for the longer chains typically gives afﬁx} of IDS1.07.5“ A
suitable extrapolation must be devised when the ratio is dependent
on n in the range covered. In the case of the discrete enumeration
of all short selfavoiding lattice chains.2 extrapolation to 1 [n =
0 produced a limiting value of turf/a} = 1.075 d: 0.007. Similar
extrapolation for the results obtained in a Monte Carlo study of
freely jointed chains with hardsphere interactions would produce
limiting values in the range l.046—l.060.‘ While these investi
gations have not produced exactly the same result, they do agree
that at,2 is larger than (2,2 for an infinitely long perturbed chain.
They imply a} exceeds or.2 by 5%—8%. The behavior of ref/or} is examined here by two methods. The
common starting point is a realistic rotational isomericstate model
for the unperturbed polymethylene chain.“ In the first method,
ensembles of representative unperturbed chains are generated” (1) Debye. P. J. Chem. Phys. 1946. I4. 636639. (2) Dornb. C.: Hioe. F. T. J. Chem. Phys. 1969, 51. 19151919. (3) Barr. R4 Brender. C.: Lax. M. J. Chem. Phys. 1981. 75. 453459. (4) Buuutgarluet‘, A4 Binder. K. J. Chem. Phys. 1979. 71. 25412545. (5) Wall, F. T.; Erpenbcclt, J. J. J. Chem. Phys. 1959. 30. 637—640. (5) Kron. A. K.; Ptitsyn. O. B. Vysokomol. Sordin. Ser. A 1967. 9.
“9—764. (7) Jurs. P. C.; Reissner. I. E. J. Chem. Phys. 1971. 55. 49484951. (B) McCracltin. F. L.: Mazur. 1.; Gunman. C. M. Macromolecules 1973.
6. 859—871. (9) Winnik. M. A.: Rigby. D.: Stepto. R. F. T.: Lemaire. B. Macromole
cules 1980. 13. 699—704. (10) Muttice. W. L. Macromolecules 198]. 14. 14851490. (1 l) Abe. A.; Jernigan. R. L.; Flory. P. .l. J. Am. Chem. Soc. 1966. 88.
631639. (12) Flory. P. .l. Macromolecules 1974, 7, 331—392. 1 :
D T In": Page  Tools v )1) The unperturbed chain was a realistic rotational isomericstate
model for linear mlymethylene.“ Bond angles were 112°, dihedral
angles for gauche states were lsplaced :l:120° from the dihedral
angle for a trans state. and ﬁrst and secondorder interaction
energies were 500 and 2000 cat Incl", respectively. The tem
perature was taken to be 300 K. The configuration partition
function was formulated as the serial product of a row vector. :2
— 2 statistical weight matrices, and a column vector.” Representative unperturbed chains were generated with a
pseudorandom number generator in conjunction with a priori and
conditional probabilities deduced from the conﬁguration partition
function in the usual manner.12 Chain atoms participating in
longrange interactions were those separated by eight or more
bonds. These atoms behave as hard spheres of diameter r" = 0.81.
1.01, or 1.2!. The method has been used previously to evaluate
expansion factors for linearm'14 and branched” polymethylenes,
as well as the asymmetry of the instantaneous conﬁgurations of
perturbed chains.” The number of chains generated at each n
ranged from l0 000 to 32 000. with the largest number of chains
being for the largest n. For the longest chain. hardsphere in
teractions rejected 89.5% and 95.4% of the chains when r" was
0.8! and 1.21. respectively. Comparison of the orE/a, for the three
different 1'" values at n = 750 suggests the standard deviation in
cal/oz} is 0.004 at this n. The conventional generator matrix scheme12 was used to cal
culate (23)., and (s1)... Generator matrix calculations that ap
proximate the chain expansion produced by repulsive longrange (13) Mattice. W. L.; Santiago, G. Macromolecules 1980. [3. 15601567.
(14] Mattice. W. L. Macromolecules 1981. 14. 1491—1495.
