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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_6-192_Mattice(1934).de V ‘7 X l p ' File Edit Go To Favorites Help [this ” w_ 7 7 7 7 l 7- ' ‘ __ i? <9? Ighttp:flbrennanmseufl.eduldieupiemafilSSIReferen... ‘ ' B ' Page - Tools v BSWSS‘COFY fiSearcn IL Select Qv L l l E.) 150% (:4) D‘, fi' Ym Fl A! 6492 J. Phys. Chem. 1984, 88, 6492-6494 ‘ m E (D E z E E Commerls Expansion of the End-to-End Dlstance and Radlus of Gyratlon In Perturbed Polymethylene Chalns Wayne L. Mattice Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803-1804 (Received: April 6, 1984) Two approaches have been employed for evaluation of the expansion of realistic rotational isomeric-state models of finite polymcthylcne chains. Simulations were used for chains of 100-750 bonds in which atoms participating in long-range 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 end-to-end distance and mean square radius of gyration approaches a limit that is significantly smaller than that estimated from several earlier studies of lattice and off-lattice chains with hard-sphere 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 end-to-end 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 long-range 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‘ - l-l I ‘h. 1'32} and then checked for the absence of long-range hard-sphere 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 hard-sphere long-range 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_6-192_Mattice(1934).de V ‘7 X l p ' File Edit Go To Favorites Help [this ” w_ 7 7 7 7 l 7- ' ‘ __ i? <9? Ighttp:flbrennanmseufl.eduldieupiemafilSSIReferen... ‘ ' B ' Page - Tools v BSWSS‘COFY fiSearcn IL Select Qv L l l E.) 150% (:4) D‘, fi' Ym Fl A! 6492 J. Phys. Chem. 1984, 88, 6492-6494 ‘ m E (D E z E E Commerls Expansion of the End-to-End Dlstance and Radlus of Gyratlon In Perturbed Polymethylene Chalns Wayne L. Mattice Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803-1804 (Received: April 6, 1984) Two approaches have been employed for evaluation of the expansion of realistic rotational isomeric-state models of finite polymcthylcne chains. Simulations were used for chains of 100-750 bonds in which atoms participating in long-range 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 end-to-end distance and mean square radius of gyration approaches a limit that is significantly smaller than that estimated from several earlier studies of lattice and off-lattice chains with hard-sphere 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 end-to-end 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 long-range 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-urh’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‘ - l-l I ‘h. 1'32} and then checked for the absence of long-range hard-sphere 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 hard-sphere long-range 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 7;? <9? |Qhttp:flbrennanmseufl.eduldieupiemafilSSIReferen... I I B Save a Copy fl Search as E (D E z E E Comments _.—--.. i'i 1. \,/ Go To g http:flbrennanmse.ufl.eduIdileupIemaE1BSIReferences!Chain_Expasion_Factor_PE_J_Phys_Chem_88_6-192_Mattice(1934).de Favorites Help I I IN I I using lattice chains. C ”' Introduction The dimensions of a flexible chain molecule comprised of to bonds of length i are often characterized by the mean square end-to-end 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 self-avoiding lattice chains of specified n”-3 and Monte Carlo simulations of longer self-avoiding freely jointed“ or lattice” chains. lattice chains with nonzero energies assigned to nonbonded units situated at neighboring lattice sites,3 and off-lattice chains with realistic short-range interactions and hard-sphere long-range 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—1.015’6-9 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 self-avoiding 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 hard-sphere interactions would produce limiting values in the range 1.046-1060.‘ 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 isomeric-state 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" . k or. a 4“1/ tof3 v ‘y x 1537' . new i“- I’" Y"! I. I' i o o . IA. .H'II ‘Il Ifl' < F 3 , v R and then chec ed for the aegsence of long-range hard-sphere 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 hard-sphere long-range 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.. Meat-dds D“ s 1 The unperturbed chain was a realistic rotational isomeric-state 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 long-range 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 tc-ied 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 3:? <9? |ghttp:flbrennanmseufl.eduldieuplemafil651'Referen... ‘ ' ' B Save a Copy i Attachments 5' Commerls —r r“ 150% jv G) “Search {‘3 ILSeIect Gkv L l S l G) 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 IDS-1.07.5“ 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 self-avoiding 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 hard-sphere 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 isomeric-state 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. 636-639. (2) Dornb. C.: Hioe. F. T. J. Chem. Phys. 1969, 51. 1915-1919. (3) Barr. R4 Brender. C.: Lax. M. J. Chem. Phys. 1981. 75. 453-459. (4) Buuutgarluet‘, A4 Binder. K. J. Chem. Phys. 1979. 71. 2541-2545. (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. 4948-4951. (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. 1485-1490. (1 l) Abe. A.; Jernigan. R. L.; Flory. P. .l. J. Am. Chem. Soc. 1966. 88. 