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ABSTRACT The structural information that has been published on the polydimethylsiloxane chain (mean-square end-to-end dimensions, linear- cyclic equilibria, chain motion from nuclear magnetic resonance, temperature coefficient of the chain dimensions, and-x-ray diffraction of the vulcanized elastomer) and on prototype small molecules (internal rotation in hexau dimethylsiloxane , and heats of formation of lower cyclics) is presented and discussed in detail. The purpose of this section is to establish the framework of information into which is fitted a model for the chain structure. Equations are developed for the meannsquare end-tosend dimensions (Ea/n12) of the polydimethylsiloxane chain which are valid at all values of n, the number of links in the chain, and in which first neighbor interactions of independent bonds and pairs of bonds can be included. The structural variables which must be specified are the length of the valence bond in the skeletal structure, bond angles in the skeletal structure, and the energetics and positions of the allowed conformations in the polymer chain. The equations are shown to be convergent for matrix elements 5 1/3. Calculations of 58/312 as a function of n are made for statistically independent and statistically independent pair models. It is concluded that in order to explain both the small energy differences that must exist between the rotational states available to the chain (certainly less than chal. mole) and the experimental value of Pg/nl2 = 7.25, longer range interferences must be included in the theory. All the possible endatoaend vectors of a growing siloxane chain model with three rotational states are generated and examined for n=6 and n=8 for the purpose of gaining insight into the distribution of cyclic molecules in the polydimethylsiloxane equilibrium. The distribution of 3 r rs is obtained for the three rotational state model for n=6 and 3 n=8 From the fraction of r '5 closing, axlequilibrium constant is calculated for the formation of octamethylcyclotetrasiloxane which is in good agreement with the experimental value. In Appendix I, the formalism for the dimensions of an -M-0-Nu0- chain is developed, the equations being valid for all values of n. A STATISTICAL STUDY OF THE STRUCTURE OF THE DIMETHIISIDGXANE CHAIN AND RINGS By NFL Jack B: Carmichael A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1965 ACKNOWEEDGMENTS The author wishes to express his appreciation to Dr. Jack B. Kinsinger for his guidance, encouragement and friendly counsel during the course of this investigation. Appreciation is extended to the U. S. Air Force for financial support in the fonm of a graduate research assistantship and to the Dow Chemical Company for a fellowship. Special thanks go to Dr. Donald McQuarrie, Mr. Paul G. Rasmussen and Dr. Morley Russell, for their assistance and encouragement. Thanks also are extended to Miss Barbara Miller and Mrs. Beth Unger for their help with computer calculations. The author is grateful to his wife for her help and under- standing during the preparation of this thesis. TABLE OF CONTENTS I 0 INTRODUCTION 0 o o c o o o I c o I 9 s I o o o I I I 0 O O I O 1. Structural and phenomenological information about the polydimethylsiloxane chain . . . . . . . . . . . . . . . . A. Determination of the unperturbed root-mean-square end-to-end dimensions (:8) of the chain from 115911: scattering, osmotic pressure, and intrinsic viscosity measurements. . . . . . . . . . . . . . . . . ... . . . B. Equilibria between linear and cyclic dimethyl- siloxanes.. . . . . . ...-. . . . . . . . . . . . . . . C. Molecular motion in the polydimethylsiloxane chain as studied by nuclear magnetic resonance . . . . . . . D. Temperature coefficient of the siloxane chain from intrinsic viscosity measurements and from stress- temperature measurements of the vulcanized elastomer E. Crystallinity and orientation in silicone rubber Studied ,by x-ray diffraCtiOn o o o o s c o I s o o o o 2. Structural information about prototype small molecules A. Internal rotation about the siloxane linkage studied in hexamethyldisiloxane . . . . . . . . . . . B. Investigations of the molecular structure of the lower cyclic dimethylsiloxanes. . . . . . . . . . . . C. Heats of formation of lower members of dimethyl- and methyl-isopropoxy-cyclopolysiloxanes . . . . . . . II. THEORETICAL MODELS FOR THE POLYDIMETHYLSILOXANE CHAIN A. Calculations of the dimensions of polymers B. Development of equations for the dimensions of the polydimethylsiloxane chain . . . . . . . . . . . . 1. Orthogonal transformations . . . . . . . . . . . . 2. End-to-end distance of a microscopic config- 11 12 12 1h 17 19 22 22 uration . . . . . . . . . . . . . . . . . . 5. Mean-square end-to-end distance of the chain . Application of the equations to chain models . . 22 26 32 news-urn-aoo-x bed-mi Sign Inn undo 9113 to ( I" '.-"x1 5 311:1; ".2. '- ‘5 u an to warm mm” Ins-03.54:: 31'901.-a.‘.‘.'-3*3.1E‘!:tfl1 has .exhsauq’ 91:0!an .3319”ch . amen-212mm“: . . . u I I '5‘}. . E"...“5'- ' r -’.\L" ' '51- u _ 1" .A Page 1. Intramolecular interactions in polymer chains.o...-....-.......‘,... 32 2. Statistically independent model for a polymer Chain I I I I I I I I l I I I I I I I I I O I I 32 3. Statistically independent pair model for polymer Chain I I I I I I I I I I I I I I I I I I :I I I 36 D. The prediction of stability of helical conform— ations of polymer chains . . . . . . . . . . . . . . 37 E. Long range interferences in polymer chains . ... . . 39 F. Chain configuration and motion . . . . . . . . , . . #2 G. Convergence of £2/n12 - investigation of the con- vergence of the matrix power series. . . . . . . . . A2 111' mSULTS I I I I I I I O I I I I I I I I I I I I I I I I . I I ’4’6 A. Verification of the equations. . . . . . . . . . . . #6 1. Correct computation of the ring formation at n=6 for hexamethylcyclotrisiloxane . . . .,. . . A6 2. Computation of rz/nla for the freely rotating model. I G I I I I I I I I I I I I I I I I I O I 16 B. Calculation of rz/nla for statistically independent models of the polydimethylsiloxane chain . . . . . . #9 C. Comparison of statisticall, independent bond and bond pair models for the polyuimethyl- Siloxane Chain B I I I I I I I I I I I I I I I _ I I I 5"} D. Calculation of ring-chain equilibria in.poly- dimethylsiloxane . . . . . . . . . . . . . . . . . . 56 1. The. distribution of ‘r‘l 's for a three rotational state chain . . . . . . . . , ... . . 57 2. Equilibrium constant for R4 formation. . .,. . . 59 5. Calculation of theoretical equilibrium constant I I I I I 0 I I I 0 9 a I I I O I O I I 60 “0 Discussion I I I I I I I I I Q I I ‘- 0 B I I b I 62 APPENDICES I Calculation of 562 for «mom—o- chain i 11 160A FORTRAN programs for calculation of 1'4 for ~M-OuMa chain ix 111 160A FORTRAN program for generating all end-to-end vectors of a growing three-rotational state chain xiii LIST OF TABLES Table Page I Experimental Polymer Dimensions 5 II Weight Fraction of Cyclic Species in Equilibrated Polydimethylsiloxane ((C113)2S:10)n 6 III Temperature Coefficients of the Dimensions of Polymer Chains 9 IV Structure of Hexamethylcyclotrisilomne 15 V Structure of Octamethylcyclotetrasiloxane 15 VI 52/1112 as a function of [$1081 for (cos ) =0. Com- parison with values calculated from Flo 's approximate equation 49 VII 222/1312 as a function of (coe'L>_ZSios:L=-1uo°, Zos10= 109°28' 52 VIII Application of statistically independent bond and bond pair ' models to calculating the same 13/1112. [81051 = 112.5, 40810 = 109°28' 55 1x Distribution of [E‘l's for n = 6 57 x Distribution of [5| 's for n = 8 58 XI Comparison of Experimental and Theoretical Calculations of the Equilibrium Constant for R4 (Octamethylcyclotetrasiloxane) 62 'r'..