(15) Mattice. W. L. Macromolecules 1984. 17. 415418. 00223654 / 84/ 20886492501 .50 / 0 © 1984 American Chemical Society Perturbcd Potymethylene Chains The Journal of Physical Chemistry. Vol. 88. No. 26. 1984 6493 ﬂ‘ http:lili’hrennanmse. g LectureS.ppt [Read... N O 1', 9 Unknown Zone ii Lecturedppt [Read. .. I 1, .‘ul, 9:52 AM ﬂ‘ httpzﬂbrennanmse.ufl.eduldileuplemao16SlReferences{Chain_Expasion_Factor_PE_J_Phys_Chem_88_64  Windows Internet Explorer v http:ﬂbrennanmse.ufl.edu!dileupfema6lBSIReferences!Chain_Expasion_Factor_PE_J_Phys_Chem_BB_6492_Mattice(1934)de v] $, x L:
File Edit Go To Favorites Help “If? '19? Ighttp:ﬁbrennan.mse.t.fl.edt.l.fdieup.fema61651'Referen...i “Search B Save a Copy 5 Attachments .' Commerls {3 Done 'wn @ ' ' Lig'Page  {ilTools v ,— Perturbed Poiymethylene Chains l0 IOOOIn Figure 1. Expansion factors for perturbed polymethylene chains of n
bonds. Atoms participating in longrange interactions behave as hard
spheres with the r'/! indicated for each curve. interactions were performed by making correlated alterations in
elements in the statistical weight matrices.” This approach re
produces the asymptotic dependence of a5 on n'ﬂ.”v1‘ When
properly parameterized, it correctly reproduces the following
properties of expanded chains: the expansion factor for the mean
square endtoend distance of a subchgin of 1' bonds passes through is <3 iébﬁi t ﬂ‘ http:Illu’hrennanirnse...‘ 1!] LectureS.ppt [Readu. E. Lecture4.ppt [Read... ijfIFSeled Qv 150% lv G) D" '35? 4“ ' ﬁrm Figure 1 Ratios of expansion factors for perturbed poly ethylene chains:
(A) simulation, 1“ = 1.2!; (B) simulation, r' = 0.8!; (C) generator
matrix, K = 0.28, 0.64 < beC < 1.053. Dashed lines in (A) and (B)
denote linear extrapolations of the offer} for chains of 100—400 bonds.
Dotted lines denote alternative extrapolations that take into consideration
the chains of 500 and 750 bonds. Limits suggested by three earlier
studies of selfavoiding lattice chains are denoted by DH. WE. and KP
(see text for details). vaiiueoiwsotltatexoansionjactcrsforchainswithnofLm—Isﬂ
i> 0 9 Unknown Zone 1 I t, Iv}. 9:53 AM Unis C httpzﬂhrennanmse.ufl.eduldileuplema616SlRelerences1Chain_£xpasion_Faclor_PE_J_Phys_Chem_88_64  Windows Internet Explorer File
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steps were studied on the simple cubic lattice.‘5 Linear leastsquares
extrapolation of (55/ (r2) vs. 1 fr: yields an intercept of 0.159,
corresponding to the elf/er} denoted by KP in Figure 2. Limits
suggested by the longer chains in the simt.tlation.s"'7 are somewhat
lower than those derived from discrete enumeration of shorter
selfavoiding walks. ' The most pertinent previous study of offlattice chains was
performed by Winnik et al.’ They employed a model nearly
identical with the one used in the present work. Their ﬁrst and
secondorder interaction energies were slightly different, causing
their unperturbed chain to have a slightly greater preference for
trans states. They also used a larger hard sphere. corresponding
to r"  1.751. The greater attrition arising from their larger r‘
prohibited study of chains as long as those used in Figures 1 and
2. When the temperature was 298 K. "of/at} was found to be
1.03 for the two longest chains (n of 60 and 100). That result
is only slightly larger than the offer} for the shortest chain in
Figure 2A. Therefore, we ﬁnd reasonable agreement with the
dimensions reported in a study that employed a similar offlattice
chain, even though our results give smaller elf/or} than those found
earlier with lattice chains. The present simulations of perturbed polymethylene chains
