EMA6165-lect-09 - pa ‘9 ' U " Lecture4[1].ppt...

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Unformatted text preview: pa ‘9 ' U " Lecture4[1].ppt [Compatibility Mode] - Microsoft PowerPoint .- E‘ X . '3‘ 7 t _ Home ‘ Insert Design Animations Slide Show Review Vicar fl 7 7 (Q I fl' as Cut f: Layout' _ _ M. A.‘ y) |:E _ E Eglllfi Text DIredion' \ \' DO [3 - DE {‘3 Shape rm- flFind A Copy gReset ' ' ' ' _:_] Align Text ' A _I_ ‘L, [1) 4} G - ’7 2 Shape Outline ' :2, Replace ' P t N AV. . - E E E E A ' k . a: e jFormat Painter gust; :3 Delete B I H a“ s H A; A ‘ I” E" '3 =| — ‘ Convert to SmartArt “ (’4 \\f\. { } '5'? 7 Hinge 5%}: . _ Shape Effects ' Le Select ' Clipboard '5 Slides Font =3 Paragraph T? Drawing '4 Editing Polymeric Hala'ids Moose l‘nr mung Ch-I Drrranatona Chain Dimensions Definitions " " H.“._ I I!" I may. — n—un-n—q - “mm - anal-rm ' cmruambonmnme fluctuate-nu -me-qu-Inu uni-lulu- -_l'ludl‘| - Projections of vectors - Mean dimensions - Bond angles ' Statistical Segments EMA 0165 Polymer Physics - AB Brennan 3 Clid< to add notes K' AnthonyB. Brennan El Lecturelppt [Read-... L3,"Le'l:turr3-i[1].i::pt[Comm I”: 1: 9:41AM Chain Dimensions Definitions Projections of vectors Mean dimensions Bond angles Statistical Segments Chi-h. V X2) $4205 EMA 6165 Polymer Physics — AB Brennan .- —- — Chain Dimensions Definitions Paige'H-VfiEM-P T?“ '3 C—C OJShinm - ~ » :e—ofl $31M?” "1 ii} % - . i m be c 0 fi ' PrOJectIons of mlw—EN @_ 3 o p vectors _ r3 - Mean \ dimensions \\ n-l ' Bond angles F ' Statistical segnuflfls’ $15 Guwuug,QM§fifi’AWw I] %1 ’ \J I . ‘ "SQ—SM“ g Svinjie. band)“ a. 5111:3351-ch EMA 6165 Polymer Physics — AB Brennan Q—“j‘rk 3 Chain Dimensions Definitions Projections of vectors Mean dimensions Bond angles Statistical Segments EMA 6165 Polymer Physics — AB Brennan _ I Science ' Freely Jomted Chaln gave—:an Vector Analysis E ?1:<r2>=n/2 +21712 (1+cosc9+cos26+cos36+...cosn—2¢9) +2]2 0056(1+20056 +30052 6+...n cos”—3 6’) which by combining terms and use of a McLauren Expansion one can Show that: EMA 6165 Polymer Physics - AB Brennan 9 >2: Freely Jointed Chain Vector Analysis Englneenng Frag ro+a+mj’/—\m 20m SEW-14ft 1 <l+cos¢9> awe W: <1_COSQ> {Mrs-e l‘/ H l For the Freely Jointed Model there are no restrictions on the valence bond angle, thus c, a"? are EMA 6165 Polymer Physics - AB Brennan 10 Freely Jomted Chaln gag:wa Vector Analysis Englneenng 00st =0; foriij EMA 6165 Polymer Physics - AB Brennan 11 _ I Science ' Freely Jomted Cham saw—ms. Vector Analysis E <cos62j> = O; for 1' 7i j Thus, the FREELY J OINTED CHAIN is defined as: <Fz> 2 n12 Consider how <r2> scales with Molar Mass EMA 6165 Polymer Physics - AB Brennan 11 >231 Freely Rotating Chain Vector Analysis Englneenng Mean Square End to En Dstance 2 $5 (r > = n As shown previously: — — _ 2 <7; r]. >—l <cost> EMA 6165 Polymer Physics - AB Brennan 12 Materials \ Scmfyr—k I.” _ b:-——1‘5\2'\}\. Freely Rotating ChEmginéél-[ng Vector Analysis 72 22M!2 - Freely rotating model: ’V’D (TI/Lam GEM — high temperature — solvated " W W 0Com, “)W T j QTW EMA 6165 Polymer Physics - AB Brennan 14 Materials 3:: Hindered Rotating Chain Model smell?!