A ROTATIONAL ISOMERIC STATE APPROACH TOWARDS UNDERSTANDING ELASTOMER CHAIN CONFORMATIONS IN TIRES

Rubber tires undergo viscoelastic losses at high and low frequencies. High frequency losses lead to traction while low frequency losses lead to rolling resistance. High rolling resistance tires require greater amount of fuel to travel a particular distance as compared to low rolling resistance tires, and thus they have a negative impact on vehicle fuel economy. Traction is needed for vehicle braking ability and propulsion. Maintaining a balance between reducing rolling resistance and maintaining wear resistance and traction is a technical challenge. Factors that decrease rolling resistance tend to worsen traction, and vice versa, while both types of changes reduce wear resistance. Experiments have found that strengthening interactions between rubber and reinforcement fillers can be used to maintain a balance between reducing tire rolling resistance without compromising on wear resistance and traction, but why this works is not known. Rolling resistance on the macroscale connects directly to energy losses occuring due to changes in elastomer chain conformations on the microscale. Thus, understanding the statistical mechanics of elastomer chain conformations provide us a vital molecular link towards quantifying rolling resistance. This thesis provides a first step towards this link. Molecular modeling is used to study the size and shape distribution, and characteristics of cisand trans-1,4-polybutadiene chains. Computations are conducted using Flory’s Rotational Isomeric State approach (RIS), in which energy distribution is considered over discrete rotational isomeric states. The Rotational Isomeric State approach is chosen because it allows generating a large number of polybutadiene chains in a computationally cheap manner using less resources and computation time , and also because the RIS approach allows each chain realization to be treated as an independent sample. Numerous (100,000) isolated single cisand trans-1,4-polybutadiene chains of uncorrelated random conformations are considered under unperturbed conditions (balanced attractive and repulsive polymer-solvent interactions, i.e. thetaconditions). Using a single chain in each computation is justified because a flexible polymer surrounded by the same polymer takes on the same average shape as a single random polymer chain in a theta solvent. Chain size and shape properties are computed at different chain lengths and over a range of temperatures. Characteristic ratios are in good agreement with experimental and prior computed values (cis-1,4-polybutadiene), and slightly higher than prior computed values (trans-1,4-polybutadiene). Characteristic ratios increased with increasing chain length for both cis and trans chains with this effect being more prominent for trans than for cis chains. Small absolute changes in chain size probability densities with temperature are observed. Larger relative increase in probability density of larger chains and smaller relative decrease in probability density of smaller chains result in increased average chain size with increasing temperature. This effect increases characteristic ratios with increasing temperature. The larger chains show a much higher increase in characteristic ratios with temperature than smaller chains, and this effect is stronger for trans than for cis chains. Eigenvalues of the radius of gyration matrix quantify chain shapes by providing eigenvalues along the three principal directions (eigenvectors). Average shape measures differ between cis and trans chains. With increasing chain length, trans chains are slightly compressed along the principal direction while cis chains are slightly stretched. Resultantly, trans chains are slightly more spherical with increasing chain length while cis chains are slightly less spherical. At the same chain length, trans chains are slightly less spherical than cis chains. At long chain lengths, trans and cis chains have similar spherical shapes. With increasing temperature, little or no variation in shape is computed for cis chains, whereas trans chains are slightly stretched along the principal direction, and thus are slightly less spherical. Most changes in shapes arise from changes along the longest principal direction. Cis and trans chains show similar asphericity (a parameter that quantifies deviation from spherical shape) at longer chain lengths. Little or no change in acylindricity (a parameter that quantifies deviation from cylindrical shape) is computed for either cis or trans polybutadiene chains. Relative shape anisotropy (a shape parameter) follows the same trends like asphericity as functions of both chain length and temperature for cis and trans polybutadiene chains. Joint correlation studies reveal that size and shape parameters are mutually dependent properties of chains. For asphericity, rod-like small size and spherical medium size cis chains show anti-correlation between chain size and shape. Spherical small size, near rod-like medium and large size chains show correlation between chain size and shape. For acylindricity, medium size chains of flattened cross section, and small and large size chains of round cross section showed correlation between chain size and shape. Round cross section medium size chains show anti-correlation between chain size and shape. Trans chains show similar behavior as cis chains with correlation and anti-correlation between chain size and shape occuring to a greater extent. The next use for the detailed conformation results in this work is to relate probability densities to the work done to alter chain size and shape. Cis and trans chains show different probability density distributions implying different amounts of deformation work to alter chain size and shape. When a tire revolves and deflects while in motion, affine deformation of the elastomer-filler system takes place. The deformation leads to changes in elastomer chain conformations, which results in entropy losses of the elastomer-filler system (since entropy is related logarithmically to chain conformations). These entropy losses lead to computing irreversible work, viscoelastic losses and rolling resistance. The effects of fillers on these conformation distributions thus will quantify interaction effects on loss modulus and rolling resistance.