would have led to the prediction of 1.028 :I: 0.005 as the limit for
affix} at infinite n if the only chains studied were those of
100—400 bonds. That result would have been interesting by itself
because higher limits were obtained in several earlier studies of
chains in which atoms participating in longrange interactions
behave as hard spheres?“7 The current results take on added
interest when the two longest chains, comprised of 500 and 750
bonds, are included in the analysis. They show that the true limit
must be smaller than 1.028 because 6(a,2/a,3)/a(1/n) becomes
positive at very small 1 /n. Plausible extrapolations might lead
to a limit for ref/cg: that is as small as one. Available computational resources do not permit extension of H 4 l I g Lecture5.ppt [Read... g http:ﬂbrennanmse.ufl.edu!dileupfemaﬁlBSIReferences!Chain_Expasion_Factor_PE_J_Phys_Chem_88_6192_Mattice(1934).de ILSelect Qv Lisle 150% ‘jv 6) El" 35C: “*7 may J. Phys. Chem. 1984, 88. 64946499 more} a52)/6(l /n) is positive at very small 1 /n, and the limit for
cal/a} is smaller than 1.03. Generator matrix calculations suggest
the limit could conceivably be less than one. The lower limits are
obtained when the generator matrix caICulation is parameterized
so that the perturbation is felt preferentially by subchains situated
in the middle of the main chain. Emphasis here has been on (Ira/£252 obtained in simulations of
the behavior of offlattice chains in which atoms participating in
longrange interactions behave as hard spheres and the relationship
of those simulations to the afﬁx} obtained by using an approx
imate generator matrix scheme. It is pertinent, however, to cite
two recent estimates for the limiting turf/a} that were obtained
by other methods. Using a very simple generalization from a one
to a threedimensional system, des Cloizeaux and Noda estimated
the limit to be 1.015.22 The renormalization group approach ﬁnds
a limit of 1.01.23 These two limits could be obtained by reasonable
extrapolation of the simulations reported in Figure 2A,B, taking
advantage of the curvature apparent with the chains comprised
of 500 and 750 bonds. The two limits are only slightly outside
the range suggested by the generator matrix results depicted in
Figure 2C. Summary Simulations described here find the limit for turf/at,2 to be
smaller than that suggested by several earlier studies of lattice
and offlattice chains in which atoms participating in longrange
interactions behave as hard spheres. An approximate generator
matrix scheme yields r.r,.2/ocJ compatible with those seen in the
present simulations. The limit for a,2/a,2 is very close to one.
signifying that the limit for (Fl/(52) is nearly identical with
chyc’s result for {r2)0/ (5%. Acknowledgment. This research was supported by National
Science Foundation Grant DMR 8315547. The manuscript was
written while the author was on sabbatical leave at Cornell can A ._I I n I! I l I 9 Unknown Zone ﬁi Lecture‘t.ppt [Read. .. )) I ‘v I“), 113:th AM ﬂ‘ http:Illlhr'ennanmse. File Edit GE} v httpiﬂbrennanmse.ufl.edu,fdileupfema61BSIReferencesIRadile_Gwation_Sty_Butadiene_Zhang__JAPS_96_2005.pdf Go To Favorites Help i? 49' lghttp:r:brmn.mse.trl.eamemimasissmereren...l
gESaweaCopy g; “Search glEIIbSelect LL ‘ :1 l e 150% chased from (KUMIIO, Korea). Tetrahydrofuran
(TIIF) and ethanol were analytical grade and were
purchased from Tianjing Fucheng Chemical Reagent
Corporation in China. The SBS copolymer 802 was
purified by redissolution in THF and precipitation in
ethanol at room temperature. The sample 802 was
further fractionated by reprecipitation from THF so—