-'.:io‘{ in '.;-:.' new-'1". —r-Y.-- 1-:- ':-‘-.'-al- 3an mum-19.11.!!!“ III minim :--..-- 1-: r ‘ Figure I II III VI VII VIII LIST OF FIGURES Interactions in a Statistically Independent Chain- Represented by a Plot of Potential Energy vs. Rotational Angle Structural Model of Crystalline Polydimethylsiloxane Conceivable Forms of Octamethylcyclotetrasiloxane Visual and Theoretical Intensity Curves of Octamethylcyclotetrasiloxane Conformation Potential Energy of Polyoxymethylene Considered as a One-Atom Helical Chain. Verification of the Equations ra/nl2 vs. n for Flory's Freely Rotating Model as a function oszSiOSi 52/n12 vs. n for Statis. Ind. Siloxane Models 52/12 vs. n for all cis chain, [81061 = lhO° (0310 = 109°28' Use of Stat. Ind. bond and bond pair Models to Obtain the same r2/n12)n=h0 Page 10 15 16 16 50 55 55 I. INTRODUCTION Characterization of the spatial configuration of a macro- molecular chain consisting of a succession of structural units linearly connected to one another, is prerequisite to the theoretical inter- pretation of the constitution and properties of polymeric materials. Mathematical methods for treating the configuration of linear chains have been recently greatly advanced through adaptation of the matrix method of orthogonal transformations to the linear chain problem. This method rests on specification of the chain configuration by assigning the valence angles and rotational states of each chain bond. A basis is thus accomplished for establishing a close connection between bond structure and the main characteristics of chain configuration.l Equations for the mean-square end-to-end dimensions of many types of polymers have been developed using the matrix method. These equations are applicable to a real polymer molecule only when this molecule is in an unperturbed state.2 From the unperturbed dimensions and nuclear magnetic resonance studies of the polymer, ;-ray diffraction analysis and the stress-temperature coefficient of the vulcanized elastomer, parameters are determined which can be related to the potential difference between preferred and non-preferred rotational states.3 Most of the equations developed to date are valid only for high molecular weight polymer chains. These equations adequately de- scribe the equilibrium dimensions of many hydrocarbon chains such as polyethylene, polypropylene, and polyisobutylene because only chains of 5,155 high molecular weight are present. Small cyclic molecules have not been found to appreciable extents during any of these polymerizations. This is quite a different matter for inorganic polymers, however. In 2 fact, K. A. Andrianov has said that the most pressing theoretical prob- lem in'inorganic polymers today is predicting the cyclic species that are formed in large amounts in many inorganic polymerizations. The most documented example of cyclic formation in an inorganic polymer system is the case of polydimethylsiloxane, in which cyclic molecules containing from six to eighteen atoms in their skeletal structure make up twenty percent by weight of the equilibrated system.6 No existing theory adequately explains this phenomenon because of approximations which allow equations for a siloxane-type chain to be valid only for chains. containing at least 100 links. The major purpose of this work is to described the structure of the polydimethylsiloxane chain and explain the chain-cyclic equilu ibrium in terms of this structure as completely as possible within the framework of available experimental information.* To achieve this result, equations for the meanusquare end—to» end dimensions ($8) have been develoPed which are valid for any number of bonds in the siloxane chain. These equations will be the means by which the experimental information is fitted to the best model for the chain. Much experimental information describing the siloxane chain has been published and will be discussed in subsequent paragraphs. 'Models for polymer chains will then be discussed as a prelude to the presentation of the model chosen to described the polydimethylsiloxane * Stockmayer and Jacobson published in 1950 the only attempt to date to predict the molecular size distributions of linear and cyclic molecules for polydimethylsiloxane. This paper is discussed in Section III-D and the results are compared with those obtained from a detailed structural analysis of the chain. 3 chain. This work is the first attempt to unify all of the available dlta into a consistent theory for the siloxane polymeric systev. Experimental data 1. Structural and phenomenological information about the poly- dimethylsiloxane chain. A. Determination of the unperturbed root-mean—square end-to-end dimensions (3%) of the chain from light scattering, osmotic pressure, and intrinsic viscosity measurements. The important molecular parameter obtained from these nn_‘_‘ measurements is ro/nla, where $0 = r -'f =i§li§lli' 11 where the end- to-end bond vector f is expressed as the sum of the bond vectors—Ii. n = the number of links in the polymer chain, 1 = the length of the valence bond in the skeletal structure. The determination of the unperturbed dimensions of polymer molecules from dilute solution viscosity measurements depends first on the specification of a solvent medium in which the net osmotic forces acting on the polymer molecule are exactly zero. It is currently thought that when this condition is fulfilled, the average configuration of the polymer molecule in solution and its average dimensions are "unperturbed“ in the sense that these dimensions depend only on the chemical bonds of the polymer chain and include the influences of the hindering potential to rotation about these bonds, the so—called short-range intramolecular effects. Since the intrinsic viscosity depends directly on the volume occupied by the chain, its determination in a solvent at the 6 temperature or in bulkd provides a method for finding the unperturbed dimensions of the chain. The first determination of the molecular dimensions of poly- dimethylsiloxane was made by Flory, Mandelkern, Kinsinger and Schultz.7 From osmotic pressure and intrinsic viscosity data of fractions in” methylethylketone and phenetole at 6 = 20° and 85° respectively the authors found Ffi/nl2 = 7.23. Several authors have subsequently studied the siloxane system and obtained similar results. Kuwahara, 33 al.8 measured osmotic pressure and viscosities over a wide temperature range to obtain second virial coefficients and intrinsic viscosities for various solvents. The authors found rg/nla = 6.89. Schultz and Haug9 made light scattering and intrinsic viscosity measurements on six fractions