CALCULATION OF ‘FE'EE UNF‘ERYURBED QiéfiEPéSEGNS FGR'LENEAR ATACTEC POLYMEM 03% A SQQARE LAWECE ‘E‘hesis far s'ha €33.35?» 3% M... S. MéCé‘ééGAi-‘é STfi TE Ui‘i’i‘é‘é’ifififi‘f Rabar? C. 7519is E363 1;». alias W '1 _, xv . LIBRARY Mldfigau Stat! University MECHIGQN STATE UNiVERSITY DEPARTN’EL‘IHT OF CHEMISTRY EAST LANSING, MICHIGAN ABSTRACT CALCULATION OF THE UNPERTURBED DIMENSIONS FOR LINEAR ATACTIC POLYMERS ON A SQUARE LATTICE by Robert C. Thomas Body of Abstract A two-dimensional square lattice model for linear atactic polymers of type CHZ-CHR has been formalized. The three-dimensional equations developed by Yoo and Kinsinger were reduced to a regular planar square lattice and several computer programs were written to calculate the mean—square end—to-end—dimensions for several polymeric models. The model allows the carbon atoms of the polymer chain to occupy adjacent corners of the square lattice. Each step in the polymer chain has a fixed length l and is not allowed to reverse its previous direction. Thus. the bonds are permitted to go forward 0°, left turn -90°. or right turn +90°. This model accounts for both first and second neighbor interactions and the chain config- uration can be either atactic, isotactic, or syndiotactic. Robert C. Thomas The values of (h2)7n12 for the seven polymer cases,ranged from 1.14 to 2.01 while a value of 1.67 was obtained for a polymer with randon configuration and randon conformation. Some comparisons are made between the results obtained on the two dimensional lattice and what would be expected for the three dimensional case. CALCULATION OF THE UNPERTURBED DIMENSIONS FOR LINEAR ATACTIC POLYMERS ON A SQUARE LATTICE by ROBERT C. THOMAS A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of - MASTER OF SCIENCE Department of Chemistry 1963 ACKNOWLEDGEMENTS The author gratefully acknowledges the invaluable guidance and assistance given to him by his research director, Dr. J. B. Kinsinger. He also wishes to thank Mrs. S. J. Yoo for her assistance in defining this problem. The author is indebted to many persons at the Computer Laboratory at MSU and the Computations Research Laboratory at Dow Chemical Company. Midland. Michigan. Appreciation is extended to the Nuclear and Basic Research Laboratories and the Computations Research Laboratory at Dow Chemical Company, Midland. for covering the cost of computer time on the Bur- roughs 220. ii II. III. IV. TABLE OF CONTENTS INTRODUCTION A. General B. EXperimental Model ‘CASES STUDIED AND COMPUTATIONAL PROCEDURE A. Cases Studied 8. Computational Procedure CONCLUSIONS. A. Results 8. Discussion of Results BIBLIOGRAPHY APPENDIX A. Derivation of the Transformation Matrices D2Vand D21; 1' B. Block Method for Determining the Eigenvalues of a Matrix. C. Algol 58 Computer Procedures. iii Page 44 44 LIST OF FIGURES Polymer chain on the square lattice . . Numbering system used for the chain atone; and IMJHdS . . . . . . . . . . . . Schematic diagram showing the first and second neighbor interactions for a (2W1) and a (2“?) bond. . . . . . . . Ehzhenultic (iiagitun dcqoictiiu: tin: polyiner Cilal‘n in case 6 O O I I I I O O I O I 0 Diagram used to determine the sin—cosine transformation matrices when the same coordinate system is used for both bends. Diagram used to determine the sin—cosine ti'arn5f011natian Inati'iCLn; “luau (iiiikartuit coordinate systems are used for the two bonds . . . . . . . . . . . . . . . iv Pave - 4 . 6 . 8 - 41 44 43 LIST OF TABLES Table ' Page 1. Conformational states for the square 5 lattice model. 2. Configurational states for the square lattice model. . . . . . . . . . . . . . . 5 3. Bonds of matrix 1351’]; . . . . . . . . . l4 4. Bonds of matrix if"?! 1 . . . . . . . . . . 15 5. Matrix [Dav] . . . . . . . . . . . . . . 16 6. Matrix flbzvvfl . . . . . . . . . . . . . . l7 7. Matrix .931]; . . . . . . . . . . . . . ..18 8. Matrixfawla..............19 9. Matrixfawana.f............20 10. Equalities of the U13 elements . . . . . . . 21 ll. Equalities in the first and second neighbor parts of the U.1 elements. . . . . . 22 12. Matrix §34 [5:W'] NHL cases 1- 4 . . . 28 13. Matrix 3‘; [3371'] an}, cases 5-— 7 . . . 29 14. Key to the symbols in Tables 13 and 14 . . . 30 15. Non-normalized eigenvectors. . . . . . . . . 34 16. Non—normalized eigenrows . . . . . . . . . . 35 17. Normalized eigenvectors . . . . . . . . . . .36 18. Normalized eigenrows . . . . . . . . . . . . 37 :19. Final results. . . . . . . . . . . . . . . . 33 I. INTRODUCTION ' A. General In 1958, interest in the theory of dilute polymer solutions was rekindled when Volkenstein's calculations of the relationship of discrete rotational states to the aver ge unperturbed end—to—end dimensions of polymer chains became known to western scientists. By this time, Volkcnstein and coworkers had developed their theories to include polymer chains with both symmetric and asymmetric structures with statistically independent rotational states. In addition, they had closed form solutions for the newly discovered isotactic and syndiotactic stereoisomeric polymers which related the aVerage end-to-end dimensions to chain gemnetry. Moreover, they developed equivalent equations for atactic polymers, the ccmfigurational (d.1) placements mathematically described by a single distribution parameter, and the chain with statistically independent rotations. ‘Shortly, however. it was evident that a statistically independent model could not adequately describe the unper— turbed dimensions, and Lifson2 and Nagai3 introduced the statistically dependent rotational model which was treated through the formalism12 dd ddl H :52 BBB-~33 1 1 1 11 I '53 BBA~--—~)4 1 1 1 1 d n {:4 BABd-MX) 1d 1d1 s :5 BAA'~-~)6 ld ldd H :6 ABE—~97 .d1 d11 H :57 ABA-«)8 d1 dld s 68 I : isotactic H = heterotactic S = syndiotactic 5. Tlu) nunflx3rixu; syestenitisetl fOI' the (:haixi attnns and bonds is shown in Figure 2. B (2v—1) (2v) B (sz-l) (2V+2) B or “*w.) . ‘__‘_.