lution to ethanol at room temperature to obtain three
fractions coded as 802F1, 802F2, and 802F3, respec
tively. The harvested fractions were dried under re
duced pressure. The final fraction was rotary evapo rated under reduced pressure at 35°C to give a semi
solid coded as 802P4. NMR analysis 1H NMR spectrum was recorded on a Mercury 300 NB
NMR spectrometer (Varian Inc., USA) with 300 MHz
at 25°C. The spinning speed, pulse delay, and total
numbers of scans were, respectively, 15 Hz, 15 s, and
2048. The sample was dissolved in deuterated chloro
form (CDCIB) to prepare a concentration of 150 mg/
ml. Viscometry measurement THF as solvent was freshly distilled prior to use. In
trinsic viscosities ([73]) of the polymer solutions were
measured at 25 : 01°C using an Ubbelohde capillary
viscometer. The kinetic energy correction was always
negligible. Huggins and Kraemer equations were used
to estimate the [11] value by extrapolation to concen (‘ httpzllbrennanmse.ufl.eduldileuplema61651ReferencesIRadius_Gyration_Sty_Butadiene_Zliang_JAPS_9 — Windows Internet Explorer a increments (tin/dc) were measured using an double
beam differential refractometer (DRM—1020, Otsuka
Electronics Co. Japan) at 633 nm and 25°C. The tin/dc
values of the samples in TIIF solutions were deter
mined to be 0.145 mL/ g for fourarm 888 and 0.133
mL / g for linear SBS. Astra software (Version 4.70.07)
was utilized for data acquisition and analysis. From the LLS data, we were able to obtain the
weightaverage molecular weight (MW) and the aver
age rootmean square radius of gyration ((52)1/2) of
polymer in dilute solution from the Zimm plot by Kc 1 16712112 2 . 2
12—270 + M, (5)5111 (9/2)+... +2azc (3)
3 w L 0 f
y K _ 4712112 (dn 2 4
— dC ( ) where A2, NA, 11, and he are the second Virial coeffi
cient, the Avogadro number, the solvent refractive
index, and the wavelength of the light in a vacuum,
respectively. SECLLS measurements SECLLB measurements were carried out on size ex
clusion chromatography combined with multiangle
laser photometer mentioned above combined with a
P100 pump (Thermo Separation Products, San Jose,
CA) equipped with columns of G4000H8 (MicroPak, @ ' l:_,}l=*age  v‘ijoals v q"x __‘. lpv Links ” )1) QM 4i» Moo f: http:,ii'turennaanse. LectureS.ppt [Re3d. .. 1:: Lecture'ippt [Read... a Lhkmwn Zone I u i; 10:08 0M f.“ httpzllbrennanmse.ufl.eduldileuplema61651ReferencesIRadius_Gyration_Sty_Butadiene_Zliang_JAPS_9 — Windows Internet Exploter GTE3r v httptﬂbrennanmse.ufl.edu,ldileupfema61BSIReferencesIRadile_Gwation_Sty_Butadiene_Zhang__JAP5_96_2005.pdf File Edit 31’ 49' lghttp:r:brmn.mse.trl.edtmetpimamssmereren...l
gESaweaCopy g; “Search ,‘EIIbSelect L l :1 l e 150% Go To Favorites Help MOLECULAR ARCHITECTURE OF SBS COPOLYMERS 6 8 10
Elution Volume (mL) Figure 1 SEC chromatograms for the fractions and unfrac
tioned samples from SBS 802: (a) unfractioned SBS 802; (b) 802F1; (C) 802F2; (d) 802F3. RESULTS AND DISCUSSION
Structure of star SBS Figure 1 shows SEC chromatograms for the fractions
and unfractionated sample of SBS 802. The values of
MW, (52)”2, [n], and arm number (1‘) are summarized
in Table I. From the results, the Mw of 802F1 is four . 9:33 ll @ : ’M QZYM perimental points in the low molecular weight range
have been neglected. The straight lines fitting the ex
perimental points from SEC chromatograms are rep
resented, respecuvely, by o. L
(52)”2 = 2.62 X 10( M3?" )for star, nm) (5) (52)”2 = 5.13 X 10'2 Mazwﬁor linear, nm) (6)
.50 _
Usually, the exponent a of flex: le polymers 1n a good
solvent is in the range from 0.5 to 0.6. The a values of
the star and linear SBS indicate the characteristic of
ﬂexible polymer.