of polydimethylsiloxane ranging from. 55,000 to 1.06 x 106 in molecular weight. Haug and Meyerhoff have also determined the unperturbed chain dimensions from measurements of 10 sedimentation, diffusion and viscosity. Both results agree substantially with those of Kuwahara, it al_.8 and Flory, at 937 For purposes of comparison, the data of chain geometry and structure for polydimethylsiloxane is summarized along with some other organic polymers in Table I. B. Equilibria between linear and cyclic dimethylsiloxanes. Hartung and Camiolo6 have studied the polymerization to equilibrium in xylene solution of octamethylcyclo- *'At T = 9, there exists an exact balance between the effect of mutual excluded volume of the segments, which tends to enlarge the molecule, and the effect of a positive energy of mixing, which encourages first neighbor contacts between polymer segments and, hence, a more compact configuration for the molecule. Thus the molecular dimensions are unperturbed by intramolecular interactions. Table I Experimental Polymer Dimensions * Chain Boga Chain Bond _ _ _ .1- Polymer Length, A Angle, ° rg/nla (ro/nfi92 Reference Polydimethyl- 91:109.5 siloxane 1.65 ee=1l+o 7.25 1.60 7,8,9,1o Polyethylene 1. 51+ 110° 6 . 55 1.81 11 Isotatic Poly- propylene 1.5h 11h° 7.55 1.95 12. Polyisobutylene 1.5h llh° 7.h5 1.95 2 Polystyrene 2.5 15 Polymethylmeth- Detailed structural infor- acrylate mation on these polymers is 2'0 15 Polyethylmeth- acrylate contained in the original 1.9 13 Polybutylmeth- references listed in acrylate Volkensteins' book. 2.1 15 Polyhexylmeth- acrylate 2.h 15 Polyvinylacetate 2.5 15 natural rubber 1.5 15 Gutta percha 1.5 15 tetrasiloxane using KOH as a catalyst and quantitatively reported the weight fraction of cyclic species from trimer through nonamer. Scottlu studied cyclic formation in polydimethylsiloxane but used large amounts of hexamethyldisiloxane and obtained low molecular materials. A summary of Hartung and Camiolo's data is given in Table II. C. Molecular motion in the polydimethylsiloxane chain as studied by nuclear magnetic reSOnance. * The mean-square freely’rotating dimension of a polymer chain, ng, is the dimension calculated for a chain with fixed bond length and valence angle(s) with no restriction on the rotational anglefil- For this model, therefore, = 0. ' 1_u Table II weight Fraction of Cyclic Species in Equilibrated Polydimethylsiloxane ($392510); .2. Weight Fraction Standard Deviation 3 .0017 .oooh h .100 .Olh 5 .0610 .0076 6 .0185 .ooha 7 .d38 .omj 8 .0025 .oooh 9 .0017 .0005 Only broad line nuclear magnetic resonance spectroscopy has been thus far reported for the siloxane chain. No conformation study of this polymer by high resolution nuclear magnetic resonance spectroscopy has been made. 15 16 Honnold, McCaffrey and Mrowca and Rochow and LeClair published the earliest results indicating line width narrowing from 5.0 gauss at 80° to 0.1 gauss at 225°K. l7 Kusumoto, Lawrenson, and Gutowsky studied a dimethyl-based vulcanized silicone rubber (Silastic 80), a polydimethylsiloxane polymer with molecular weight >106 (#00 gum), and a lower molecular weight fluid polymer (200 fluid). They found a line width of u.5 gauss from * The decrease in abundance of dimethylsiloxane rings was discussed earlier by Flory2 who stated on the basis of less complete data than that of Hartung that the trend is that expected if the decrease is due solely to the statis- tical decrease in probability of ring closure with increasing chain length, specific steric factors being of minor importance. He concluded that "the greater length of the Si-O bond and the large SiOSi angle may alleviate the repulsions between substituents (in this case -CH3), which in rings con- sisting of -CH2- have been suggested as being responsible for both the strong preference for five- and six-membered rings and for the severe difficulty of forming rings of eight- to twelve-members.” 77° to 150°K. and a sharp narrowing to >.10 gauss over the range 150° to 1809K. The mean values of the second moments (ABE) in the region of 77°K. were found to be 8.h, 8.h, and 7.8 gauss for Silastic 80, #00 gum, and 200 fluid respectively. They attribute a slight narrow- ing of line width at about 90°K. to translational motion and the.maJor narrowing in the region of 160°K. to the onset of rotation about the 61-0 bonds. The minimum in the spin-lattice relaxation time for #00 gum at 190°K. is explained by a torsional oscillation of large amplitude along the Si-O-Si axis. Huggins, St. Pierre, and Bueche18 measured the proton magnetic resonance over the temperature range 60°-500°K. of a sample of polydimethylsiloxane with weight average molecular weight 7.5 x 104. The low temperature (60°~80°K.) peak to peak line width was 4.9 gauss, in satisfactory agreement with the results of Gutowsky, st 21. The second moment was found to be 7.0 * 0.2 gauss. It has been shown that an isolated rigid methyl group should exhibit a 0H3 of about 21.5 gauss. Free rotation about the C-Si bond would reduce this to about 5.h gauss. The authors presume then that the second moments for silicone polymers at 80°K. consist of contributions of 5.h gauss due to freely-rotating methyl groups and an inter-group broadening of 1.6 gauss. A sharp line narrowing to .019 gauss was recorded at approximately 190°K. The motion causing this narrowing was assumed to be a rotation about the chain axis made favorable by the low torsional force constant of the Si-O-Si linkage. The major conclusion of these studies is that the polydimethyl- siloxane chain is very flexible with low potential barriers to rotation about the chain axis, Gutowsky, et al., calculated an upper bound of of 3.5 Kcal./mole and shallow potential wells. D. Temperature coefficient of the siloxane chain from intrinsic viscosity measurements and from stress-temperature measurements of the vulcanized elastomer. A linear polymer molecule is characterized by the large number of conformations possible through internal rotation around the axis of the chain bonds. For free rotation all these conformations have equal energy and, therefore, the mean-square end-to-end dimension of the polya mer is not a function of temperature. If different energy levels exist in the function describing potential energy vs. rotational angle, however, then the unperturbed dimensions of the molecule will be a function of tempera- ture. A study of the temperature variation of the mean-square end-to-end dimension of linear polymer molecules affords a method for studying the nature of the intramolecular interactions in the chain which cause the variations in the potential energy profile of the internal rotational angle. The temperature variation of‘FE is accessible to experiment. Ciferri has determined d lnrg/dT from the equilibrium tension~tempera- ture coefficient at constant volume and length for a cross-linked poly: 19 dimethylsiloxane network. This result is compared with d lnEo/dT determined by Ciferri for the free polydimethylsiloxane chain through intrinsic viscositywtemperature measurements carried out in athermal polydimethylsiloxane solutions.20 ' '- |-.