__> or >' __ _ __ > or A A A (2Vk2) (ZVLl) (2V9 (2V31) (2V32) Figure 2. Numbering System Used for the Chain Atoms and Bonds.* 6. The tmxnwiinate systenxtuaxi is the sanmkzus that described by Yoo and Kinsinger. For a bond vector ter— minating at an asymmetric chain atom, a right-handed system is used for a bond terminating at a d configuration (A site) and a left—handed system is used for a bond terminating at an 1 configuration (8 site). For a bond vector terminating at a methylenic type chain atom, a right—handed system is used if the CHR-CH2 bond originates from a d configuration (A site) and a left—handed system is used if the CHR-CH bond originates from an 1 configur_ 2 ation (8 site). 7. The equations used are those described by Yoo and Kinsinger. We consider here only first and second neighbor interactions. The generator (or state) matrices are defined on the following pages. *The numbers for the bonds appear at the top of the figure directly above the bond which is shown as an arrow. The numbers for the chain atoms appear at the bottom of the figure directly under the chain atom. {U0 (W) o (W) 0 on 0 mm U11 U12 U13 U14 0(W) 0(h) 0(W) 0(h) U21 U22 U23 U24 . UNDJ (‘1) 2V1 0(h) 0(h) 0(W) 0(W) . U31 U32 U33 U34 0(W) 90*) 0(W) 0(W) _U41 L42 U43 U44 1 where w 1 when A —*-~— 0 ———~--~ A w ~ 2 when B --~---0 —-—~-------B w — 3 when 8 ---—-o ---—-—-- w — 4 when A —-—~—o-'-~—B Matrix (1) is a generator matrix for one of the conforma— tional matrices. The elements of matrix (1) are defined in equation (2). U0(W) (W) (1) . (W)( SH) (1‘) Urt “ EXP [ {% (GZV'l) E 27 vwlik//;T fir st neighbor intei action terpj second neighbor inter action teim In U2Y+1 the 2101 signifies the rotation of the 2Y3}; bond about 2101' U(:i is a term proportional to the probability that the last bond will be in state r when the previous bond is in state t and the configuration state is (W). The first and second neighbor interactions are depicted schematically in F1 gure 3 . II“"~.;C .r" ~11 V . a (2""+1)bond a (2") bond Figure 3. Schematic diagram showing the first and second neighbor interactions for a (2W+l) and a (21’) bond. r. 1 0 0 ~ D2"+l 0 ,. . 0 DZ_ 0 0 t . 7 2“"+1 *D‘“)i ~ (3) .- 2W1; 3 0 0 112m1 0 t l 4 t- 0 O O D2),+1 Matrix (3) is a transformation matrix which is part of the transformation matrix EDOT+L . LO F. (1) ( [12%1 0 ) o (2) 0 U2Wl 0 o 2v=1 - (4) .(3) 0 0 b2v;l 0 (4) if.) 0 0 U2Y;L Matrix (4) is the conformational matrix for 2V+2 bonds about Z‘WI bonds. I... .. Cl 0 CG 0 O 0 C3 C7 2V4 .- (:3) 0 ()4 0 C8 C2 0 C5 0“ t... Matrix (5) is the configuration generator matrix. C1. CXp —(i/RT (6) where €i is the free energy of activation for a given configuration as defined in Table 2. o l 10 F’(1) ‘fi D2Y+1 O O O (2) O D2v+1 0 0 [DZ‘V'"1:l - O O D(3) O (I) 2Vil (4) 0 0 0 D . _..' ZV'EJ Matrix (7) is the transformation matrix for transforming the coordinates of a 2Yt2 bond into the coordinate system for a 2Y1]. bond and each element on the diagonal is a transformation matrix as given in (3). Up to this point we have been considering the bond type A o ——«-- or __ 8 Now in a similar fashion we will con— A or~--~o sider the bond type B __ Then: FfiE(w) UE(w) UE(W) UEWJI ll 12 13 14 E(w) E(w0 E(w) E(w) U21 U22 U23 ”24 (W) ,. U2v ” (b) E(w) E(w) Etw) E(w) U31 U32 U33 U34 E(w) ' E(w9 E(w) E(w) _P41 U42 U43 U44 _ Matrix (8) is a generator matrix for one of the conforma- tional matrices. where w - 1 when 0 —--—~—-— A ~—-——-«--I w r 2 when o---— B--m-o 11 (w) . ,. _ (W) (F) J, (w) t r “ (- Uft exp[ {6 Egy- ) ' E (BZV-l eZV/R'J (.1) first neighbor interaction term second neighbor interaction term r 1 1 D21" 0 o 0 2 0 . 92v 0 0 [Dix/2] - 3 (10) 0 0 D 0 2v 0 0 0 1)4 L. 2V11 Matrix (10) is a transformation matrix which is part'of the transformation [DZV] , " (1) U2v 0 0 . 0 (2) - . 0 U 0 0 ; . i’ j . 2Y (11) 2v .- (1) . 0 0 ”W 0 0 0 0 111%.: Matrix (11) is the conformational matrix for 2Val bonds' about 2V bonds . 12 P 1 11.5%,} 0 0 0 (2) 0 112?. [P23 ("> (12) 0 Dd;- (4) L0 ”21d Matrix (12) is the transformation matrix for transforming the coordinates of a 2111 bond into the coordinate system for a 2Thond and each element on the diagonal is a transformation matrix as given in (10). The final equation used to calculate the unperturbed dimensions of the various cases is shown in equation (13) as adopted from the three dimensional formalism of Yoo Q l! ‘ and Kinsingei. 2 2 . *, * - n —l 1 n (11),:11 E) 1] [112+ A INA A (532+A)(t32—M) (M+Dzv)aa] (1.)) where m . final-fag; [921;] anagliawdjxa N = L§?3;[Dawl[§awaa @MJYXD A? [I]; a): m2 = the largest positive eigenvalue of [E313 [22143 [2 271:] q the eigenvector corresponding to the above eigenvalue the eigenrow corresponding to the above eigenvalue b as s 13 E2 = a 2 x 2 identity matrix EH2 = a 32 x 32 identity matrix Q The bonds considered in matrices [£24 and f are shown in Tables 3 and 4. The matrices D , 27% 2 [D2,+fl, [EZQQ' [§21-+J2, and [22%) 2 x 4 are shown in 7, 8 * Tables 5, 6, , and 9. The derivations of the two transformation matrices are given in Appendix A. 8. In this square lattice we only allow the growing chain end to go in three directions. We disallow it to reverse its previous direction. This causes the follow— ing elements of the U matrices to be zero. U13 1‘ i U231“; ”331 l T Uer JI—a U31 $ J. U32 , U349? Due to symmetry many of the elements within U are identical to other U elements. These identities are shown in Tables 10 and l. l. {V d -. L... .w.