“— llustration of RIS filofink‘ Englhée'mg EMA 6165 Polymer Physics — AB Brennan 15 Materials BE: Hindered Rotating Chain Model Assumptions “gm mg - Freely rotating model: - high temperature — solvated — 0< 8 <360 — 0< ¢ <360 EMA 6165 Polymer Physics — AB Brennan 16 M 9 Hindered Rotating Chain Model Assumptions - Hindered rotating model: —torsion (valence) angle dependence —first order interactions —high level interactions — 8 <= 109.5 for a 0-0-0 bond angle — o< ¢ <360 f f,“ EMA 6165 Polymer Physics — AB Brennan ‘ 17 M . Science ' Chaln DimenSIons gsfinx. Characteristic Dimensions E 2 2 r O=_Cn[ Where: <r2>0 - chain dimensions at Theta conditions C - Characteristic Chain Ratio n - number of segments 1 - segment dimension (length) EMA 6165 Polymer Physics - AB Brennan 21 Chain Dimensions Chain Expansion Factor D a - chain D \V - interaction expansion factor entropy - C - characteristic D G - Theta condition chain ratio temperature D T — temperature EMA 6165 Polymer Physics - AB Brennan 22 I u I Materials E Chaln DlmenSIons q L-x.---—1“r-\:\‘_"§:fi Chain Expansion Factor Engihée'nng Da=lzwhenT=9 1 » £24. 110. F >-“ 2D“ _> I] l 9 T > 0 :3 (1,0059, |+¢D$zfi$> <r7~>: n2 0 <r2> = <r2>0 at T = q W59 \n460}¢> - Chains behave as phantoms 3‘ ‘l <‘/ > 0 PL 5“ f j *' 6 l rm) 3_ 3 _ _ A came < D a aocgyl TM wife’s) Cwl'tTLa-wb EMA 6165 Polymer Physics — AB Brennan 23 f: http:l!brennan.mse.ufl.eduldileuplema6165IReferencesIPaper1_Chain0imensions.pdf- Windows Internet Explorer 61E} v http:flbrennanmse.ufl.edu!dileupfema61BSMeferencesiPaper1_ChainDirnensions.pdf rVi ‘? X 7 .- — — i p ' Fiie Edit Go To Favorites Help Unis ” )1) @ ' |:_;}Page - -@-Tools v a; 4% 88 ' @Anthonyfljrmilesearc... ighttptflbremmmsmufle... x SaveaCopy L: “Search :‘flflbsawg gqv lele 150% v G) _va 13;} an. m HTI'I'IL HBSTRHCT “’ LII'IHS 3| JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 7 15 AUGUST 2004 Contributions of short-range and excluded-volume interactions to unperturbed polymer chain dimensions Hiromi Yamakawa and Takenao Yoshizaki r\ “a \_ Department ofPolvmer Chemistry; Kyoto Univeizsig; Katsm'a, Kyoto 615-8510, Japan ' (Received 9 March 2004: accepted 26 May 2004) Q A Monte Carlo study is made of the mean-square radius of gyration ($2) for the freely rotating chain with such fictitious excluded-volume interactions that the Lennard-Jones 6-12 potentials at the (t) temperature act only between the foru‘th—tln‘ough (3 + A)th—neighbor beads (A a 1) along the chain. The behavior of the asymptotic value ((Szfln)ac of the ratio (52)!” as a function of the munber n of bonds in the chain in the limit of n—:-1 is examined as a function of A. It is shown that the approach of ((5 2);” ")1. to its value for the real unpertiu‘bed chain with A =06 is so slow that the interactions between even up to about 100th-neighbor beads should be taken into account in order to reproduce nearly its dimension. The result implies that the unperturbed polymer chain dimension as experimentally observed at the (d temperattu'e depends not only 011 short-range interactions but also to a considerable extent on the long-range excluded-volume interactions. and that the asymptotic value C; of the characteristic ratio C}, for the rotational isomeric state model in the limit of n—z-x. which is determined only by the very local conformational energy. cannot be directly compared with the corresponding experimental value. © 2004 American Institute of Pitt'sics. [DOII 10.1063/1.1774155] Attachments I. INTRODUCTION that the unpertlu'bed ((9) state of a polymer chain may be determined only by short-range interactions. Otlieiwise. such A Monte Carlo (MC) study1 has recently been made of a notion Should be altered. effects of chain stiffness and chain ends on the gyration- _ _ _ Now. in the rotational isomeric state (RIS) model.3 radms expausmn faCtor “5 Of a pOIymer Chm“ as demled as (higher-order) short-range interactions consist of those be- £114:- n41.411- ‘1; tells. min a; £114. .nALsn a; "' 'utbbl leHifia 0 Unknown Zone f: l'ItIpJIfII'lJtESFInaT'IJI'ISEL... LectureB.ppt [Read-... 