--- . ’.1' . These data for the polydimethylsiloxane chain are listed in 21,22 A the table below along with similar data for polyethylene. structural interpretation of this information follows the table. TABLE III Temperature Coefficients of the Dimensions of Polymer Chains Polymer Method d ln'i'~%/<1'r,¢ieg.'1 Ref. Polydimethylsiloxane tension-temp. 0.h6 (10.1h3 x 10"3 19 Polydimethylsiloxane intrinsic vis.-temp. 0.53 (t0.05) x 10-3 20 Polyethylene tension-temp. -l.l6 (30.10) x 10"3 21 Polyethylene intrinsic vis.-temp. -l.2 (30.2) x 10-3 22 For both polymers there is excellent agreement between the un- related methods. This lends substantial support to the assumption that the molecular conformation in the amorphous state is not appreciably affected by interactions between neighboring chains. When the rotational states in adjacent links in the polymer chain are statistically independent (that is, the motion of the groups about the chain axis is not a function of the position of the groups on any following or preceeding chain bonds) and the potential function for each given by a function of the type V03) = V°/2[x(l-cos¢) + (l-x)(l-cos3¢)]23, where V° is the value of V03) for o = tn, and x is the probability para- meter which determines the potential energy differences between the minima, then the sign of d lining/am is easily predicted. ‘3' .91“: as units: scrim-ea: sauna-um III man-1' _ :1". .'-'-:w{,£0'i '35 L1; .- r 2....”3 -::-.L" 5:. '.jflfill: :T'NGC- 'J'WJmuqme'E ' I —. F .-. -- '... :..'.' Inn'fi'n. . .. ..- i 10 Figure I Interactions in a Statistically Independent Chain-Represented by a Plot of Potential Energy vs. Rotational Angle 1. Model for the polyethylene Chain U(¢) e 60' 180° 590° 560 ¢i ¢2 @e 2. Model for the Polydimethylsiloxane Chain c». is a States bl and ¢3 represent gauche configurations and state ¢2 represents the trans configuration. AU = U2 — U1. Consider now the case of the polyethylene model. Let a,b,c, be the probability of states $1, $2, $3 respectively. The point of zero potential energy is chosen to be U2. Then, 11 e-AU/k‘l‘ e-AU/kT a. 8 ———=——7T— = . "' C. E eU11 k l + 2e'AUJRT n=l ' 1 b a 1 + Qe’AUIkT -AU/kT As T increases, e increases. Therefore, a and c increase while b decreases. Since the all-tranS'configuration is that of maximum extension of the polymer chain, when b decreases, the mean-square endato- end dimensions of the chain decrease. Thus, d lnrg/dT is negative. Since for the model of the polydimethylsiloxane chain, the trans state occurs with lower probability than the gauche states, as T increases b also increases while a and c decrease. Thus, the mean-square end-to- end dimensions of the chain increase and d lnrg/dT is positive. However, Hoeve has already shown that the dimensions of the polyethylene chain cannot be predicted using a statistically independent 4 model.2 Later in this work a statistically independent model will prove to be inadequate to interpret all the experimental information on the siloxane chain. Thus, the temperature coefficient of the mean» square endetoeend dimension is a function of more than one variable and cannot be simply interpreted as above, E. Crystallinity and orientation in silicone rubber studied by x—ray diffraction. Ohlberg, Alexander, and Warrick in 1958 studied the x-ray diffraction pattern of a silica-filled, vulcanized polydimethylsiloxane polymer at different temperatures and elongations.25 The crystalline fraction of unstretched and stretched rubber samples was determined. 12 From this work and a supplemental study of density data, tension-temperature measurements, and normal stress-strain data taken at low temperatures, Warrick26 concludes that stretched silicone rubber seems to be able to crystallize chain segments in all orientations in space. This contrasts markedly with the behavior of natural rubber which forms crystallites oriented almost exclusively in the direction of stress. He further states that this and all the other differences in crystalline behavior between the two rubbers can be founded upon a much greater chain segment mobility for silicone rubber than found in natural rubber. The latest x-ray diffraction study reported for polydimethyl- 27 siloxane-based rubber was by G. Damaschun in 1962. His study was carried out at -90°C with an extension ratio of h. The unit cell was reported to be monoclinic with a = 13.0A, b = 8.3;, c = 7.75;, B = 60°. This cell contains six dimethylsiloxane groups with H .wfih oomv .m “wows map Mo Show oaomno we: mo>hdo hpfimsopnfi Hmofipoh Iowan op pfl% pmom Aooadmmmv l .23 838% em mdoa m + oi no; eremmAnmo: .oohoxodm we mean when arson mm moa To: mo; eremonmo: nanossoo ..Hom .oflmov .nmoav .m 0.8 esooasOU onmxoaflmanpopoaohoahmposopoo Mo oadpodApm . > flees .nmomam an op when oopeomom an .m H mma .m H mfl no.0 H 84 naoemmanmo: .nmnmam on op when oopnooom on .oma .43 SA voHmmAnmo: mammov/ \manmov I I I \ em] o/ \o 16/ . . e nosed on mm .e + one .e + mos no o + so a o/Hmlo\.m/o Ism\o 3 when voAmv \. / map oopnomom NAomoV NAomov npoossoo ..eom emoemv oemov 3: ohm oseoosoo osmxoafimanpoaohoahneostom Mo madpodapm 3 Sons ma l6 Yokoi has constructed several models which give theoretical intensity curves closely approximating that of the experimental curve. Pictured below are diagrams of these structures along with the intensity curves. Figure III Conceivable Forms of Octamethylcyclotetrasiloxane5h Figure 1v Visual and Theoretical Intensity Curves of Octamethylcyclotetrasiloxaneju prerimental 17 1 Model 310:0(11) 8i::g) were derived and re- ported° On the basis of the assumption that the linear polysiloxanes considered have no strain in their molecular structures, Tanks concludes that the ring structures of the cyclic tetramer and cyclic pentamer are free from strain and the cyclic trimer has strain energy of somewhat less than 9 Kcal./moleo The conclusion that strain exists in the six—membered ring agrees with the smaller value of —2‘ + MHZ)? + (A2A1)2 + n-h n-2' A2(A1A2)—2‘ +(A2A1)T + t t t §l§ Elé “'6 A2A1 + A2 + I + ---(A1A2) 2 A1 + (A1A2) 2 + (AlA2) 2 A1 + n_-& n_-A_ (A1A2) 2 + (A1A2) 2 A1 + ---- n-6 n—6 n-8 [(A1A2)TAl 1t + [(A1A2)—2_]t + [Omar—Al 11: + I + A2 + AaAl + A2(A1A2) + (A2A1)2 + n-h n-6 n-6 [(A1A2)'2‘_]t [A2(A1A.o_)"2_1t + awe)? 1t + A2 + I + A1 + A1A2 + (A1A2)A1 + 2:2. n_—2t 31;}: t [(A1A2) 2 Afl + [(AeAi) 2 ] + [(AiAe) 2 Ad + --- +[ (A2A1)2]’ + [(A1A2)A11t + [(A2A1)]t + A? + I} (3 0 Here again the 5,5 element in (At)n will be the same as the 5,5 element in An. (c) For n even, r2 n'1 E —— = n +v TE: [(n-k+l)(A1A2) 2 A113,3 I k=1,5,5.“ n- .12.; + [(n—k—l) A2(A1A2) 2 13,3 k=l,5,5 n- E + [(h'k)(A1A2)2 13,3 k=2,h,6... n—2 E + 2: [(n'k)(A2Al)2] 3,3 51 (d) A change in summation notation follows: Let Mike 2 B, AeAl a D n_-2 n-l k-l 2 k E (n-k+1) [3‘2 Ana. = 2 01-21:) [(13) A113,. k=l,5,5 k=0,l,2 2;“ n-5 k-l 2 k (n-k-l) {A2132 113,3 = 5: (n—2k~2) [A213 13,3 = ,5,5 k=0,l,2 n-h 2 k = 2 (11—2102) [D A2: 3,3 k=O,l,2 n-2 n-2 2 2 . z (mask/13,3 = Z for the continuous and discrete models. 