tive impact on vehicle fuel economy. Traction is needed for vehicle braking ability and propulsion. Maintaining a balance between reducing rolling resistance and maintaining wear resistance and traction is a technical challenge. Factors that decrease rolling resistance tend to worsen traction, and vice versa, while both types of changes reduce wear resistance. Experiments have found that strengthening interactions between rubber and reinforcement fillers can be used to maintain a balance between reducing tire rolling resistance without compromising on wear resistance and traction, but why this works is not known. Rolling resistance on the macroscale connects directly to energy losses occuring due to changes in elastomer chain conformations on the microscale. Thus, understanding the statistical mechanics of elastomer chain conformations provide us a vital molecular link towards quantifying rolling resistance. This thesis provides a first step towards this link.
Molecular modeling is used to study the size and shape distribution, and characteristics of cis-and trans-1,4-polybutadiene chains. Computations are conducted using Flory's Rotational Isomeric State approach (RIS), in which energy distribution is considered over discrete rotational isomeric states. The Rotational Isomeric State approach is chosen because it allows generating a large number of polybutadiene chains in a computationally cheap manner using less resources and computation time , and also because the RIS approach allows each chain realization to be treated as an independent sample.
Numerous (100,000) isolated single cis-and trans-1,4-polybutadiene chains of uncorrelated random conformations are considered under unperturbed conditions (balanced attractive and repulsive polymer-solvent interactions, i.e. thetaconditions). Using a single chain in each computation is justified because a flexible polymer surrounded by the same polymer takes on the same average shape as a single random polymer chain in a theta solvent. Chain size and shape properties are computed at different chain lengths and over a range of temperatures.
Characteristic ratios are in good agreement with experimental and prior computed values (cis-1,4-polybutadiene), and slightly higher than prior computed values (trans-1,4-polybutadiene). Characteristic ratios increased with increasing chain length for both cis and trans chains with this effect being more prominent for trans than for cis chains. Small absolute changes in chain size probability densities with temperature are observed. Larger relative increase in probability density of larger chains and smaller relative decrease in probability density of smaller chains result in increased average chain size with increasing temperature. This effect increases characteristic ratios with increasing temperature. The larger chains show a much higher increase in characteristic ratios with temperature than smaller chains, and this effect is stronger for trans than for cis chains.
Eigenvalues of the radius of gyration matrix quantify chain shapes by providing eigenvalues along the three principal directions (eigenvectors). Average shape measures differ between cis and trans chains. With increasing chain length, trans chains are slightly compressed along the principal direction while cis chains are slightly stretched. Resultantly, trans chains are slightly more spherical with increasing chain length while cis chains are slightly less spherical. At the same chain length, trans chains are slightly less spherical than cis chains. At long chain lengths, trans and cis chains have similar spherical shapes. With increasing temperature, little or no variation in shape is computed for cis chains, whereas trans chains are slightly stretched along the principal direction, and thus are slightly less spherical. Most changes in shapes arise from changes along the longest principal direction.
Cis and trans chains show similar asphericity (a parameter that quantifies deviation from spherical shape) at longer chain lengths. Little or no change in acylindricity (a parameter that quantifies deviation from cylindrical shape) is computed for either cis or trans polybutadiene chains. Relative shape anisotropy (a shape parameter) follows the same trends like asphericity as functions of both chain length and temperature for cis and trans polybutadiene chains.
Joint correlation studies reveal that size and shape parameters are mutually dependent properties of chains. For asphericity, rod-like small size and spherical medium size cis chains show anti-correlation between chain size and shape. Spherical small size, near rod-like medium and large size chains show correlation between chain size and shape.
For acylindricity, medium size chains of flattened cross section, and small and large size chains of round cross section showed correlation between chain size and shape. Round cross section medium size chains show anti-correlation between chain size and shape. Trans chains show similar behavior as cis chains with correlation and anti-correlation between chain size and shape occuring to a greater extent.
The next use for the detailed conformation results in this work is to relate probability densities to the work done to alter chain size and shape. Cis and trans chains show different probability density distributions implying different amounts of deformation work to alter chain size and shape. When a tire revolves and deflects while in motion, affine deformation of the elastomer-filler system takes place. The deformation leads to changes in elastomer chain conformations, which results in entropy losses of the elastomer-filler system (since entropy is related logarithmically to chain conformations). These entropy losses lead to computing irreversible work, viscoelastic losses and rolling resistance. The effects of fillers on these conformation distributions thus will quantify interaction effects on loss modulus and rolling resistance.