Flory viscosity factor is represented by ¢ = [a] MilM66203” (7)
________ The values of qb for the samples were calculated to be 1.78 X 1023 mol‘1 for star SBS and 1.08 X 1023 mol—1 for linear SBS from MW and [n] in Table 1. Reported d) values of P8 in benzene and cyclohexane lie in the range from 1.6 X 10:13 to 3.0 X 10 3 mol‘1 and decrease
first sharply and then gradually as radius expansion
factor as (=(Sz)/(Sz)o). Our 4) values are lower than
that of PS, owing to the heterogeneous structure of
SBS and different solvent. The d) value of star SBS is
larger than that of linear SBS because of the relatively low radius expansion factor as of star one. The (t: is
ahnnﬁmdpnpndpnlnflm in ﬂmlimjipd M 1:211:10? QM 4?» “‘09 f: http:,i,I':urennan.rnse. LectureS.ppt [Read. .. Ell Lecture'tppt [Read... a Lhkmwn Zone @ ' l:_,}l=*age  v‘ijoals v q"x __‘_ lpv Links ” )1) I u 1: 10:15AM File Edit GTE3r v httpiﬂbrennanmse.ufl.edu,fdileupfema61BSIReferencesIRadiIJs_Gwation_Sty_Butadiene_Zhang__JAPS_96_ZDDS.pdf Go To Favorites Help i} 49' lgMp:ffbranan.mse.dl.e¢ddiem1m6165fﬂeferen...’
:Emacwv of. e. tiearm ginIrsm iii 2% Lin MW, (52)”2, [n], and arm number (f) are summarized
in Table I. From the results, the MW of 802F1 is four
times that of 802—P3, while the 802—F2 is twice that of
802F3. This indicates that the sample 802 contains
single, two, and fourarm copolymers and has been
successfully fractionated to obtain three fractions, hav
ing different arm numbers. The predominant species
of 802 is fourarm starshaped SBS 802F1, and its
content has been estimated using the division princi
ple of the SEC chromatogram area to be above 05%.
The 1H NMR spectrum and the assigned chemical
shifts of the 802F1 are shown in Figure 2. The signals
around 4.96 and 5.42 ppm are assigned as vinyl pro
tons of the butadiene unit and those at 1.45, 1.58, and
2.05 ppm are assigned as methyl of the components a,
b, and c, respectively. The schematic structure of the
fourarm starshaped SBS is shown in Figure 3. Molecular weight dependence of (82)”2 and [1]] The SECLLS is an absolute method, since, from the
SEC chromatogram detected by LLS, we can obtain
MW and (52)!” values of numberless fractions, which
have been estimated from many experimental points
in the SEC. Figure 4 shows the comparison of the
molecular weight dependence of (52)” for SBS 802F1
and that of linear SBS 1401 in THF. For low molecular
weights, often the mean square radius has uncertain
ties larger than the values themselves and has not
been considered in calculating the results. So the ex eﬁveoév; f.“ hitpzllbrennanmse.ufl.eduldileuplema61651ReferencesIRadius_Gyration_Sty_Butadiene_Zliang_JAPS_9 — Windows Internet Explorer ' 'arger' an 1a 0 inear I ecauseo ie rea 1ve y low radius expansion factor o:5 of star one. The d: is
almost independent of M“, in the limited M“, range.7
[1]] can be calculated accurately by the Flory viscosity
equation with the substitution of measured (52) and
Flory viscosity constant calculated from the deter
mined (52) and [1]] of the same copolymer. Since, by
substituting the data of (52) into eq. (7), a set of [n]
values was estimated roughly as shown in Figure 5,
and the Mark—Houwink equations for both star and
linear 538 in the MW range from 1.0 X 105 to 3.0 X 105
are the following: [11] = 3.68 X 10‘2 Ma.“ (for star, mL/g) (8)
— M " [n] = 3.73 x 103 M9,” (for linear, mL/g) (9) TABLE I Experimental Results of Molecular Weights and intrinsic
Viscosity of the 838 Star Copolymers in THF at 25°C SECLS LS
Sample [1;] (mL/gl M“. X 10—“l f M“. X 10"1r 802 I98.8 15.8 Mixture (4 / 2/ 1) —
802F1 114.7 16.0 4 14.7
802F2 65.2 8.2 2 —
802F3 34.9 4.3 1 —
1401 69.2 9.6 Linear 10.1
4402 03.7 20.5 4 —
YSOS 120.8 22.0 4 — QM 4i» Moo f: http:,iihrennan.rnse.._. LectureS.ppt [Re3d... Lecture'lppt [Read... a Lhkmwn Zone q"x __‘_ lpv Links ” ﬁ' éngagevﬁToolsv )1) I u 11 10:17AM git? v [éihttpzﬂbrenrtanmse.uf.edu,fdileupfema61BSMefarenceszadiIJs_Gyration_Sty_Butadiene_Zhang_JAPS_96_ZDDS.pdf
File Edit '11,} 49' lghttp:ﬂbrennan.mse.ufl.eduldiemiemaﬁl65fReferen...’