12‘] Lecture‘1[l].ppt [Com... I 'U__ 10:07 AM f: http:llbrennan.mse.ufl.eduldileuplema6165IReferencesIPaper‘E_ChainDimensions.pdf - Windows Internet Explorer lp' 59 v léihttpzflbrennaiimge.ufgu‘fdileupfemafilBSMeferencesfPaper1_ChainDtrnensions.pdf File Edit Go To Favorites Help ‘\ a; ‘ifi 88 ' ngwnyBjrmResearc... lghttpwbremmmsmufle... x 'ESaveaCopy “Search E‘fllIbSelect gqv 150% Attachments n on s. eac 1 o engt 1 unity. an 0 11+ 1 entica ea" 5, whose centers are located at the n— 1 junctions of two suc- cessive bonds and at the two terminal ends. The beads are munbered 0.1.2.. . . .n from one end to the other. and the ith bond vector connects the centers of the (i—1)th and ith beads with its direction from the (i - 1)th to the ith bead. All the 11—1 bond angles H (not supplements) are fixed at H = 109°. so that the configtu‘ation of the entire chain may be specified by the set of 11—2 internal rotation angles {(b,,_2} = ($2 ._ (b3 . . . . _ d),,-1) apait from its position and orientation in an external Canesian coordinate system. where (b; is the internal rotation angle around the ith bond vector. The total (excluded-volume) potential energy U A of the fictitious chain with the interactions only between the fourth- through (3+A)th-neighbor beads (331) as a function of {(b,,_3} may be given by n-4 min(n.1'+3+d) Unwrap; J E '=1'+4 II ( Rf l l 1 ) with R ,- J- the distance between the centers of the 1' th and jth beads. The pair potential "(12) is the cutoff version of the LJ 6-12 potential given by utR_}=="~ for 0$R<co =NUIR) for CU$R<3U =0 for 306R. (2) where 1:”(12) is the LJ potential given by (3) with 0' and 6 the co 131011 diameter and the depth of the . . . L] . ' We I ‘ a .‘ . ‘ I I a E Lecture3.ppt [Re-3d-. .. f: http:,f,I'l:Irennan.rnse.... . G) Lit. 14 4'» >1 12'] Lecture4[l].ppt [Com... inc} 1 , vfl fl“ - m ' " I ' II II out a MC run. The mean-square radius of gyration (52) has been evalu- ated from (5) where the 511111 has been taken over the 105 sample configu— rations. and the squared radius of gyration S 2 as a function of {(15% 2} for each sample configuration has been calculated from _,.. 2—> n rd)- 1 o $2=n+l j; ll‘r‘_rc.m.l“ (a (6) Here. 1',- is the vector position of the center of the ith bead. and rum is that of the center of mass of the sample chain and is given by l H 2 r.» (.7) r = “’1' "+1j=0 All MC runs have been carried out at the 0 temperature T* = 3.72 previouslyl detennined. For a generation of pseu- dorandom numbers. we have used the subroutine package MT19937 { based on the Mersenne Twister algorithm) sup- plied by Matsumoto and Nishimura.9 III. RESULTS AND DISCUSSION 0 Q 9 Unknown Zone @ ' |:_-,}F’age - ffi-Tools v links ” )1) rwwufb flatten I 'u__ 1; 10:14 AM f." http:llbrennan.mse.ufl.eduldileuplema61651Referenceslpaper‘i_ChainDimensions.pdf - Windows Internet Explorer git? v httpzflbrenrianmse.bfledu‘fdileupfemafilBSMeferencesfPaperl_ChairiDimensions.pdf lp' File Edit GD To Favorites Help ‘\ a; 49' 88 ' gmhonyfljrmllesearc... lghttpwbremanmseufle... x jESaveaCopy “Search Atlachrnents (1+cos H) 2 cos2 1‘) 1—('—cosl'a'}”+1 (8) (1+cosfil)| (n+1)2 with 6': 109°. For each A. (52)!” increases monotonically with in- creasing » and approaches its asymptotic value. which we denote by ((551705. . as in the cases