DEDICATION
I dedicate this dissertation to my parents.
Ma and Baba, thank you for instilling in me virtues, teaching me that there is no alternative to hard work and how important it is to be patient in life.
Everything I will ever achieve in life will always be dedicated to you.
viii PREFACE The following work is presented in manuscript format in accordance with the guidelines set by the University of Rhode Island Graduate School. The thesis consists of one manuscript which is prepared for submission to the journal Polymer.  Characteristic ratio vs. inverse of chain length n for cis-(filled) and trans-(unfilled) 1,4-polybutadiene using r 2 0 (circle) and 6 r 2 g 0 (square). Literature results at 1/n = 0 indicate models [10,11] (+) and experimental values [7,8]   i.e. good wear resistance and iv) has low rolling resistance leading to greater fuel economy [3,4].
Traction depends on the friction properties of the tire tread and is determined by the road conditions and temperatures i.e. wet traction, ice traction and winter traction. Steering control depends on the stiffness properties of the tread. Wear and tear depends on the abrasion resistance of the tread compound. Rolling resistance depends on the viscoelastic losses or loss modulus of the tread compound [3,4]. Thus the tire tread, which is the rubber covering the circumference of the tire, plays an important role in determining tire properties.

Tires and Viscoelastic properties
Rubber tires undergo viscoelastic losses both at high as well as low frequencies.
During rolling, the strains on a tire tread exert stresses through both elastic and Rolling resistance, traction and wear resistance comprise the "magic triangle of tires" [5]. It is a technical challenge to reduce one aspect of the magic triangle without compromising on the others. In the 1980s, thick and hard tires were designed in order to reduce tire rolling resistance [3]. While they achieved the purpose of reducing rolling resistance, the traction was greatly compromised.
Balance between reducing rolling resistance and maintaining traction and wear resistance can be achieved by using reinforcement fillers with rubber tires. Reinforcement fillers have been found to lower rolling resistance of tires, increase tensile strength (higher storage modulus), improve wear resistance and durability [4]. Carbon black and silica are the two most prominent and widely used reinforcement fillers with rubber tires. Our collaborators at Ford Motor Company, Dearborn, MI USA, are looking experimentally at several novel filler systems such as Silanol, BR-Acrylate Terpolymer, Hybrid-CB Silica, Treated Aramid Fiber Granule and Broad Aggregate CB as reinforcement fillers with the rubber elastomer system [6].
Parameters such as the geometry and type of filler, rubber-filler adhesion, and so on need to be considered in rubber-filler systems. Non-linear interactions between the rubber (elastomer) and filler make it difficult for experimentation alone to optimize the system and thus models need to be develped to understand elastomer-filler interactions. Our work involves developing computational models to understand how molecular-level changes in elastomer-filler interactions affect rolling resistance and viscous losses.