-...---..LJ , a 1.9 d- .a -. .. . - _..--.,. _._-~.._...-...—o.~ ,-o - I ~.-..- -.-—— -Wu m.--~_,HMWM . . 4‘! “-0 .. .— V—r‘ov-*.W-——o - ‘ «——- 5.4..-- s-i. “a.”uu l3 \ I- it! {In . It'll-i). ..( I... \f .'l l.ao'.|| hilt t'u)..ll\§ to? ll -_ 2...-.. .. . ”a ...-....--- ....._..-.....- _ .-- [0 ..._... -.....-.. -..., .m AW. (MIN 41 M \--' n---¢a--wd—-uo.—-—o¢“ ‘.. a...” . all ,. m .i i 1:11....1311043195 1.1- 1, s -Ill I! 1‘? I.) 15 VF? . I I 1_ if“ '1 i («9, R 1 ‘R f“ } R- T ' ‘mrmz 1. K It * “ k {33 f4g A 1 “K i . hi “i _ VI! ii 1‘\' .1 2 ' J A (it ’1" w I t ’3 “‘13 . — - ‘ -12 « I ‘ "f “k f t ‘1 I? L fig? I 7 R 1‘7? 1N ; _ ‘ K I . a- X f‘ 7' i 9%.]! ’3 ‘t, a ‘1' g l '1‘ It. ,,,. A. 1 1 3. ‘* 1 . / A- 5 1 1 t ’57 St E (t ~+ '1 y - Lzlfi? “TH z 1 ’ K .g {M , . q A 1” :31 t )1. I a I I k“. A ‘ . by we h” ‘ : , ,. 4g 5 _ ‘1') I (-121 it 4‘ 1“; . V ‘ I [’19 L1 1....) \K K‘If} figfl tg/I Eb} ("~- .2,“ “-1 . I" )1 T J’ I?“ 1'1 1’ . “'1’: IT...) m .' fl ‘ A Alf” “'T - :5 LI! 4‘ -- t J» L M ‘7)" A" A ~ t as 1 . ,g 2. f2 ’TLS 53‘ 2'5 I a 3 7? .fi‘ 5' 7 E? ‘1" (a I! at .1 s‘ a “_ ()'\ -_.~) -c - - J4‘ ow“ --. ~_ L... ... a”... _. (4.... ... "fl MAI~-w_ ~———--_.. . . .,- ‘) .. ) g; .L... J. St) ~--. .- R-h-Au -— - --——-v-i ' "'M W'sfifl- ., ) \II 1 ll _ t’ til I. /l\ l \1 . ( ‘1 (x t 1'. 2t) ./ s v \’ /\.\ ll I V; 1|! 4 \I u I‘ . er (in!) 1 6-.)153E 1 . c . . h. 1 'Al til. . _ L .l-ul \u \ | .I N \ | (in I ) )4 .) \\J i) 11.1 . ~.\ I (All In x ( l s . ,)4 II“ \I ). \l D) )b; I? m l a 0 A I u d L ( ( \ \ q 2‘) ’ ) 5.1‘ and ttlltz l ()() 1H) ‘7‘. ' ( ~< ,(11 .(l1 7 ()1 ' ' l 1 W1 1 ()1 (1 .(1': I 1 I )1 ‘;" I “of ,(): ,(H 1.,1 1.... I. , ‘5 )Z ( \ 1 1‘," 1‘... 1'5). )1 II, T) '1‘ f) ()1 ,0 () ‘ I 11 [““ l .t ,(H ,.()1 .U‘ 1‘ [“ |_ .4) 1.. i 1 t . ' ) ‘-" () :( [(2 , "U )j‘ 2‘ )T' H) '( [A '( 11) 3 t (1 .(A‘ (1‘ H 1 I .t): t)" ,()f' [.2 1 ,,l t , t. i I} Lititi 1»? - o o- t)” l) ' l t. I i 1 c ) ,\. :_ H, l'( 1,0 1 131 and 22 :3:; 11(?" 1'(S;: t (?‘S , Iltfy 2.9 a lid. IN) It; It“ lix' .' . if)?“ Equal i t i 0. El 11 El 12 El 14 E1 U21 121 22 El 24 £1 41 m 42 E] 44 U U U U U U U U .04 L11 ()2 14 0-1 14 “02 H ()4 J 2 02 41 ,04 '11 ()2 44 U U fi '71 ‘- TABLE 10. in the Ujj olemonts. L'11 £2 14 E2 12 E2 «11 ,1a2 ’44 E2 42 L‘21 E2 U04 .— U U U U Um 22 E2_ C.‘ “‘23 C‘. . C: rem Hm Hm r—‘m F r\ fiVv C‘ r— 2.; C mm mm (0;; AC.) N'J . i 22 Table 11 Fh1u1111.11( s ir1 tlzc 111'st a11d 540(11ncl n(\igl1hcu' {Dux'ts ()1 131.. Ian L1) Clements. __ 6 . ' 1 04 3 ’. p ' - firsi neighbnr U‘c RF ELI ‘(UL1){U‘ . 11 11 [w «— 51(‘('(>11(1 11(11 1:111)(11' ~12 1<22 wrxs .154 «12 .422 41x: 1-24 «En -<22 141;; .114 T ' 1 ' 1 " T ; — I —_ _; _ Y r _.‘ f 1 , L11 L11 L11 l41 U12 L12 IJ12 U12 LL4 [14 L14 11-1 "I‘I 1 _ 3.132 "PI-'1 ,"E4 1 ,-#E1 @122 «15:: HE! #131 «E? -,* E3 ; ‘ - 1’ IJ42 L24 42 24 42 24 L42 f(n_ £02-_/ 0:: (’04 01 2(11 tTH_ £02 41 "21 ‘L41 ’121 ”21 “L41 L'21 41 01_./02 Ir01; ‘204 ,203_,w04_,f01u,;02 f44 L22 44 '122 L'22 "'44 ””22 ”L44 23 II. CASES STUDIED AND COMPUTATIONAL PROCEDURE A. Cases Studied A description of the seven cases studied is shown on the following pages. Case. 1 ’ei/RT' .r . v. _. . ,1 Ci _-; (1 / t1 v C2 —: 63 —: t4 *— t5 C6 = t7 '1 C8 7.000 C12] 1110113 —"-' RT . . 1. U -= 0 “here C is the sum oi the 111'81’. and second neighbor interaction energies. In case one the polymer has a random configuration. l)_ (11_1‘111'11_1:111;_1_t ion Element Rotat ion a 1" __Ener_.: y, Assunajd” .Y 0' s = «15' s 1000 Ca 1," mole }-()'s [515's 1000 Cal/mole where 9‘0 is the 21411 iirst neighbor interaction energy «E is the 2 t iirst neighbor interaction energy 7 O is the 2111 second neighbor interaction energy E is the 21’second neighbor interaction energy This spet-iiies that the polymer in case one has a random Contormation in addition to a random configuration. :1: The rotational energy values correspond to the energy differences between an arbitrary energy level and the lowest point in the rotational potential well. 24 Case 2 el 4 6.2 :- 63 2 E4 : 65 26.6 Z 67 = 68 = 7,000 cal/mole This polymer has a random configuration. U Conlormation Element Rotational Energy assumed *_ .— o-co 5 =41: S 1000 cal/mole Irv-1 I f El11 lOO cal/mole . all otherpE S except P2111 200 cal/mole 5 i )9 0:111 100 cal/mole i all 7803 S that have a bent 750 cal/mole [ conformation " all 301 S that have a bent 1250 cal/mole conformation that permits the R groups to be further apart all 7901 S that have a bent 2000 cal/mole conformation that permits the R groups to be closer together T“e preierred co1 ~rmations have the lowest rotational / energies. flEl11 is the 2Vsecond neighbor interaction energy 0f 910mm“. Uii) which is g-“ p0311 is the 27411 second neighbor interaction ‘ which is R- energy of element Uii) Case 3 61 r 62 2 63 = (,1 : 65 3' 66 = 67 == E8 :- 7,000 cal/mole This polymer has a random configuration. U Conformation Element Rotational energy - assumed ——.-—.._..-—-. _._.. .——- “-0— _._-. ,_.____ d0 5 = dB 5 500 cal/mole ’0 S 23E S 500 cal/mole Tfliis polymer has a Case 4 / r -— 2: 9'. L‘ ‘ ~1 ~2 k2 'Fliis polymer has a U Conformation random conformation. ‘-. —» .- “'4 5 L'3 L"7 random con f igura t ion. first neighbor first neighbor second neighbor second neighbor second neighbor The (:(nnparing the first \Vliich can be seen in Tables 3 Element H . . . H H R H . . . H H . . . R R . . . R and second neighbor zintl 4. A ,. . P. '— P "‘ ~ ’ \ r. 7,()0() czll/nn)1£‘ Rotational Energy assauned .——.—.. .—.-.--.——-_.——.- -.