of A = x and also the ideal chain (dotted line). The values of (32)/n for A =1. for which only the interactions between the forum-neighbor beads are taken into account. are seen to be smaller than those for the ideal chain. This may be regarded as arising from the fact that in this case the attractive tail of the LJ potential rather than the repulsive core has a dominant effect on the chain dimension to make it smaller than that of the ideal chain. Such a tendency for A = 1 seems to be exhibited by any freely rotating chain having the interaction potential with an attractive tail between beads if the diameter of its repulsive core is not very large compared to the bond length. Although the asymptotic value ((.S'2)/n),c (or C.) as a func- tion of A increases monotonically with increasing A and ap- proaches the value of ((Sz)/n)m for A = 96. i.e._. the value of ((52)0ln)gC for the real chain. the approach of ((52)/n)I to it is seen to be unexpectedly slow. The interactions between even up to about 100th-neighbor beads should be taken into account in order to reproduce nearly the real unpertru‘bed chain dimension. Thus it is concluded that the excluded- volrune interactions at the (9 temperature are of long range rather than of short range. In other words. the rurpertru‘bed polymer chain dimension as experimentally observed at the (7) temperature depends not only on the short-range interac- II.| I.I'll f: http:,l,I'|:urennan.rnse. E Lecture3.ppt [Re-3d-. .. fllItsm :Q'lelelE' © @333 El Lecture1[1].ppt [Com... 'ncli , the constant differential-geornetrical cruvature K0 and torsion 7'0 of its characteristic helix taken at the minimrun zero of its elastic energy. and the shift factor M L as defined as the mo- lecular weight per unit contoru' length. The (unperturbed) HW chain with values of these parameters so determined may be considered to mimic the behavior of any real poly- mer chain of whatever length even at the (9 ternperanu'e. since it is clear that effects of both short-range and long- range interactions as considered above are already reflected in the determined values of A“. K0 . and To. Thus the present problem does not concern the analysis based on the HW chain model. IV. CONCLUSION Possible contributions of long-range excluded-voliune interactions to the unpertru'bed polymer chain dimension (S 2) have been examined by the MC simulation of the freely rotating chain with such fictitious interactions that the LJ 6-12 potentials at the l“) temperature act only between the fourth-through (3 + A)th-neighbor beads (A21) along the chain. It has been found that the asymptotic value ((52)/n)1 of (52W?) in the limit of N—.‘ x as a function of A approaches very slowly the value of ({Sgfln) I for A = I. indicating that the unperturbed polymer chain dimension as experimentally observed at the l") tenrper‘atru'e depends not only on the short- range interactions but also to a considerable extent on the long-range excluded-volrune interactions. The result implies that the value of C x. for the R18 model. which is determined only by the very local conformational energy. cannot be di- rectly compared with the corresponding experimental value. 6 Unknown Zone Links ” @ ' l:_-,}l=*age - ffiToals v I '2 1: 10:17 AM )1) f: hitp:flbrennan.mse.ufl.eduldileuplema61 651Reierencesl1466-entanglen1ents‘3-320at‘9-a20polyme[9-3205urfac 7 Windows Internet Explorer 61;} v [E http:flbrennanmse.ufl.edu!dileupfema61BSMeferences!1466-entarig|ement5%20at“£120po|ymer°102Elsurfaces%20and°fo2l3interfaces.pdf File Edit i} 4% % :lglttpwbrmmsqmflm... X lgmtpwbrenpmmseufledul... SaveaCopv Lt." “Search :‘fllIbSeleot gqv L l :3 l e 150% Atlachments Go To Favorites Help Macromolecules, V01. 