52 For a chain model having a continuum of rotational states, 231 / e'u(¢)/kT cos¢ d dg o 211 / e-u(¢)/k'r d b 0 where: u(¢) is the rotational potential energy function. For a chain model consisting of a finite number of discrete rotational states, 3" = 2 : Pi c0543i i = l n 2 , P1 1 = l where: Pi = the probability that the chain assumes the 1th rotational state. The representation of the polymer chain as a mixture of rotational isomers was recently experimentally justified by V. N. Nikitin and B. Z. Volchek?5 who found spectroscopically a transition of certain rotational isomers into others on stretching a polymer. 3h The concept of statistically independent rotational states in polymer chains with discrete states is perhaps mOSt clearly illustrated by the use of probability models. Consider a polymer chain viewed along the bond axis from the ith atom in the chain to the (i+l)th. This chain will be allowed to assume four possible rotational states (in the case of the siloxane chain these are gauche right, trans, gauche left, cis;) The rotational states ¢T ) ¢Gl will be designated as ¢G l n-l n- , , b with probability of n-l C l occurrence a, b, c, d respectively. This model is pictured below. a + b + c + d = l Succeeding links will move outward from whichever of these four states it occurs. If the polymer chain is statistically independent, the probabilities of chain paths do not change with choice of the rotat- ,ch , $0 are still a, b, c, d respect- H n ional state. That is, from ¢G n-l , t. ively. The probability model is illustrated below. ¢G' , ¢C the probabilitis n—l } n-l n-l of going to ¢Gn , QTn 55 36 Since this probability model continues through the polymer chain, a convenient shorthand notation is: i. i. n-l a man 91.4 . hm c 96 ' n-l <1 (#G In+m PCn-l ¢cn+m where m = 1,2,5... 3. Statistically Independent Pairs of Bonds Model for a Polymer Chain The concept of statistically independent pairs of rotational states in the polymer chain means that the probabilities of the ¢G , ¢T , *3. , ¢C states will depend on the rotational state previously assumedn by :he pglymer chain. That is, the chain will assume a preferential orientation depending on the outcome of previous events. For the most general case, there will be a different probability attached to each rotate tional state. The restrictions thus imposed are that (l) ] and (2) = = . . . for n chain bonds. The arguments of Section II C. 2. regarding statistically independent chain bonds can be applied to this case where now the repeating independent chain unit is a bond pair. 57 D. The Prediction of Stability of Helical Conformations of Polymer Chains. DeSantis, Giglio, Liquori, and Ripamonti5h have approached the problem of predicting the configurations of polymer chains from fundamental principles of non-bonded interactions within the chains. These workers have calculated rotational potential energy curves for several simple polymer chains using a Van der Waal's potential to describe the interactions between pairs of non-bonded atoms. The most stable helical conformations calculated predict the previously known preferred rotational states with surprising accuracy. In several cases less preferred rotational states appear at angles not previously suspected. No calculations were made on the polydimethylsiloxane chain. Calculations made for the structurally similar polyoxymethylene chain H H (-!=0=é=0=) assuming a one-atom helical chain (statistically independent 4 J. model) with [000 = LCOC = 110°53' predict symmetric gauche states at 67° and 295° and a less preferred trans state at 180°. The calculated potential energy curve is pictured on page 58. In a note added in proof the authors state that no new rotational states were predicted when the conformational potential 3.8 '0 (M Cal. I Mole 9 .0 0' 40' 00' I20' I60‘ 200’ 80‘ 200'-— 560' <1) FIGURE V Conformational potential energy of polyoxymethylene considered as a one-atom helical chain. energy of the polymer chain was calculated using a statistically dependent, two-atom helical chain model. The authors state that the lack of side-groups attached to alternate atoms of the chain is responsible for the fact that the gauche conformations are more energetically favorable (have a higher probability of occupancy) than the trans conformation(s) found for both polyethylene and polytetrafluoroethylene, in which side-groups are attached to every atom in the chain backbone. They apply an analogous reasoning to explain the marked flexibility of the polyoxymethylene chain and the existence of a number of crystalline helical conformations. The polydimethylsiloxane chain is structurally similar to the polyoxymethylene chain since in both cases the oxygenv atom is an alternating atom in the chain and has no side groups attached. In the view of this author the results calculated for polyoxymethylene and the observed structural similarities to polydimethylsiloxane lend weight 39 to the three state approximation (G, G', T) for the allowed conform- ations of the polydimethylsiloxane chain. E. Long Range Interferences in Polymer Chains The average configuration of real polymer molecules in the unperturbed state is markedly influenced by the requirement that two elements of the polymer chain are forbidden from occupying the same location in space. This requirement is often implicitly neglected in calculations of chain dimensions and has been neglected in all prior derivations of this work. The long range interference effect means that a given configuration calculated from a period one or period two model will be acceptable only in the case that none of its segments is assigned to a site occupied by another segment. Only a small fraction of the calculated configurations may be realized by the actual chain moleculess. Thus, even if one uses an accurate assignment of rotational states and their occupancy and correct bond angles and lengths, the calculated dimensions will be smaller than the actual dimensions because unallowed configurations have been included in the calculations. G; W. King has provided much insight into the theoretical considerations necessary to take account of the long range interference effectsq King reifterated the generally accepted principle that the thermodynamical properties of long chain molecules depend only on the configurations of segments of the chain very short compared with the whole molecule. The geometrical icomplexion of a segment, say, 20 atoms long are independent to a very large extent of segments more than 20 atoms away. The smallest tolerable subsystem necessary to accurately to describe such a system is thus a sliding segment of Zf%-2O chain links, whose configurations are independent of atoms outside the segment but dependent on those within it. Each chain link occurs in Z1 + 1 different segments obtained by sliding. What King proposed is, in the terminology of this work,a chain of period Zr Let F1(Zl) be the number of configurations of a segment of Zl atoms. Then the Markov matrix, P, necessary to describe the chain is of order F3(Zl). Consider now a property P which has a value pi in the ith. segment. P could be the vectorial distance between the ends of the segment weighted by the Boltzmann factor of the energy of the combined configuration. The elements of the Markoff matrix are pi,j = pixpj. An element pij is the value of the property P of the segment of length 221 obtained by the concentration of a segment of Zl atoms of configuration 1 with a segment of Zl atoms of configuration 3. The most