—— 50 cal/mole 250—1000 calfmole 50—600 cal/mole 200—2000 ca 1,”!111‘1le 8(H3—25H30 (nil/nu)le rotational energies assumed were assigned after interactions ”most probable“ (“11ergy was assigned to each conformation based on our 1)(?st interpretation of a real chain. 1, in" _ "m7 ' ., Case This Ca s e This 213 61 :62 :EJ :64 65 :26 polymer has a random configuration. -‘ E7 :68 4 1.000 cal mole 13 U Conformation Element Rotational Energy a s s ume d .—. - _-..._ ......_-._._..—.._..- _-- ....... n U 0‘0 5 W‘E b 500 cal/mole j 3001!} a .10 cal/mole $0144Rfi , 00 cal/mole all otherflO S 1000 cal/mole 1111 71: S 1000 cal/mole 6 El 62 =- 63 = 64 = 66 = 67 = 7,000 cal/mole ES 2 6% = 2,000 cal/mole polymer has a predominantly syndiotactic configuration. 27 H£9Bf213"i‘-t2_i_.‘,’_‘}__§lf?}i'_el‘_t_ Bgligieml £32.93). A“““"-_EE‘ 040‘s =:°(E"S 500 Cal/mole fl 0311R-f. 50 Cal/mole fl 0144 “H" 50 Cal/mole all otherfio's 1000 Cal/mole all fl E' s 1000 Cal/mole Case 7 62 7' 64 65 "Z 66 :67 :68 = 7,000 Cal/mole 61 63 2.000 Cal/mole This polyn‘er has a predominately isotactic configuration preferred. g genierfimjtionnl‘j‘lerfht Rotational Ener gLAssumeg (X 0'5 =O‘E’s 500 Cal/mole fl 03113.? 50 Cal/mole F ()1 $1., 50 Cal/mole 4MP All otherfiO's 1000 Cal/[role A11}? E's 1.000 Cal/mole Tables 12, 13, and 14 show the reduced 12 x 12 matrix Dim [fwd 1.33mi! B. Computational Procedure Case 1 will now be used as an example and the methods of computation will be described. The first step after assinning energies and calculating the various U. j elements was to calculate the 16 x 16 niatriXL-Ez JBZV+JJ ELY-+1 q 2 (31189 28 Table 12 Ma trix [I33 Eafl1E’J-3sos 1-«1 case 1 a FppTIfla.;s Tiifimflxaé s CEEVUIT; . U 5me 7.70 _ a p or. H i -...- ._ waw Y B C E E 13-3-.. ,6 T. I m! 9. .3 M In \A ’5 EDD [3&1 . i,LfiRa H .36# E p p .. WW; a WWO.tuik-m N FDDGII m LRNXdré CEEGIIl t USPWYX .QunUrnVr+UMHHHH ,WHVflMUM.MVchflV .PLcL It 70_T‘TL nunP./ ‘1» o< ts $3 "l ED 33 < 'c: >6 :2 £3: ET: 3: :t u n ') s} C»: .51x10‘ .32x10— .52x10“ .96x10‘ .88x10- .24x10- .66xIO— .02x10’ .42X10- .81x10’ .85x10- .03x10- .24x10— 8 8 7 6 7 6 6 6 7 6 6 7 7 15.£N‘ (p IT\“$ \(. <3 Ln 13!‘- hi 'I i U: .35x10‘ .80x10- '.37x10- .52x10- .87x10’ .76x10‘ .66x10- .87x10’ .72x10’ .78x10' .24x10- 8 7 .51x10’7 6 n .20x10-4 6' 6 7 6 6 6 7 8 . .- - ,. . ., - ., . . .- / Matt 1X [ZaflJV is an expanded foim of man 1x Q39”) in which each element of the 4 x 4 matrix is multiplied by E4, a 4 x 4 identity :atrix. Because we do not allow the bond to reverse its direction, rows 3, 7, ll, 15 and COlLunnS 3, 7, ll, 15 are all zero. This then allows the rmxtrix to be reduced to a 12 x 12 for the purpose of cal- culating the largest positive eigenvalue and the corres— ponding eigenvector and eigenrow. A machine language pro— gram to do these calculations was written for Michigan States' digital computor which was called MISTIC. This program calculated the 16 x 16 natrix [331a [EJWJKJ 1"qu irom the 28 distinct Ujj elements and the 8 Ci elements. Michigan State University computer program MASM was then used to determine the characteristic polynomial of the 12 x 12 reduced matrix. Michigan State University computer program J2 was then used to determine the roots ol this polynomial. which would he the eigenvalues of the 1: atrix. This approach gave erroneous answers because of its method of calculating the characteristic polynomial. Program MASM uses the N + 1 points method which will give erroneous results when the eigenvalues lie close together. Case 1 has only two different rows in its reduced 12 x 12 matrix which indicates that it has not more than two non—zero eigenvalues. Various other methods were investigated until a program 9.10 was written to use the iterated vector method. The rmuaining programs were run on a Burroughs 220 digital computer at Dow Chemical Company's Computations Research Laboratory at Midland, Michigan. These programs were written in Burroughs Algol 58. Some hand calculations using the iterated vector method gave the largest positive eigenvalue for Cases 1, 2, 3, and 5 belore the computer program was written. A method of blocking the matrix was used to solve for all of the eigenvalues of the matrix for Cases 1 and 3. The procedure is described in Appendix B. The eigenvector and eigenrow output from the first brogram in Appendix A had to be normalized so that 5 X; L?! 1. Then the next step was to calculate matrices LN]. I'M d T" Ami—1 r1 ' 1 t . + D . an — l . ' 11s (ata was out n on cards because of a limit, of 8000 words of core memory available on the computer. This data could have been put on magnetic tape and used later but since the time required to punch out the natrices was reasonably short. it was decided to take the easiest approach. The final program in Appendix A 2. . 2 . . . calculates (h ) nl . The new Burroughs B 5000 whicn is at the Corputations Research Laboratory now could very easily handle all of the total calculations in one step. This would require only the 28 Uij variables and the 8 Ci variables to be input. III. CONCLUSIONS A. Results l) A 2 The eigenvalues. eigenvectors, eigenrtms and 61 >.‘ul are shown in Tables 15. 16, 17, 1.8. and 19. Table 19 also includes the values of two 2 x 2 matrices which are called MATE and MATIB. MA'I‘2 is the product of 1:;ultiplyintgg A* N A while MAT3 is the product of A“ [1332+N] [E332 —- Ila—1 [M‘DZHA (see equation 1.)). One can see upon analysis- . . . .2 2 . . . of equation (1.5) that (z > nl is equal to 1 a MAT3(2,2) 4 MAT3(2.2). NON—NORMALIZED EICHNYECTOR ><><><>< % ~1'D with care H ><><><><><>< (.0 >< pas—a t-‘C l X >< H 5—4 L3 \3 X l-' “'3 >< Hm Hm news a |0|| .HO©:.+OH .HOCC.+CH .Onzoc..+omo .HOOO.+OH .scoo.+oo .Hoco.+oH .aoao.