This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ChI
Drrranatona Chain Dimensions Definitions " " H.“._ I I!" I may.
— n—unn—q  “mm
 analrm ' cmruambonmnme
ﬂuctuatenu
mequInu
unilulu
_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:turr3i[1].i::pt[Comm I”: 1: 9:41AM Chain Dimensions
Definitions Projections of
vectors Mean
dimensions Bond angles Statistical
Segments Chih. V X2) $4205 EMA 6165 Polymer Physics — AB Brennan .
—
— Chain Dimensions Definitions Paige'HVﬁEMP 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 \\
nl
' Bond angles F ' Statistical
segnuﬂﬂs’ $15 Guwuug,QM§ﬁﬁ’AWw I] %1 ’ \J I . ‘ "SQ—SM“ g Svinjie. band)“ a. 5111:3351ch 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 SEW14ft
1 <l+cos¢9> awe W:
<1_COSQ> {Mrse
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 $5
(r > = n As shown previously:
— — _ 2
<7; r]. >—l <cost> EMA 6165 Polymer Physics  AB Brennan 12 Materials \
Scmfyr—k
I.” _ b:——1‘5\2'\}\.
Freely Rotating ChEmginéél[ng
Vector Analysis 72 22M!2  Freely rotating model: ’V’D (TI/Lam GEM — high temperature
— solvated " W W 0Com, “)W T j QTW EMA 6165 Polymer Physics  AB Brennan 14 Materials 3::
Hindered Rotating Chain Model smell?!“—
llustration of RIS ﬁloﬁnk‘ Englhée'mg EMA 6165 Polymer Physics — AB Brennan 15 Materials BE: Hindered Rotating Chain Model Assumptions “gm mg  Freely rotating model:
 high temperature
— solvated
— 0< 8 <360
— 0< ¢ <360 EMA 6165 Polymer Physics — AB Brennan 16 M 9
Hindered Rotating Chain Model Assumptions  Hindered rotating model:
—torsion (valence) angle dependence
—first order interactions —high level interactions
— 8 <= 109.5 for a 000 bond angle
— o< ¢ <360 f
f,“ EMA 6165 Polymer Physics — AB Brennan ‘ 17 M . Science '
Chaln DimenSIons gsﬁnx.
Characteristic Dimensions E 2 2
r O=_Cn[ Where:
<r2>0  chain dimensions at Theta conditions C  Characteristic Chain Ratio n  number of segments
1  segment dimension (length) EMA 6165 Polymer Physics  AB Brennan 21 Chain Dimensions
Chain Expansion Factor D a  chain D \V  interaction
expansion factor entropy  C  characteristic D G  Theta condition
chain ratio temperature D T — temperature EMA 6165 Polymer Physics  AB Brennan 22 I u I Materials E
Chaln DlmenSIons q
Lx.—1“r\:\‘_"§:ﬁ Chain Expansion Factor Engihée'nng Da=lzwhenT=9 1 » £24.
110. F >“
2D“ _> I] l 9 T > 0 :3 (1,0059, +¢D$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'tTLawb
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  [email protected] 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 shortrange and excludedvolume interactions
to unperturbed polymer chain dimensions Hiromi Yamakawa and Takenao Yoshizaki r\ “a \_
Department ofPolvmer Chemistry; Kyoto Univeizsig; Katsm'a, Kyoto 6158510, Japan
' (Received 9 March 2004: accepted 26 May 2004) Q A Monte Carlo study is made of the meansquare radius of gyration ($2) for the freely rotating chain
with such ﬁctitious excludedvolume interactions that the LennardJones 612 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 100thneighbor 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 shortrange interactions but
also to a considerable extent on the longrange excludedvolume interactions. and that the
asymptotic value C; of the characteristic ratio C}, for the rotational isomeric state model in the limit
of n—zx. 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 shortrange 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 (higherorder) shortrange 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}
= ($2 ._ (b3 . . . . _ d),,1) apait from its position and orientation
in an external Canesian coordinate system. where (b; is the
internal rotation angle around the ith bond vector. The total (excludedvolume) potential energy U A of the
ﬁctitious chain with the interactions only between the fourth
through (3+A)thneighbor beads (331) as a function of
{(b,,_3} may be given by n4 min(n.1'+3+d) Unwrap; J E '=1'+4 II ( Rf l l 1 )
with R , J the distance between the centers of the 1' th and jth
beads. The pair potential "(12) is the cutoff version of the LJ
612 potential given by utR_}=="~ for 0$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 [Re3d. .. 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 meansquare 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  fﬁTools 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 forumneighbor 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 100thneighbor 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 shortrange interac II. I.I'll f: http:,l,I':urennan.rnse. E Lecture3.ppt [Re3d. .. ﬂlItsm :Q'lelelE' © @333 El Lecture1[1].ppt [Com... 'ncli , the constant differentialgeornetrical 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 shortrange 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 longrange excludedvoliune
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
612 potentials at the l“) temperature act only between the
fourththrough (3 + A)thneighbor 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
longrange excludedvolrune 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 651Reierencesl1466entanglen1ents‘3320at‘9a20polyme[93205urfac 7 Windows Internet Explorer 61;} v [E http:ﬂbrennanmse.ufl.edu!dileupfema61BSMeferences!1466entarigement5%20at“£120poymer°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 onehalf 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 nearsurface 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 nearsurface region may be different
from that of the biilk.2°‘2“ There is evidence that the
nearsurface 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—toend 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 ...
View
Full Document
 Spring '08
 Brennan
 Polymer, Polymer Physics, Jointed Chain, AB Brennan, Wormlike chain

Click to edit the document details