important feature of this model to note is that long range interferences are dealt with explicithrand directly. An element Pi,j always has a factor which is 0 if the combined segments cannot exist, 1 if the combination is allowed. The immense complexity of the formalism necessary to describe the period Z1 chain makes calculations prohibitive. The important feature is the insight which the model offers into the long range interference problem. If the chain can be represented by a sliding segment of twenty atoms, the model of King would account for long range interference effects. Tractlble solutions to the long range interference problem lie in finding approximate methods to eliminate unallowed chain config- hl urations. In a classic paper, E. W. Montroll has treated a chain on‘a square lattice in which right and left steps are required at each lattice point, and short range overlapping is eliminateo.5? Those over- laps which occur when a monomer four removed from a given monomer returns to the position of the given monomer are called first-order overlaps ( Ll ). Overlaps of monomers l2 removed are called second-order over laps ( r{ %7), etc. The first order overlaps are much more probable than those of higher order and Montroll has chosen as his approximate treatment the elimination of interferences in the chain due to first order overlaps. Montroll is able to obtain in closed form, the difference equation for the number of configurations for ths very restrictive model. When n(the number of links in the chain) = 10, the fraction of config- urations without first order overlap — 0.18. When Montroll's method is applied to a real polymer chain off a lattice, as, for example, the polydimethylsiloxane chain, the number of configurations available to the chain is increased many times over the number available to the square-lattice chain and predicting a priori those configurations which lead to overlap becomes a much more difficult matter. A numerical approach is taken in this work as follows: The siloxane chain is assumed to have G, T and G' rotational states available. The value of the end—to-end bond vector is calculated for all possible configurations for small values of n to numerically obtain the paths that lead to interferences. The number of configurations available to this chain model is —l in , A further interest of this calculation is to gain insight l . .- p . '- . . I . . -_-. .--_-._:_-'. _,,_ . .:.'.--'-1. ... ... . . . . =' . ._ J .. n . l I . I. I. I. ‘ . .- .- 1 . — j.-— .' 1:3.1. I I l -: .t- . _ ln'a. M2 into the necessity of postulating a cis state to explain the formation of hexamethylcyclotrisiloxane. F. Chain Configuration and Motion Grubb and Osthoff58 have studied the kinetics of the KGH-catalyzed polymerization of octamethylcyclotetrasiloxane as a function of catalyst concen- tration and polymerization temperature. In all cases, the reaction was found to be first order in monomer concentration. At 152.6°C and 0.0h3$ by weight KGE, equilibrium is reached in approximately 90 minutes. The frequency of intramolecular rotations in the polydimethylsiloxane chain has been studied by nuclear magnetic resonance. Huggins and co-workers18 found that the transition temperature, arbitrarily defined as the point at which the line width has changed by one-half the total change in log AH, is 180°K. The temperaturewdependence of N.M.R. line narrowing has been explained in ternm of the onset of intramolecular rotations by Slichterfig . The frequency of chain=twisting and torsional oscillation is concluded from.line narrowing data, to be greater than several tens of kilocycles per second. Since the rate of intramolecular rotations is many orders of magnitude ' greater than the rate of monomer addition to a chain, the chain configuration is taken up independently of the reaction mechanism and rate. G. Convergence of ra/nla as Investigation of the Convergence of the Matrix Power Series The conditions under which the equation for ra/nla converges will be determined. Consider now the following matrix power series*: * Multiplying each term in a matrix series by a constant matrix does not alter the conditions of convergence. Two of the matrix series in ra/nla are multiplied by a constant matrix, but are otherwise identical with la) and lb). Therefore, establishing the conditions of convergence of la) and lb) is sufficient to establish the conditiOns of convergence of rz/nla. 3 la) ; (1 - ak/n)3k; 1b) '2 (l - 2k/n)Dk; where B - A132 k. ,2' k: ,2 13 ' Ask; Both of these series can be expressed as: 2) S c 2 csAe, where A is an m x m matrix and the ca are mmbers. . 6:0 The power series of S can be shown to converge for each element of S if z: |c5| (na)e converges, where [an] _S a; i, J = l, 2......n 60. The subject of interest thus becomes that of the convergence of ordinary number power series 61 . For the case of series la): n-2 T w 2= 2 we as W on k=0,1,2 We examine the ratio of any two arbitrary terms in the series. Consider the terms where k = m, n+1: 1 E 2 m+1 m+1 h) n x 53a; a: test ratio. 1 .. ~23 5“ n Simplification of it) gives: n=2m=2 5) W x 5s = test ratio. 11 — 2121 We wish to consider the lim (:‘fm:) E L n——>(I) '1 2 1m 8312 6) (mfg-2) . 1 - :1. - 3-2- ;s— - ........... n - 2m 5 7)mm[1-§--%§-%§~mm] -1=1. n—> ‘o The series 5) will converge whereverzé‘ 8)--J-'-—<3a<—l-— a -1<3a = > = O) that —2 2 _ l - cosG l — c0592) 1) r /nl — g1 - c0591 c0592; Results from 1) are compared in Table VI with the computor results for the freely rotating model. For the calculation of r2/nl2 vs. n for a period 2 chain the matrix elements in A1, A2, AlAe, and A2A1 are the data from which calculations are made. Figure\fll is a plot of r2/nl2 vs. n for the freely rotating model. From the plot in Figure [Bjone can see that the computor value of r2/n12 at large n approaches that value calculated in 1). mammov H l O\. m O/H / \- H I «Ammov.m 0 mm mm mmnmmov now a .m> we soapmadoamo mQOflpMSwm map mo QOfiPmOfltflam> Q o m s m .H> magmas ML 1&8 on m manna mom was ..nmoflmw .wo sowposa .m we Hoeoa wawpwpon maven.“ minnoah no,“ a .m> I mm mm ma G +1“ mm OH m .HH> gunk was ‘9 Table VI. r2/nl2 as a function of lSiOSi for = 9. Comparison with values calculated from Flory's approximate equation* r2 0 F2 52 %d ‘t' f zsos- — n=2l — n=l+o — Flor '5 ”la 10‘] 0 1 l n12) n12) n12)eq. y Flory's results from computer value Case 1 1.50° 2.650 2.71** 2.79 2.9 Case 2 1h0° 2.96 5.07 5.16 2.9 Case 5 lh2.5° 5.02 5.lh** 5.25 5.5 * [0810 = 109°28' for all cases ** Extrapolated from n = 21. B. Calculation of ra/nlz for statistically independent models of the polydimethylsiloxane chain. Because of the symmetry due to all identical substituents (the same symmetry as exists in the previously discussed polyisobutylene chain), U(¢i) = U(¢i+l). In all models considered here the rotational states are symmetrically disposed about the trans state. Therefore, = O. From the chain symmetry; = . For stat— istically independent models = >. In the series of curves of r2/nl2 vs. n in Figure X, decreases from Case 1 through Case 5. It should be recalled at this point that is a function of two kinds of variables - the rotational states available and their respective probabilities. For the