+oo .Hboo_+ou .HOOC.+OH .HCCC.+OH .anoc..+omv .Hooo.+OH .HOQO.+O~ .HQOO.+OH .OOCO.+OO .HCOO.+OH Av . #511111: swarm pm. O>mm N Afivcuw.lavs .omwm.|ow .occo.+oc uomq.nc~ scou.io~ mopm.:cw oooo.+oc ou~m.uop .mowm.rcc .aL@©.+oc .ccoo.+oo me©©.+cc Hooo.+oH m¢®©.+oo oooc.ioo wA©®.+CO .1 l l. n.‘ 11.1! it“. nsmsem . 5:03;: .HOOO.+CW .OOOO.+OO .Hooo.+oH .HOOO.+OH .wcoz.iow .COOO.+OO .Hooo.+OH .#QOO.+OH .HOOO.+O~ .OOOO.+OO .HOOO.+OH .HOCO.+O~ .HOCO.+OH .OOOO.+OO .HOOO.+OH 73:!32fi3 :wsu a do 03< .mowm. nemmli .Quwm Hmcu mVAcrvso mcHo .moa» .wsbq OCOC. .mouu. .oomo .HMQO COCO .wmow .pocc .maom .OCOO swam. .+oo .ICH .+co .+00 .+00 .+00 +00 lacy .+OC .IOH .+OO .+OC .+c~ .+00 .+00 lop 9mm, m .Hcco.+o~ .Hooc.+CH .oooo.+co .Hocc.+ow .Hazoc.+AC~ .Hooc.+ow .Azoomv.+Aoc .wooo.+©~ . H occ e :1 . H Avsvsv . +.mo H .oooo.+co .Hooo.+OH .Hooo.+op .Hoco.+oH .OOOO.+OO .Hooo.+oH .Hooo.+OH .Ccco.+oc .HOCO.+OH .qmsc.lou .qmeo.10u .cooo.+oo .umgo.lou .qu;.icq .Huug.lou .OOCO.+OO .smua.|0q .Hmue.lou .Hmwa.low .coco.+oo .wmmL.iou osmm a .H :00 . +0., HCCC.+OH OCCC.+OO .Hcoc.+OH .qugsc.lfloq .umso.loq .Cm:.0.+goc M®¢O.IO< .qua.iou .Umq».10q Caz:v.+m:u HWM¢.IO< Hmms I u Hmm@ I u ocoo.+oo .gmma.l u NON—NORMALIZED EIGHNVETTOR ><><><>< L.~20Uiur.. JIOH ( ><><><><><>< D X y—‘s—a l-‘C l >< >< >4 >< F—I H- -,._. A L: '0 '1 H H0 >< news a "I .Hm;:0.+0H .H000.+0H .06:00..+0m0 .H000.+0H .H000.+0s .H000.+0H .H100A0.+~ru .H000u+0H .H000.+0H .H000.+0H . 0000 .. +00 .H000.+0H .H000.+0H .H000.+0H .0000.+00 .H000.+0H 0>mm 0000 .UHHQ. . AVAVAVAV. mama. saou. uuHm. 0000. uuHm. .womm. .aieo. 0000. swarm m .I0H 10H +00 I0H I0H :0H +00 10H +00 +00 +00 .wH0©.+00 .0000.+0H .m00©.+00 .0000.i00 .wA©®.+00 70 7:31:22: 3:50 m;.m.0...mm 0 ‘I I‘ll! Il'l’. . 0 HOHOH0 .H000. .0000 .H000 .4000 .qmec .0000 .qu0.I0u . Hqu.»._iH0q .qug .0000.+00 .0mus.104 .Hmm$.|00 .Hmwa.low .0000.+00 .Hme.100 .+00 +0H .+00 .+0H .Iou .I0q .+00 .qu 0>mm u .HHXEO.+00 .0000.+00 .0000..00 .H000.+0H .1500: . 10% .qmgo.l0q .0mz00..+0n0 .4000.10q .qus.l0u .H0u0.!0q .0000 . +00 .kug.l0q .Hmms.l00 .Hmmo.l w .0000.+00 .omms.- u IUIC NROW 1 NON- NORMAL I ZED 1‘ A ) | r):- ._.>wrm 00 . 2331:3750. sued n.H.,...C:..C¢.m. msmm.w osmmim nsmw.c npmm s n»Mm-m w»mm m mmmfi:a as .socs.i:~ .oooo.+oH .scc:.+oH .sau0.+:c .Hocc..c0 .acoe.+co .secs.+co «m .Pooo.+co .sces.+cc .Hooc.+c~ .oooo..oo .umoo.+cc .seoo.+co .seos.+co «a .occc.+co .occo.+cc .oooo.+cc .ccco.+cc .cooc.+oc .cocc.+oo .scoc.+cc as .sccc.+o0 .smoa.+oc .Hccc.+ca .oooo..o_ .occc.+ca .cooo.ics .Hopo.+co «m .Hoco.+o_ .oeum.+co .ooco.+o~ .swoo.+cc .Hoco.+ou .momm.-oq .acmm.-oa «a .Hcoo.+oH .saom..co .Hooc.+oo .m:qc.+oo .Hcoo.+co .qmmm.-oq .qoma.-ou <4 .oocc.+oo .cccc.+co .ocoo.+oc .oc:o.+o: .cccc.+cc .cc:c.+oc .ocoo.+oc «a .Hooc.+o_ .scem.+cc .Hcco.+cs .mcmm.+cc .umpo.+cc .usam.-oq .osmm.-oq «o .Hcoo.+oH .coam.+co .Hooo.+c~ .saqs.+co .Hooo.+oH .emso.-cu .mmsu.-cu <00 .Hooo.+oH .sces.+oo .Hccc.+o_ .Hmoa.+co .eaow.+oc .mmse.-cm .owsu.-cm «op .oooo.+co .o:oc.+oc .oooc.+oo .occo.+oc .ccoo.+oo .cooo.+oo .cooo.+oo <_m .Hooo.+cH .saom.ico .Hcoo.+co .oeam.+co .Hcoo.+ce .masm.-oe .moso.- u «so .Hcco.+oH .moqm..oo .Hooo.+oH .samo.+:s .ocoo.+ca .maom.-oq .mmom.-oq «us .Hooo.+oH .saom.+oc .Hooo.+op .mqu.+oo .oooo.+oH .Hoqm.-om .Hqu.lom «Hm .oooo.+oo .cocc.+co .ccoo.+oo .ccco.+oo .oooc.+oc .occo.+co .coco.+oo a .Hooc.+oH .soum.+oo .Hooo.+oH .mcmw.+co .umow.+oc .Hsacalom .somo.-om H0. f) t) EXVECTOR ‘ I NORMALIZED EI( coo eHs sHm 0H0 nsmm-H N110. . +3: mmxu.+00 A0.0a0~0. +.HOH0 Nqu.+00 .mxaq.+00 mmru.+©0 . AVHOHOH0 ..+ AOA0 mqu.+00 mrmq.i00 mxmq.+00 ,0000.+00 mmmqw+00 mwm<.+00 mmm<.+00 0000.+00 wmmu.+00 a. '11:»-qu. s. I I.l.lll..-udlLl'Il-l.q %ymfim 0w. 03mm m oymm-e wuux.I00 n.01aw.I~0H «VAVHvAV. . HVHV 400... . 000 1030.100 hawxw.100 . HOAOAOA0. + AOA0 <003.ICH Amm0.+00 ume.+00 0000.+00 wmoH.+00 0MMO.+00 umw~.+00 0000.+00 ume.+00 m11u..00 mmau.+00 AOAOHOA. . + a0.0 .mrqu . + 00 .oocc.+oo Q twmu.+00 mode . i 00 0m200..+0H0 mmmu.000 mqu.+00 mmmu.+00 wwmqe+00 mmmq.+00 Nmmq.+00 0000.+00 mmmu.+00 2273;00xfe 6. H JCS/.01.; Cu.” 0pm0 ; “0:3..00 curs.|:_ .0000.000 .Hz0ma.+az0 .AUHH.+00 . “WHOm0Hw . i H0a0 .A0fi.sx . +HOH0 .0A00.|0H .0000.+00 .0000.I00 . 0000 . + 00 .muoq.+00 .Amm0.+00 .msuu.+00 .0000.+00 .w0U$.l0H Uttw.hCC Nmau.+03 fivnvrvaV. +,n0A0 NIQN..A:0 + AVAV I [HJHFJN . f flvc 06:20.+H:c Maid +00 +00 mmmu.+00 0000. +00 .+00 mmmq.+00 mmmq.+00 0000.+00 mmmq.+00 qsmm a wqux~.+s:0 mqq¢.+00 .oooo..oc .mqqs.+oc Ammu.I0q HQNQ.I0Q AO0H:0.+g00 .sumq.I0q some.-ox coae.-om oooo.+oc meme.-om «New.-os assm.-¢¢ oooo.+oc qmsm.-0s A0m0HOA0 . .mqqg. .Awmq. .Awmu. 0000. .ewMQ. .0000. .0000. .0000. .womo. .qwam. .qmam. .0000f .QNAM. Iillilt. \ witwar '\ r' v $ I ‘* 4 a JH'QIN 3*Ul*$¢*L NROW A F .1 i Q“ n *- l Cd 0 *H*H*©*CX*‘~I~I~ NORMALIZED EIG H 1‘» ‘ Q", *- b—J [\J nwmm.w .mwmq..00 .mxxu.+00 . 0000 . .+00 .waaq . +00 .maaq.+00 .Mwwu.+00 50HOH050. +.nvmv owlq.+00 t mmma.+oo mmma.+oo .oooo.+oo mmma.+oo .mmma.+oo mmwu.+00 .0000.+00 .mqu.+00 Com -m . ”UHW‘WHV . . AVAV .mumrw.+mx0 .0000.+00 .pr©.+ax0 .0000.r00 .mgmm.+00 .0Hz00.+g00 mcm0.+00 0000.+00 .mum0.+00 .0000.+00 .M¢m®.+00 .wmmm.+00 .mewm.+00 .0000.+00 . um0.+00 .mxmu.+00 . HOH0m0m0. i.HOH0 . mchq . +00 max<.+00 mmwa.+cc .0H:00.LA00 .mmmq.+oo mmmq.+00 .mmmu.+00 .oooo.+oc .mmma.+oc .mmmu.+00 .wmmu.+00 .0000.+00 .mmmu.+00 03?: Gem. 0p: m1.-. ,0 .mu0m.+00 .0qm;.+00 . AOAOHOA0. i.H0a0 .mqq0.+00 .0Uub.+00 .0000.+00 .H400.+00 .wq00.+00 .Hquu.+00 .0000.+00 .000u.+00 mqu.+00 umqs.+oo .oooo.+oo .Hq00.+00 0&mMEU Y».+A00 ..+.0H0 400 .+00 .+00 +00 +00 +00 .+mz0 +00 +00 .+00 .+00 +00 +00 .+00 neMm.m .uzfiwu.i;00 . 30.7w . + 00 . ~0«0HOH0 ..+.HOH0 .mmiu . +00 .wwms.ioq .mqwq.l0q .0000.