jESaveaCopv 9,; “Search; Atlachrnents f." httpzllbrennanmse.ufl.eduldileuplema61651ReferencesIRadius_Gyration_Sty_Butadiene_thang_JAPS_9 — Windows Internet Explorer Go To Favorites Help swam Q“ a 7 6 5 4 3‘6 150% v © life; Figure 2 'H NMR spectrum and the peak assignments of the sample 802F1. The filled marks in Figure 5 represent the experimen
tal values of Mw and [n] in Table I. The MW values of
the sample 802F1 were measured by LLS and calcu
lated from Zimm plot as shown in Figure 6. It can be
seen that the experimental points for the fourarm star
838 samples 802, Y805, and 4402 are close to the
MW—[n] relationship. This indicates that the devel
oped method, which can be used to establish Mw
dependence of [1]] of star copolymers from one sample
by SECLLS techniques, is feasible. The MarkHou
wink exponent of 0.64 for star 835 exhibits a some—
what lower value in the same solvent compared with
linear 588, which shows considerably higher [1]] vale
ues for the same molecular weight. In recent studies
on hyperbranched polymers, the compact and globu
lar shape are consistent with unusually low values of
[n] and the MarkHouwink exponent (o).13 It appears
that the relatively low values for [n] and a in our f: http:Illu'hrennan.rnse.._. E LectureS.ppt [Re3d... El Lecture4.ppt [Read... samples are a result of a compact starshaped struc
ture, leading to small hydrodynamic volumes. Branching factors The ratios 3 of (52)” and g’ of [n] for the star SBS to
those for the linear 5138 of the same molecular weight
are represented, respectively, by c? = (52): star/(52): linear 5M Liwu at“ (Tl = [Tl]star/[n]linear 3’ = g” (12) 6 Unknown Zone @ ' l:_,}t=*age  fﬁToals v lp' links ” )1) I '2 fr; 10:20AM f: httpzllbrennanmse.ufl.eduldileuplema61651ReferencesIRadius_Gyration_Sty_Elutadiene_Zliang_JAPS_9 — Windows Internet Explorer ﬁx ‘. p
 L 2 v httpzﬂbrennanmse.ufl.edu!dileupfema61Bartleterences!Radius_Gyration_Sty Butadiene_Zhang_JAPS 96 2005.de
K.) _ _ _ File Edit Go To Favorites Help links ” 7 )1) '11,} 4% lghttp:ﬂbrennan.mse.ufl.edufdiet.ip!ema6165fReferen... l ' :_,}l=‘age  Tools v
'ESaveaCopv “Search E‘QTjIIbSelect gqv 150% v G) _:5'fv ‘ﬂv .7 m wink exponent of 0.64 for star 838 exhibits a some at
what lower value in the same solvent compared with (3' = [n]5m/[1]]hnear linear 888, which shows considerably higher [1;] val ues for the same molecular weight. In recent studies 3' = gt on hyperbranched polymers, the compact and globu— lar shape are consistent with unusually low values of [n] and the MarkHouwink exponent (0:113 It appears that the relatively low values for [n] and a in our 5.6 . . 5.4 5.5
logM_ M w Figure 4 Molecular weight dependence radius of gym
tion for fourarm star 838 (—) and linear SBS (  ) in THF at Figure 3 Schematic structure of fourarm starshaped SBS. 25 °C. Attachments MOLECULAR ARCHITECTURE OF SBS COPOLYMERS 9 Unknown Zone f.‘ l'lttp:,l,l'hrenrian.rose.m E Lecturesppt[Read.. 1:] Lecture‘tpptmeadm I '2 1; 10:20 AM ...
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