29, No. 2, 1996 For simplicity. consider a polymer with molecular weight M where M» M. In such a situation. chain ends can be ignored since their concentration is small. The volume pervaded by a single chain is given by10 v3 = mega” V A,( 3/2 3 (l) = — 1 0 m c where A and A’ are constants. a is the size ofa segment with molecular weight m. and (REZW2 is the root mean square radius of gyration of the chain. VP. the total volume occupied by a chain. is given by _ _ 3 Vp— ma (2) Now. the entanglement molecular weight. M. is defined as the molecular weight where 1/0le is a constant. B. Therefore 1149 = mgr (3) With respect to a bulk melt. the dimensions of any chain sections of molecular weight Mcentered at the interface are unperturbed in a direction parallel to the interface and are compressed normal to the interface. It is not obvious how to define the dimension of the chain normal to the surface in a quantitative manner. Qualitatively, however. the dimension will be approxi- mately halved. Consequently. the pervaded volume is roughly one-half that in the bulk. which is equivalent to dividing A’ in eq 1 by 2. Hence. from eq 3. M in the - G) _va 1:2; 5* - m Communications to the Editor 799 the chain, that one would expect to see some perturba- tions to the average volume pervaded by the chains. The increase in Me. for the arguments presented above. are strictly valid for the modifications to the chain packing in the vicinity of an impenetrable inter- face. Similar increases in 1M.3 should. therefore. be found in any situation where the reflection hypothesis is applicable. Two examples immediately come to mind. The first is at a free surface which effectively acts as an impenetrable interface clue to the energetic costs of having a rough interface. The second is at the interface between two immiscible polymers where enthalpic interactions between the polymers force the interface to be sharp. As discussed by Muller et al..'9 the minimization of the number of contacts per chain between the two polymers should perturb the chain configuration at the interface. This should. also. be relevant at the interface between two identical polymers during the initial stages of interdiffusion. The implication of a reduced entanglement density at the free surface of a glassy polymer on the mechanical response of the near-surface region of the polymer is profound. This situation is complicated. to some extent. by the fact that the glass transition temperature. T8. of the polymer in the near-surface region may be different from that of the biilk.2°‘2“ There is evidence that the near-surface 7;; may be as much as 50 °C lower than the bulk value. In general. entanglements significantly influence the high strain properties of a material. Both the “natural draw ratio" and the extension of craze fibrils are controlled by the ratio of the contour length of a chain of molecular weight 1Me to its root mean square end—to-end distance. These extension ratios inpmm Luifln M “2 and mnnmmnflJJ—dJnLdLI harm i4 4 iv f: http:,l,I'l:urennan.rnse. E Lecture3.ppt [Read-. .. 0 Unknown Zone 12'] Lecture‘1[l].ppt [Com... l'u__ fr; 10:22AM ...
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EMA6165-lect-09 - pa ‘9 ' U &amp;quot; Lecture4[1].ppt...

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