+00 .m000.I0u .qwmm.100 .qm00.|om .0000.+00 .H00 .100 .M000.10q .Hoom.lox .0000.+00 .00H0.100 mstlw .umqm.+00 . 000.0. + 00 . H0a050A0. i.AOH0 .mugu.+00 .UH00.10< .0u®<.10q .0000.+00 .mom0.10q .qua.100 .qmwm.l00 .0000.+00 .Hmom.-ou .000@.I00 .0000.+00 .00H0.300 38 Amny000.0m Hm» . m.» so.__ ~.n.m.Hs_ 0 +.. Iosmwlw- -nWmm m- ln»Mmrg: nommzm: -n»mm-t: n»Mm.m osmm,q .aus.|c: .oua.loa .sem.-:u .seu.-2a .Hmc.-cm .mxs.-cm .muc.-:m asamfls.ov ..Haa.+:o ..sum.-:0 +.~aq.+:c ..cc:.t:c +.sqs..:: ..uma.-:s ..psw.+:c abawaHhmv +.H0,.N..+00 i .01m...00 +.00<.+00 +.wc...“..+00 i 7:01.000 4. 00,0.I0.“ +.MKH .00 {\3 LC 2.93.”:va +.H0q.+00 «.mimr..00 +.00.N.+00 + .+00 .HH3100 I.00C.I0L I...L.Z..00 [‘0 v .+_ L—a Ch \1 2.90.0 m. +00 4 .an.I0H .. .+00. . +00 i . Hm“... +00 +. HHu. .00 .. 0001.0.» + . ”the r00 Z>H3A0.00 +,000.t00 i.000.+00 + 000.+00 +.000.+00 ..asq.+00 +.m0m.t00 n.3um.10H Z>%0AH.MV +.H0<.+00 +.00m.+00 +.HGQ.+00 +.H00.+0_ +.001.+00 +.wm~.+00 +.mmm.+00 Z>stm.Hv +.00w.+00 +.smq.i00 +.HOQ.+00 1.000.10H +.w01.+00 I.MmH.+00 I.Nmm.+00 ass Au.m0 .a:;.+co 4.mee.+cc +.moc..eo + + 000.+00 +.000.+:0 +.rHN.+00 I.0MU.I0H ‘1 h J srxssr H.mq o.mm H.mu m.cH H.9m “.mH H.0s B. Discussion of Results The final. results for the unperturbed dimensions of the seven cases are shown in Table 16. Case one is an example of both totally random configuration and random conformation. The value of .(h27/nl2 is entirely in— dependent of the energies assumed for the. totally random case. This results from the fact that matrices M and N (see equaticnnsil-l and 1?) are d vided bx . tJn’ large st positive eigenvalue of [IZTJE‘WF J [Ear-134.1ad that the normalized (.‘igcnvector and eigenrow are used. Case three is simply a check on case one. Case two has totally random configuration. totally random first neighbor interactions, all second neighbor 2Vinteractions except one are random, and a distinction is made between only the second neighbor 2Wf» l inter- actions (see part A section two). This does HUI notice— . 2 2 , . ably change the quantity (h >/nl from that obtained for the totally random case. This implies that for this model the second neighbor interactions must be quite far apart to noticeably effect the end—to—end dimension of the polymer In case four a distinction is made in both first and second neighbor interaction energies while the configuration is still kept random (see part A section two). This has a significant effect on the result. The rotational ener— gies assumed were ”most probable” values assigned after I w...n._--__. ~i . ‘ . 4O C(nnpflrlEMJH (xf tin? liiwst atnj scxxind 1n>ightxn~ inttu'aetitnis whitli aiw: dcq3icttui sclnnnaticnilly :in TYHiles I} anci‘4. ’rhe V 2 , 2 _ . . . _ value oi h >gnl obtained in case tour (2.01) is es— sentially that obtained by considering a freely rotating polyimgthylene (flurin consisnjan; of rllMJHdS of icknitieal length l joined at fixed valence angles 0. ~For large and a tetrahedrally bonded chain (6 =— 109.5() a value of -2, 2 . . . . r nl = 2 is obtained. Hence on a two d1mens1onal lat— tice a value of 2 represents a reaSonably expanded chain. Cases 5. 6, and 7 are identical with respect to con—- formation but case 5 has a random Configuration. while ll .ur _‘-_..n.. case 6 has a preferred syndiotactic configuration. and case 7 has a preferred isotactic configuration. The re- sult for case 5 is very close to the result obtained for the totally random case. One would probably predict from the inspection of the!z preferred“ cinfigurations for cases 5. 6. and 7 {1’ n 1.3 - / OR I c that case 6 with a preferred syndiotactic configuration . . . 2 would have a higher value oi 0’3 Inl than a random case while case 7 with a preferred isotactic configuration would have a lower value than a random case. The result for case 7 is in agreement with this reasoning while that for case 6 is not in agreement. Be- cause of the disagreement of case 6 with this theory, case 6 was run thru the various computer programs a second time. The same result was obtained in both determinations. 41 Fin-ther study on the model in case 6 led to the idea that time polyner was actually running in a straight line for a while and. then turning around and going back in the (direction in which it previously came. A schematic dia- grzun depicting this is shown in Figure 4. v /t\ \V /" V A t Y A V A v i V H A V )is Y A V A V )i V A V to Figure 4 Schematic diagram depicting the polymer chain in case 6 42 It three dimensional model would allow the polymer to audio many more moves than the square lattice and hence ttm3 results would be different in many cases. If the isu)tactic polymer in case 7 were allowed to proceed in tJrree dimensions it would undoubtedly coil up and form t1 helix with a longer end—to-end dimension. These preliminary analyses have aroused more problems amui questions than they have answered. It would be ex- f tremely interesting to determine the effect of temper— ature on the end—to—end dimension. There are many more different cases which would provide more information about the polymer model. It would be quite beneficial to have the matrices M and N (equations 14 and 15) multiplied out in symbolic form so that one could see the positions of the favored conformations in the two matrices. IV . BI BI. IOGR APHY M. V. Volkenstein. J. Polymer Sci.. 1’0. 111 (19.38). 01 6 . 10. 11. J. Yoo and J. 1371 (1962) . S. Li fson, J. Chem. Phys . '30 961 (19.39). K. Nagai, J. Chem . Phys . L11, l 169 (19.38)). S. Lifson, J. Chem. Phys. 29, 80 (1053). K. Nagai. J . Chm”. Phys. {30. 660 (lilofl). C. A. J. Hoeve, J. Chem. Phys. (1?, 68.“; (1960) . C. A. J. Hoeve. J. Chem. Phys. 11:). 