four state model of G, T, G' and C states with respective probabilities a, b, c d ) (and a=c from chain symmetry), 1) = l/2(l-5b+d) 50 m: .HH> magma mom .maoeoa mquoaflm mm an on mm mm H omdo N omwo n ammo a omso m ommo ma . UQH .mwpdpm mom a :H OH .m> was mm .mmomOH " oO:H n w a r—{Nrfl—tufi .HHH> mmDOHm Oflmow flmoflmu m \\ ammo ammo ammo cmmo ammo [fl \O was 51 However, the gauche rotational states might actually be shifted several degrees from their ideal values in hydrocarbons of 60° and 500°. Liquomh calculations indicate minima shifted in polyoxymethylene (G=67°, G'=295°).6O Two other sets of assignments for the gauche states in the siloxane chain will be presented to discuss the implications of shifted gauche states on the parameter of the chain dimensions. (2) G=50°, G'=510° = .6h28 - 1.62% + .5572d (5) G=70°, G'=290° = .5h20 — 1.51201) + .6580d. For a given value of , the magnitude of the probabilities b and d will depend on the choice of rotational angles for G and G‘. The normal set of gauche states G=60° and G'=500° are chosen so that the curves may be considered to be a function of only probability variables, thus making interpretations simpler. But the choice is somewhat arbitrary and further experimental work may indicate that those of (2) or (5) better approximate the real chain. None of the curves in Figure VIII approach the experimental value of rz/nl2 = 7.25. In addition, the energy difference betWeen the trans and gauche potential well OSE) in each of the Cases 5-7 in Table VII far exceeds the allowable few hundred calories per mole commensurate with the data of Scott, et. a1.28 From these two results it is concluded that the di— mensions of the polydimethyl siloxane chain cannot be described using a statistically independent model. One explanation for the difference between r2/nl2 obtained from a model with a reasonable AE and the experimental value of 7.25 is that included in dimensions of the model 52 are many configurations of the entire chain that are physically unallowed due to long range interferences. These interferences are from a segment of the chain occupying a space previously occupied by another chain segment. Chains in which this occurs have dimensions much lower than chains which do not have long range intramolecular interferences. Thus, the effect of including "unallowed" total chain configurationsis to lower the calculated chain dimensions.* The nature of the long range interference effect in the siloxane chain will be examined in detail in III-D for chains with six and eight chain bonds. Table VII- ;2/n12 as a function of zs1081=iu0°, 10310:109°28‘ —2 Probability §i2)n=uo of trans * AE+, 25° Case 1 .2 2.22 .200 750 Case 2 0 5.07 -555 0 Case 5 -.2 5.95 .467 —650 Case u -.u b.85 .600 -1250 Case 5 -.6 5.21 .755 -1950 * Assuming prob. of cis = 0. Then :%(l—5b). + Measured with E increasing upward from reference point of E=O at deepest gauche minima at 60° and 500°. The purpose of the graph in Figure IX is to investigate the necessity for postulating the availability of a cis state to the siloxane chain in order to explain the planarity of the hexamethylcyclotrisiloxane ring. The model for this graph is [SiOSi = 140°, [OSiO = 109°28' (the * In the case of polyethylene, the experimental re/nle = 6.55. Using a statistically independent model with rotational states, G, T, G' and AE = 500 cal/mole (the energy difference between trans and gauche obtained from spectral data of normal paraffins), E2/n12 = 5.5M. Perhaps here as well as in the siloxane case one of the major contributory factors to the low model value is the inclusion of configurations which are unallowed due to long range interaction. 55 _wmoQOH .oOdH II *II ofimow meHmV Hscone mflo ado so.“ a .m> .2 NH N «W .NH mmbUHm 51+ bond angles accepted in this work as valid for the polymer chain) and = 1 (corresponding to the cis state being occupied with probability =1). At n=6, the value oikfr27l2 represents how far the chain ends are apart giving a direct indication of how readily they might be expected to join to form the planar ring. From the graph, fire/12 = .19. Thus the chain ends are slightly less than half bond length apart at n=6. If the cis state is available to the chain then the sequence of cis conformations seems to be a plausible mechanism for the formation of hexamethylcyclotrisiloxane. C. A comparison of statistically dependent and statistically independent models for the polydimethylsiloxane: chain. In TableVIII the values of and - are listed for statistically independent pair and statistically independent modem for the polydimethylsiloxane chain. The purpose of these models is to illustrate that two rather widely differing models can be employed to generate approximately the same value of r2/nl2 at large n. Case 1 is a statistically independent model of the same form as presented in Fig- VIII. In Case 2, case 2< case 1. This has the effect of‘increasing the chain dimensions of Case 2 over Case 1. However, Case 2>‘2case 2. Introducing this statistical dependence into the chain has the effect of reducing the dimensions from what they would be if the chain were statistically independent. At large 11 the two changes in Case 2 nearly balance giving the result that for Case 1, rQ/n12)n=h0 = h.2h, for Case 2, rz/n12)n=b0 = h.28. Note from Figure X that the curves begin rather closely together, spread apart in the region of n = 6—20 and become closer together m: 55 H ammo N omdo . NHQ O “G I: e A mm mm HHH> oHnt 00m meow esp swopnc op mHoeoa pqoosomoo .popw can .oafi .popm mo mob :n on mm mm mH dH OH .x mmbOHaH _mmomoa .m.mea w m H OHmO V HMOHm V m 56 as n approaches ho. The greater spread in the curves from n = 6-20 is a possible indication that the statistically dependent model of Case 2 more accurately re- fleets the cyclic formation in the polydimethylsiloxane chain which takes place predominately in the region of n = 6-20. Table VIII. Application of statistically independent agd'statistically dependent models for calculating the same rZ/nla. ' 4.s1031 = 1h2.5,.20s10 = 109°28' -2 '2 (008» represents 1! 37: 2 / n12 = 6.5 i . . ' T't_'.-. \/ 65 In no case is there good agreement between theory and experiment. However, K calculated in this work represents an improvement over the earlier Stockmayer— Jacobson result. The only difference between the two derivations is in obtaining P and Vs' Stockmayer and Jacobson assumed a Gaussian distribution with an effective link length parameter, b, and we obtain P from a conformation model with equally probable rotational states. Also, we find it necessary to calculate va whereas Stockmayer and Jacobson found K independent of vs. we must remember that Hartung's data, upon which the calculation of Kexperimental was based, is data extrapolated for the undiluted polymer from solvent polymerizations. Certainly, Hartung meant this extrapolation as a guide to be used for predicting cyclic contents at solvent concentrations beyond the limits of those he used. We are, no doubt, subjecting the extrapolated concentration of R4 to undue scrutiny in Table XI. ' Certainly the model developed here is a primitive beginning to the ring closure problem. Evaluating preferred conformations and relating these to ring formation in siloxanes with other organic side groups could help provide fundamental groundings in this hitherto undeveloped area. No mention has been made here of specific solvent effects which are, at present, impossible to evaluate theoretically. No published data is available on the effect of media viscosity on cyclic concentra= tion. It is hoped that this reuexamination of the ring closure problem will lead to further critical experimental studies. U120: 10. ll. 12. 