631-639. (12) Flory. P. .l. Macromolecules 1974, 7, 331—392. 1 : D T In": Page - Tools v )1) The unperturbed chain was a realistic rotational isomeric-state 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 second-order 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 long-range 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. hard-sphere 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 long-range (13) Mattice. W. L.; Santiago, G. Macromolecules 1980. [3. 1560-1567. (14] Mattice. W. L. Macromolecules 1981. 14. 1491—1495. (15) Mattice. W. L. Macromolecules 1984. 17. 415-418. 0022-3654 / 84/ 2088-6492501 .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 'w-n @ ' ' Lig'Page - {il-Tools v ,— Perturbed Poiymethylene Chains l0 IOOOIn Figure 1. Expansion factors for perturbed polymethylene chains of n bonds. Atoms participating in long-range 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 end-to-end distance of a subchgin of 1' bonds passes through is <3 iébfii t fl‘ http:Illu’hrennanirnse...‘ 1!] LectureS.ppt [Read-u. 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 self-avoiding 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 7:? <9? lghttp:flbrennan.mse.uf|.eduldieupiemafilSSIReferen... l B Save a Copy an E (D E z E E Commerls _,..—..y n, A v '\.,/ Edit Go To a""'\ Favorites Help _' fiSearch 6494 by WE in Figure 2. Self-avoiding five-choice walks of 197-«1997 steps were studied on the simple cubic lattice.‘5 Linear least-squares 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 self-avoiding walks. ' The most pertinent previous study of off-lattice chains was performed by Winnik et al.’ They employed a model nearly identical with the one used in the present work. Their first- and second-order 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 off-lattice 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 long-range 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_6-192_Mattice(1934).de ILSelect Qv Lisle 150% ‘jv 6) El" 35C: “*7 may J. Phys. Chem. 1984, 88. 6494-6499 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 off-lattice chains in which atoms participating in long-range 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 three-dimensional 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 off-lattice chains in which atoms participating in long-range 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 83-15547. 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 (KUMI-IO, Korea). Tetrahydrofuran (TI-IF) 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 802-F1, 802-F2, and 802-F3, 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 802-P4. 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 TI-IF solutions were deter- mined to be 0.145 mL/ g for four-arm 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 weight-average molecular weight (MW) and the aver- age root-mean 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 _ 471-2112 (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. SEC-LLS measurements SEC-LLB 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 [Re-3d-. .. 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 GTE-3r 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) 802-F1; (C) 802-F2; (d) 802-F3. 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 802-F1 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] Mil-M66203” (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 [Re-ad. .. 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 GTE-3r 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. tie-arm ginIrsm iii 2% Lin MW, (52)”2, [n], and arm number (f) are summarized in Table I. From the results, the MW of 802-F1 is four times that of 802—P3, while the 802—F2 is twice that of 802-F3. This indicates that the sample 802 contains single-, two-, and four-arm copolymers and has been successfully fractionated to obtain three fractions, hav- ing different arm numbers. The predominant species of 802 is four-arm star-shaped SBS 802-F1, 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 four-arm star-shaped SBS is shown in Figure 3. Molecular weight dependence of (82)”2 and [1]] The SEC-LLS 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 802-F1 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 10-3 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 SEC-LS LS Sample [1;] (mL/gl M“. X 10—“l f M“. X 10"1r 802 I98.8 15.8 Mixture (4 / 2/ 1) — 802-F1 114.7 16.0 4 14.7 802-F2 65.2 8.2 2 — 802-F3 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 [Re-3d-... 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 802-F1. The filled marks in Figure 5 represent the experimen- tal values of Mw and [n] in Table I. The MW values of the sample 802-F1 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 four-arm 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 SEC-LLS techniques, is feasible. The Mark-Hou- 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 Mark-Houwink exponent (o).13 It appears that the relatively low values for [n] and a in our f: http:Illu'hrennan.rnse.._. E LectureS.ppt [Re-3d-... El Lecture4.ppt [Read-... samples are a result of a compact star-shaped 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 Mark-Houwink 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 four-arm star 838 (—) and linear SBS (- - -) in THF at Figure 3 Schematic structure of four-arm star-shaped SBS. 25 °C. Attachments MOLECULAR ARCHITECTURE OF SBS COPOLYMERS 9 Unknown Zone f.‘ l'lttp:,l,l'hrenrian.rose.m E Lectures-ppt[Re-ad-.-. 1:] Lecture‘t-pptmead-m I '2 1; 10:20 AM ...
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