1266 (1981). S. B. Kinsing‘er. J. Chem. Phys. 30, E. Bodewig. ”Matrix Calculus". North—Holland .‘—..—._—- A... _ Publishing Conpany. Chap. 2. C. Lanczos. "applied Analysis", Prentice Hall. Englewood Cliffs, P. Flory. "Principles of Polymer Cheri:-'-~tt'y". Cornell Universi ty Press. I Amsterdam. 1959. Part IV, J., 1961, Chap. II. Ithaca, New York. 1953. Chap. X. Inc. _‘ 1.... s! ”-331.14..- Ir. 44 APPENDIX A FIGURE 5 ‘Ditugram used to determine the sin-cosine transformation rmatrices when the same coordinate system is used for I both bonds. Y ‘Y (r ”‘7 For d d Cos = A/Y Sin = B/X Sin = C/Y Cos = D/X ‘ For 1 1 exchange X's and Y'S X =C+D = XCOS + YSin X'= XCOS — YSin y'zA—B =-XSin + YCOS Y'- XSin + YCOS Cos Sin Cos -Sin -Sin Cos Sin Cos FIGURE 6 Ditugram used to determine the sin-cosine transformation nurtrices when different coordinate systems are used for the: two bonds. For d 1 Cos = A/Y Sin = B/X , Sin = C/Y Cos = D/X For 1 d exchange X'S and Y's X'=A—B = ~XSin + YCos X'= XSin + YCos Y'=C+D = XCOS + YSin Y'= XCOS — YSin —Sin Cos Sin Cos Cos Sin Cos —Sin The values of the sin and cosine of the four bond atnxles are shown below. tungle 0 w/Z w 3w/2 sin 0 1 O -1 cos 1 0 —l O . W . . . . Matrix: 0313) is d d therefore the Sin—cosine matrix used is cos sin -sin cos sflxigg bond angle matrix 1 0 1 O 0 1 2 377/2 0—1 1 O 3 a —1 O O -l 4 w/2 O -l 631 Matrix 037*, is l 1 therefore the sin—~cosine matrix used is cos -sin sin cos state bond angle matrix 1 O 1 O O l 2 3w/2 0 1 —l 0 3 w —1 O O —l 4 n/2 -1 46 47 Matrix [Davfljl is l d therefore the sin—cosine matrix used is sin cos cos -sin _§tate bond angle 1 0 2 3W/2 3 n 4 77/2 at Ma trix [132714] is used is -sin cos cos sin state bond angle 1 0 2 dw/Z 3 w 4 fl/Z matrix 0 l l O —1 0 O l 0 —1 -1 0 l U 0 —l d 1 therefore the sin-cosine matrix matrix 0 1 l 0 l O O -1 O -l -l O -1 0 O 1 But for matrix [03.3 [021.] = [031». [03917] -‘ [Dgl‘fll] 48 APPENDIX B BLOCK METHOD FOR DETERMINING THE EIGENVALUES OF A MATRIX A A A O 0 0 Let B = A A A and O = O O O as shown for A A A O O 0 case one in Table 10. The 12x12 matrix then becomes rB O B 6W 0 B O 8 Upon solving for the eigenvalues(Y). O B O B the matrix becomes U3 0 B Q“ "B—Y o B o “ O B-Y O 8 Which upon multiplication yields 0 B -Y B _r3 0 B -YJ fl B—Y 'B-Y o B T -+ B 70 B-Y B = 0 B —Y B O B B L0 B —Y_' LB 0 -Y ‘ d Upon multiplying out further (B—Y)Z(Y2—82) + (B—Y) B 32 + B (B-Y)(+82) + 82(—82) : 0 9 1 ’ " L B“Y2 —28Y3 + Y1 - B4 + 283Y -82Y2 + B4 - BZY + B4 -B”Y—B x0 —2BY” +Y4 =0 Y”(Y—2B) x0. Y=0,0,0,+ZB A A A 1 1 1 But B = A A A , so 2B = 2A 1 l 1 A A A l 1 1 The next step is to solve for the eigenvalues(x) of the I 3hy3 nuitrix. ‘ 17* R- 2'. AW \ 4 9 Upon solving for the eigenvalues(x) the matrix becomes r.1—x 1 1 w Which upon multiplication becomes 1 1-x 1 L1 Ll l-xJ m~ (1—x)[(l-x2) :1] -l[(l-x) 4:] +1E1— (1— x] 0 E (l—X)(1-2X+X2-l) +X +x =0 ~2x+x 2*Lx2—xs +2x ‘ 0 } 3x2—x3=0 -x2(x—3)TO x=0,0.+3 Therefore the twelve eigenvalues of the original 12 by 12 ; matrix are eleven zeros and 6A. Thus the largest positive ; eiu'envalue of [@37'] Hay-+1 [2231‘01‘ case 1 is 6A or 6(2 57x10 8) which is 1. 54x10 , APPI‘TNDIX C ALGOL 58 COMPUTER I‘ROCIJDUUCS COMMI'I‘JT ITERATI‘ID VECTOR METHOD FOR CALCULATING THE LARGEST POSITIVE LIGI‘IXVALUE OF A TWELVE BY TWICLVIC MATRIX. THIS PROGRAM ALSO PRINTS OUT THIS CORRESPONDING FIGI‘NVECTOR AND ILIGi‘lNROW, R. C. THOMAS: INTEGER 1,J,K,L; ARRAY TMATRIX( 12 , 12) .MATRI .\'( 12 , 12) ,PRESVECTOR( 12) , parvvrcron<12>,r<12>; 1.0. . K L—‘O; FOR I (1.1.12) PREVVLCTOR(I)*0.0; PRESVECTOR(1)~1.0; FOR J=‘-(2 , 1,12): PRESVECTORLIWO .0: INPUT MAT (FOR I=(1 , 1,12): FOR J: (1 ,1 , 12): MATRIX(I,J)): READ (;;MAT): L1. . K=K+1; IF K EQL 100: GO TO LO; FOR I (1,1,12 : BEGIN Y(I)'=0.0; FOR J=(1,l.12): Y(I) '-‘- Y(I) + MATRIX (I ,J).PRICSV1§CTOR(J) IND: L2 . . I..RG‘~'0.0; FOR I (1,1,12); BEGIN IF Y(I) GTR LRG; BEGIN LRG "—' Y(I) ILI‘ID END; ._‘_" .‘to . ‘. M” A ...____ ‘ La.. I.(3 . . L7.. WRITF(:;LARGE,FORM1); OUTPUT LARGE (LRG); FORMAT FORM1(B7,F11.5,W); FOR I=(1,l,12); PRFVVRCTOR(I) 2 pRFSVRCTORz FOR 12(1,1,12); PRESVHCTOR(I) e Y(1)’LRG: FOR I=(1.1,12): BEGIN IF (ABS(PRESVECTOR(1) - PRKVVLCTOR(I))GTR 0.00001): GO L1 END: FOR 1-(1,1,12): BEGIN WRITF(;;VFCRO~,FORM1); OUTPUT VFCROW \JRHSVECTOR(I)) END; L=L+1 ; IF (L FQL 2); GO TO LO: FOR I=(1,1,12). FOR J-(1,1,12); TMATRIX(J,I) = MATRIX(I,J): FOR I=(i,!,12); FOR J*(1,1,12): MATRIX(I,J) : TMATRIX(I,J): GO TO Ll: FINISH; A_‘=-.O A 1.1 n! 52 (NDMMENT THIS PROGRAM CALCULATES MATRICES M AND N. IT AINSO PUNCHES OUT ON CARDS IN THE FOLLOWING ORDER MATRICES DI, (M+D2V), AND (E32—M)-1. (PHI2V)2.(D2V+1) AND (IUII2V+1)2.(SIGMA2V+1)2X4 AND LEIG ARE READ IN ON CARDS IN THAT ORDER. R. C. THOMAS; TARRAY’MATA(32,32),MATB(32,32),MATC(3‘,32),MATD(32,32), MA’I‘E(32,32); INTEGmII,JJCN; PROCEDURE INVERTI (N ,A( ,) : :ERRI): BEGIN '- Fa...