15. lh. 15. 16. 17. 18. 19. 20. References P. J. Flory, "Solid-State Macromolecules, Lectures in Materials Science." W. A. Benjamin, Inc., New York, (1965). P. J. Flory, "Principles of Polymer Chemistry" Opt. 10, Cornell University Press, Ithaca (1955). c. A. J. Hoeve, J. Chem. Phys., 55, 1266 (1961). s. Lifson, J. Chem. Phys., g9, 80 (1958). s. Lifson, J. Chem. Phys., 59, 96h (1959). H; A. Hartung and S. M. Camiolo, Papers Presented at the Washington Meeting, Am. Chem. Soc., Div. of Polymer Science, pp. 268-72 (1961). P. J. Flory, L. Mandelkern, J. B. Kinsinger and W. B. Schultz, J. Am. Chem. Soc., 13, 556A (1952). N. Kuwahara, Y. Miyake, M. Kaneko and J. Furuichi, Kobunshi Kagaku, 12, 25 (1962). G. V. Schultz and A. Haug, z. Phys. Chem., 55, 528 (1962). A. 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Mayer, "Statistical Mechanics," John Wiley and Sons, Inc., New York (19%). APPENDICES APPENDIX.I CALCULATION OF E FOR -M-O-N-O— CHAIN Presented here is a derivation of an expression for r2 for a polymer chain with two different bond lengths of the form 'M‘O’N‘O’M' R R | For purposes of illustration, the system - O - Si - O - Sn — will t. t O\ \/ \::/ Let ll)l be the first bond of length 11 be used: 11 ll)n_2 be the (n—l)th bond of length 11 ll)n/2 be the nth bond of length 11 12)l be the first bond of length 12 12)n_2/2 be the (n—l)th bond of length 12 l2)n—2 be the nth bond of length 12 A1 = Matrix associated with bond angle 91 and rotational states ®l)i. A2 = Matrix associated with bond angle 92 and rotational states ¢2)i. A3 = Matrix associated with bond angle 63 and rotational states ¢3)i. Each bond vector is written now in terms of the preceeding bond vector using the associated transformation matrix. Matrix relations are then written for the first eight bonds in the chain in terms of all prior bonds. ll)l = 11 lst bond ll)2 = Alll)l 2nd bond l2)1 = A211): + 12 " ll 3 A2Alll)l + (12 ' ll)* 5rd bond 12)2 = Aala)1 = A3A2A111)1 + A3(12 — 11) 4th bond ll)3 = A2'12)2 ' (12 ' ll) = AeAeAaAlll>l + AeAe(12 - ll) — (12 — 11) 5th bond ll)4 = Alll)3 = AlAeAeAeAlll). + A1A2A3(12 - ll) - Al(12 - 1.) 6th bond ‘ 12). = A2 11). + (12 - l.) = A2A1A2A3A2Alll)l + A2A1A2As(12 ' 11) - A2A1(12 ~11) + (12 - 11) 7th bond 12)4 = A312)3 = A3A2A1A2A3A2Alll)l + A3A2A1A2A3(l2 - ll) ‘ AeAeAl(12 - l.) + Ae(12 - 1.) 8th bond For n = h, 8, 16, 20’ n-h 12)n/2 = (AeAeAlAe) 4 AeAgAlll n—h + (ASAZAlAZ) 4 A3(12 * ll) n-8 — (ABAZAlAZ) 4 A3A2Al(12 ' ll) * The difference in bond lengths, 12 - ll, must be included for the matrix equality to hold. This is the salient mathematical feature of this chain model. ii n—4 law/2 = (Agnew—4‘ AeAll. n-h + (A2A1A2A3)T <12 - 1.) n_-8 - (AaAlAaka) 4 A2Al(l2 - ll) n— ll)n/2 = (AlAeAsAe) 48 Alll n— + (A1A2A3A2)"Z_ A1A2A3(12 — ll) £32 ' (AlAeAaAe) 4 A1(12 ' ll) Consider the summations for the four types of bonds. k—h n 1’1 _ Z 1.)n /2 = fl (AeAeAlAe) 4 AeAeAlul) k : u,8,12... k=!,8,l2... n 15.1% + Z (n/lL — k/lL + l)(AeA2A1A2> 4 A3<12‘11) k=u,8,l2... n k_—8_ + Z (n/h - k/2+ + l)(AeA2AlA2) 4 AeAeAlul - l2) k=8,l2,16... n “ 1:}. Z 12)” /2 = Z (AgAlAeAe) 4 AeAlul) k=A,8,l2... k=u,8,l2... n E + X (n/u-k/u + Comm...) 4 <1. - 1.) k=u,8,l2... iii In- n ‘ n-8 + 25: (n/4 - k/4 + l>(AeA.A2Ae)’Z~ A2A1(ll - 12) k=8,l2,16... n n 'k- 4 IE: 11)n/2 = IE: (AlAeAeA2)T All. k=u,8,l2 k=u,8,l2.. n .k_-8_ + SE: (n/h - k/h + l)(AlA2AsA2) 4 A1A2A3(12 - 1.) k=8,l2 ..... n k_-8 + SE: n/h — k/h + l)(AlA2A3A2) 4 A.(ll - 12) k=8,l2 ..... n n k-u Z 1) =Z (AeAA2A>Tl k: u, 8 l2 “ 2/2 k=u,8,l2... 3 l l n as + 25% (./h — k/h + l)(AeAeA2Al> 4 A2Ae(12 - 11) k: ,l2... k8 (n/LL - k/h + l>CAeAeAeAl)T (l. - 12) rvq s k—8,l2... A The expression for r is the sum of the contributions from each of the four types of bonds. n I]. e - kZE: 12>n/ + 23:. 12) + ii: 1.) r ‘ (l O 0) M8 l2. k=u,8,l2... "“2/2 k=u,8,l2... “/2 Zn 11) k=h,8,l2... “’2/é] Substitution yields: k- A n k-u ? = (l O O) :E:h (A3A2A1A2)— 4 A3A2Al + 2:; (A2A1A2A3) 4 A24% 1&8 l2. k=u,8,l2... n k-MA n k u + SE: (AlAeAeAe) 4 + ZE:W (AeAsAaAl) 4 :] k=u,8,l2... k=u12. iv n E + (12 - 11 o 0) Z (n/4 - 14/4 + 1>(AeA2A.A2) 4 A. k=u,8,l2... n k-A + Z (n/4 - k/4 + 1)(A2A1A2Ae)—4‘ k=u,8,l2... k-8 8 (n/4 - 14/4 + l)(AlA2AeA2)T AlAeAe ,12... 4 AeAl k=8,l2.ne \ .9— §:§ + Z (n/4 - k/4 + l)(A.A2%A2) 4 A1 k=8,l2... 3i .k_-§ + ;3_ (n/A - k/A + l)(A2A3A2Al) 4 ; n 1.2-1. + (l2 0 O) [1%)B)12.FI.1/)+ _ k/u + l)(A3A2AlA2) 4 A3 n k—A + Z (n/lt ‘ k/4 + l)(A2AlA2A3)—4— k=A,8,l2... n k—8 + Z (n/4 — 14/4 + l)(AlA2A3A2)T AlAeAe k=8,12... n k—8 + Z (14/4 — k/4 + l)(A2AeA2Al) 4 AeA. k=8,12... [D]; k—8 I’l ~ Z (n/A — 14/4 + l)(AeAeAlA2)—4“ AeAeAl k=8,l2... ' n k-8 - Z (fl/Lt ' k/Lt + l)(A2AlA2A3 )7 AlA2] k28,12... vi k—8 — Z (n/4 - 14/4 + l)(A.A2AeA2)T A. k=8,12... k_-§ - Z (n/4 - 14/4 + 1)(A2AeA2A.) 4 1 k=8,12... With the notation introduced above, ??= (11 0 o) [c] + (12 o o) [D] 4> — — — t r2 = r . > > > ~ t T?2 fill o 0)[C] + (12 o 0)[nflx[(ll 0 0)[C] + (12 o 0)[n] — t t i2 = (11 0 0)[c] + (12 0 0)[Dlx[01 1. + [D] 12 . O O ‘ O O —> t t r2 = 1% [C][C ] l,l + lg [D][D] 1,1 t t + 1112 ( [D][C] + [C][D] ) 1,1 1,1 As an illustration of these equations consider a chain with eight links. —> T = (ll 0 0)[1 + A1 + A2Al + AGAZAl + A2A3A2Al + A1A2A3A2Al + A2A1A2A8A2Al + AsAaAlAeAsAzAl] + (l2 — ll 0 0)[ l + A3 + A2A3 + Al A2A8 + A2A1A2A3 + l + A3A2A1A2A3 + A3] + (ll—12 O O)[ l + Al A2A1 + ASAZAl] ?> = (11 o 0) [l + Al + A2Al + AgAeAl + A2A8A2Al + AlAgAaAgAl + A8A2A1A2A3A2Al - l - A3 ' A2Aa - AlAaAs - AaAlAeAa - l - A3A2A1A2As - A3 + l + Al + A2A1 = ASAZAl] +(l2 O 0) [l + A3 + AaAs + AlAZAG + A2AlA2A8 + l + A3A2A1A2A3 + As - l - Al ' A2A1 ' ASAZAl] Simplification gives: vii ? 4 (ll 0 0) [2A1 — 2A3 + 2A2Ae + 2A3A2A. - AlAeAe + A2A3A2Al - AaAlAaPe + A1A2A3A2A1 - AsAaAlAaka + A2A1A2A3A2A1 + AsAzAlAaAaAaAl] +(12 0 0) [l + 2&5 - A1 + AzAa - A2A1 + A1A2Ax'5 - AsAaAl + A2A1A2A3 + AsAzAlAaAs ] viii APPENDIX II 160A FORTRAN Programs for Calculation of r2 for -M-O—M— Chain Case one - n even 0001 format (12, (E8.A)) :0002 format (12HN; even; case; i2) 0005 format (E15.7) OOOHOdimension ala2(3,5), 00041a1(3,5).a2(3.5).a2al(3.5),d(3.5).w(l°).x(l°).y(l°).z(l°) 0005 subroutine matpow (B,c) 0006 nonlocal d 0007 do 10 i=1,5 0010 d(I;J)=B(I;l)'C(l;3)+B(I;2)'C(2;J)+B(I;5)'C(3;J) return end 0011 subroutine matmul (B,c,e) 0012 e=B(l,l)’C(l,l)+B(l,2)‘C(2,1)+B(l,5)'C(3,1) 0015 return end 0014 read 1,ncase, ala2, a1, a2, a2al 0015 pause 0005 0016 do 50 3:1,5 0017 do 50 i=1,5 0050 d(I,j)=AlA2(l,j) do 55k=2,10 0051 call matmul (D,al,e) x(K)=E 0032 CALL MATMUL (A2d,e) y(K)=E 0033 CALL MATPOW (A1A2,d) 0055 w(K)=D(l.l) @056 J=l.5 d056i=1,5 ix 0056 d(I,j)=A2Al(I,j) do 570 k=2,10 0057 call matpow (A2Al,d) 0570 z(K)=D(l.l) 0058 punch 2, ncase pameomfi x(l)=Al=0.0 504 TEMP3(IoJo1)=OoO ___ 35 CALL MULT