“ _ i ccmmunn‘ INVERT A SQUARE MATRIX, IN PLACE, BY THE NBS PIVOTAL ROW METHOD. INPUT CONSISTS OF THE ORDER AND NAME OF THE MATRIX, OUTPUT IS THE INVERSE, WITH THE SAME NAME. REFERENCE MUST INCLUDE A STATEMENT LABEL TO WHICH A TRANSFER MAY BE MADE IF THE MATRIX IS SINGULAR, A PRINT OUT STATING INVERSION FAILURE WILL BE MADE. NOTE THAT ARRAY DECLARATION ALLOWS FOR A MAXIMUM MATRIX OF ORDER 30. IF LARGER ORDER IS NEEDED, CHANGE ARRAY DECLARATIONS ACCORDINGLY. AUTHOR - C.D. ALSTAD; INTEGER IHJNIJMLU5N: ARRAY ORDER<32),SAVE(32),SHIFT(32); FORMAT FINVl(BlO,*MATRIX SINGULAR, INVERSION FAILED*,WO): FOR I=(1,1,N); BEGIN ORDER(I) = O: SHIFT(I) = 0 END: Ml . Mi! . MIL. ONSFICK. . END MI} LND M2; Z M4 . . MS” COMMENT MAKIUIIK NOW INVERTED BUT SCRAMBLI‘LD, SO UNSCRAMBLIC: o» L- FOR L = (1,1,N); BEGIN CHAMP = 0; FOR I = (1,1,N): BEGIN z s A(I,1): IF ABS(Z) GEQ ABS(CHAMP): BEGIN K 2 1; FOR J v(1,1,L); IF ORDER(J) EQL K; GO ONSEEK: CHAMP = Z; ORDER(L) '—= K: H CHAMP; K ORDER(L): IF CHAMP EQL 0: BEGIN WRITE (:1FINV1); GO ICRRl 31 FOR J s(1,1,N-l):A(K,J,) FOR I =—'(1,1.N); BEGIN IF I EQL K: GO MS: ML=LT A(I,1): H FOR J =(1,1,N—1); A(I,J) A(I,N) . —MULT.A(R,N); END M4 END Ml; COMMENT UNSCRAAHSLE ROWS; Kit): MR1 . . MRI? . . MR3 . . FOR L = (1,1,N); IF SHIFT(L) EQL 0; GO MR2: IF ORDER(L) EOL L; BEGIN SHIFT(L):L;K w K+1z IF K EQL N; GO MC4: GO MR1 END; FOR J:—'(1,1,i\v ; SAVIXJ) =- A(L,J): SJ L; I = ORDER(L); FOR J=(1,I,N); A(L,J) < A(I,J); SHIFT(L) = I; LmI; R=K+1; IF K EQL N; GO MC4; IF ORDER(L) NEQ SJ; GO MR3; FOR J (1,1,N); A(L,J) = SAvE(J): SHIFT(L) = SJ; K : K+l; IF K NEQ N; GO MR1; A(K,J+1),'/.: A(K,T')‘ I“ 1“; D : 1,/.: A(I,J‘i1) " hIUIJ'I‘.A(K,fJ): . '-A.l (:OMMENT NOW UNSCRAMBLE COLUMNS; MC4.. FOR I=(1,1,N); SHIFT(I) = 0: K a 0; MC5.. FOR L=(I,1,N); IF SHIFT(L) EQL 0; GO MCG: MC6.. IF ORDER(L) EQL L; BEGIN SHIFT(L) : L; K K+l. IF K EQL N; GO MCD; GO MCS END MCG: FOR J:(1,1.N); SAVE(J) = A(J,L); SJ 2 L; FOR I=(1,1,N); IF ORDER(R) EQL L; Go MC7: MC7.. FOR J=(1,1,N); A(J L) = A(J,I): SHIFT(L) A I: L = I: K = K+1; IF K EOL N; GO MCO; FOR I =(1,1,N): IF ORDER(I) EQL L; GO MCS: MCS.. IF I NEQ SJ; GO MC7; FOR J=(1,1,N); A(J,L) = SAVE(J); SHIFT(L) : SJ; K = K+l; IF K NEQ N: GO MC5: MC9.. RETURN END INVERT1(); L01.. FOR I=(1,I,32); FOR J (1,1,32); BEGIN MATA(I,J)«0.0; MATB(I,J)=0.0; MATC(I,J)=0.0; MATD(1,J) 0.0; MATE(I,J)=0.O END L01; L02.. FOR I=(1,8,25); MATA(I,I)—1.O; FOR I:(2.8,26) MATA(I,I)=I.0; FOR I=(5,8,29); MATA(I,I)=—1.v: FOR I=(6,8,30); MATA(I,I)r-I.O; MATA(4,3)=I.0; MATA(3,4)=—1.O; MATA(7,8):1.0: MATA(8,7)*—l.0: MATA(ll,l2)rl.O; MATA(12,11) -1.0: MATA(16,15)=1.0: MATA(15,16)=—1.0; FOR I=(1,l,32): MATB(I,I)*1.0: L03.. INPUT MAT1(FOR I=(l,l,8); FOR J<(I.1,8); MATC(1,J)); READ(;:MAT1); L04. L05.. O] (,1 INPUT MAT2(FOR I=(9,1,16); FOR J:(9,1.16); MATC(I,J)); READ(;;MAT2); INPUT MAT3(FOR If(l7,1,24): FOR J:(I7,I,24); MATC(I,J)); READ(:;MATS); INPUT MAT4(FOR I=(25,l,32); FOR Jr(25,1,32); MATC(I,J)); READ(;;MAT4); INPUT MAT5(FOR I=(l,l,8); FOR J:(I,1,E): MATD(I,J)); READ(;;MAT5): INPUT MATG(FOR I=(1,I,8); FOR J=(17,1.24): MATD(I,J)); READ(;;MATG); INPUT MAT7(FOR I (9,1,16); FOR J (9,1,16); MATD(I,J)); READ(;;MAT7); INPUT MATE(FOR I=(0,1,16); FOR J=(25,l,32); MATD(I,J)); READ(;:MATS); INPUT MAT9(FOR I=(I7,1,24): FOR J=(9,l.16): MATD(1,J)); READ(;:MAT9); INPUT MAT10(FOR I:(I7,1,24): FOR J=(25,1.32): MATD(I,J)); READ(:;MAT10); INPUT MAT11(FOR 17(25,I,32); FOR J-(I 1,8): MATD(I,J)); READ(;;MAT11): INPUT MAT12(FOR I=(25,1,32); FOR J (17,1,24); MATD(I,J)): READ(; MATI2); INPUT LEIGV(LEIG); READ(;;LEIGV); FORMAT FORMG(W5): COMMENT INITIALLY AFTER INPUT MATAxDEV, MATRrLJZ, MATC=(PHIEV)2(D2V+1), MATD:(PH12V+1)2(SIGMA2V+1)2x4; LOG..FOR I:(1,1,32); FOR J=(1,1,32): BEGIN S 0.0; FOR Kx(1,1,32): S S+MATC(I,E).MATD(K,J); MATE(I,J)=S END: L07.. FOR I=(1,1,32); FOR J (1,1,32): BEGIN Sr0.0; FOR K;(1,1,32); S SJMATA(I,K).MATF(K,J); MATC(I,J):S/LEIG END; COMMENT MATC NOW IS MATRIX(M); FOR I (1,1,32): FOR J=(l,1,32); MATD(I,J):MATE(I,J) /LEIG: COMMENT MATD NOW IS MATRIX(N); WRITE(;;FORM6); WRITE(;;N,FORM2); OUTPUT N(FOR I:(l,1,32); FOR J (1,1,32); MATD(I,J)); FORMAT FORM2 (*5*,6F13.5,W5): FOR I=(1.1,32); FOR J”(1,1,32); MATD(I,J)=MATA(I,J) + MATC(I,J): COMMENT MATD NOW IS D2V+M; WRITE(::FORM6); WRITE(;;D2VPM,FORM2); OUTPUT D2VPM (FOR Ir(1,1,32); FOR Js’l,l,32); MATD(I,J)): LOS.. FOR IL(1,1,32); FOR J=(1,1,32): MATD(I,J)rMATB(I,J)-MATC(I,J): LOU . . L10.. 57 FOR I=(l,1,32); FOR J:(1,1,32): MATB(1,J)=MATD(I,J); INVERTI (32,MATD (,);; L01); FOR Ir(1,l.32); FOR J;(l,l,32): BEGIN 570.0: FOR Kr(1,1,32); S=S+MATB(I,R).MATD(R,J): MATC(I,J):S END; WRITE(;;FORM0); WRITE(;;E32MMM1,FORM2); OUTPUT E32MMM1 (FOR I;(l,l,32); FOR J-(1,1,32): MATD(I,J)); WRITE(;;UNITIFINV,FORM3); OUTPUT UNITIFINV (FOR I (1,1,32): FOR J (1,1,32); MATC(I,J)); FORMAT FORM3 (E(XO.5),W); GO TO L01: FINISH; COMMENT THIS PROGRAM CALCULATES H2/NL2. MATRICES M+D2V, (E32—M)-1,N, DELV, AND DELR ARE INPUT IN THAT ORDER. R. C. THOMAS; ARRAY MATA(32,32), MATB(32,32), MATC(32,32). MATD(32,S2), DELV(32,2),DELR(2,32), SUM1(2,32), MAT1(2,2), MAT2(2,2), MATU(2,2), MAT4(2,2), M 1(l,2), MT2(2.1), MT3(1,2): INTEGER I,J,K,L: L01.. FOR I=(l.l,32); FOR J=(1,1,32): BEGIN MATA(I,J) 0.0: MATB(I,J):O.0; MATC(I,J)=O,O; MATD(I,J) 0.0 END: FOR I=(1,1,32): FOR J (1,1,2); DELV(I,J):O.0: FOR J~(1,1,32); FOR I:(1,1,:): BEGIN DELR(I,J) 0.0: SUMI(I,J) 0.0 END: LO2.. FOR I=(1,1,2); FOR J=(1,1,2); BEGIN MAT1(I,J)<0.0: MAT2(I,J)=0.0: MAT3(I,J):0.0; MAT4(I,J) 0.0 END' MT1(1,1):0.0; MT1(1,2)=1.0; MT2(1,1):0.0; MT2(2,1) 1.0; MTU(I,1):0.O; MT3(1,2)=0.0: MAT1(I,I)~1.0; MAT1(1,2)=O O; a T1(2,1)r0.0; MAT1(2,2)~1.0; L03.. INPUT MPD2V (FOR I=(1,1,32); FOR J:(l,1,32); MATA(I,J)): READ("MPD2V); L04.. INPUT E32MMM1 (FOR Ir(1,1,32); FOR J (1,1,32): MATB(I,J)); READ(;;E32MMM1); L05.. INPUT N (FOR I=(1,1,32); FOR J~(1,l,32); MATC(I,J)); READ(;;N); _L06.. INPUT VECTOR (FOR I:‘(1,1,32); FOR J'—*(l,1,2): 59 DELV(I,J); READ(::VECTOR): L07.. INPUT ROW (FOR Ir(l,l,2); FOR J:(l,l,32): DELR(I,J); READ(;;ROW): L08.. FOR I=(l,1,32); FOR J=(l,l,32): BEGIN S=0.0; FOR K=(I,1,32); S~S+MATB(I,K).MATA(K,J); MATD(I,J)=S END; L09.. FOR Iz(l,l,32): FOR J=(1,1,32); MATA(I,J)=0.0; FOR I (1,1,32); MATA(I,J)=1.0; FOR I"(1,1,32): FOR J—(1,1.32): MATB(I,J) MATA(I,J)+MATC(I,J): L10.. FOR I=(1,1,32); FOR J (1,1,32); BEGIN s 0.0: FOR K;(1,1.32); S=S+MATB(I,K).MATD(K,J): MATA(I,J)=S END; COMMENT MATA NOW IS (E32+N).(B32—M)-1.(M+D2V): Lll.. FOR I=(l.1,2): FOR J'(l,l,32); BEGIN S+0.0: FOR K=(1,1,32): =S+DELR(I,K).MATA(K,J): SUMLII,J)