Hypothesis for This Work and Relevant Prior Literature
The hypothesis of our work is quantifying deformation force and estimating tire rolling resistance as a result of changes in elastomer chain conformations.
During rolling, the tire tread flattens against the road and as a result, the elastomer chains undergo a change in their conformations and affine deformation of the elastomer-filler system takes place. This deformation changes the number of ways the elastomer chains and filler particles can be arranged and thus affects the entropy of the system, which is logarithmically related to the number of confor-mations [7]. The original distribution of the chains is restored after an entire cycle of tire rotation by random fluctuations after the deformation and this change in entropy requires work which is dissipated as heat, leading to rolling resistance.
One approach to decrease the work dissipated as heat will be to reduce the extent to which the polybutadiene chains change their shapes under deformation. This will be accounted for within the simulations by directly bonding the polybutadiene chains to the filler particles. Favorable elastomer-filler interactions will lead to lower rolling resistance and conversely, poor elastomer-filler interactions will lead to higher rolling resistance. Thus studying the chain conformations and the changes in the conformations under deformation and presence of filler particles is of utmost importance in our work.
Mohsin, Berry and Treloar [8] determined viscoelastic properties (storage and loss moduli) of polybutadiene samples using an experimental approach known as the torsional pendulum method. The samples studied were "high cis" containing 98% cis by weight and "cis-trans" containing 52% cis, 48% trans by weight. In Moraglio [9] and Abe and Fujita [10] used experimental viscosity measurements to compute characteristic ratio [11] (an important chain conformation property which is being discussed in detail in chapter 2). Using Mark-Houwink's equa-tion [12] for theta-solvent, Moraglio and Abe and Fujita predicted the K factor (Mark-Houwink parameter) for cis-1,4-polybutadiene in n-heptane and diethyl ketone respectively. Obtaining the K factor allowed them to compute the characteristic ratio of cis-1,4-polybutadiene under theta-conditions.
Mark [13,14], Abe and Flory [15] studied random conformations of cis-and trans-1,4-polybutadiene using Flory's Rotational Isomeric State approach (RIS) [11]. Details of the RIS method and a brief summary of the conformational properties obtained by Mark, and Abe and Flory are being discussed in chapter 2.
Mattice and Li [16] used molecular dynamics (MD) simulation method to simulate single chain and bulk amorphous cis-1,4-polybutadiene systems. The low energy states computed were in accordance with the ones suggested by Mark [13] and by Abe and Flory [15]. Different population probability distributions about bond angle supplements and torsion angles were observed for single chains and bulk structures. They also computed cohesive energy of the bulk system. Cohesive energy can be defined as the energy needed to remove a molecule from the bulk system and it was found to be around 4100 cal/mol.

Overview of this Project
I have used cis-and trans-1,4-polybutadiene as the elastomer systems in my research. The polybutadiene can exist as the single elastomer in the rubber tires or could exist as a constituent of the styrene-butadiene rubber (SBR) co-polymer [3]. Numerous (100,000) isolated single chains of uncorrelated random conformations of polybutadiene are generated at different chain lengths and over a range of temperatures. Using a single chain is justified since a flexible polymer surrounded by the same polymer takes on the same average shape as a single random polymer chain. These chains are generated under unperturbed conditions (attractive and repulsive forces balanced between polymer-solvent i.e. theta conditions) using the RIS method. The RIS parameters suggested by Mark [13,14] and later on used by Abe and Flory [15] are used in our work. Probability density distribution of different chain conformations of cis-and trans-1,4 polybutadiene are obtained. These probability densities are related to the deformation work done in order to alter chain size and shape, leading to mechanical and viscoelastic properties of chains and ultimately to rolling resistance.
My Master's thesis looks at random chain conformations (chain size and shape), probability density distribution of cis-and trans-1,4-polybutadiene chains under unperturbed conditions, and also at joint correlations between chain size and shape. Amongst several findings, a key finding of this work is explanation of chain swelling on heating occuring due to increase in average chain size, which is attributed to the "taut conformation effect" (discussed in detail in chapter 2).

Graphical Abstract
Conformation Distribution Taut conformations expand more with T Sizes and Shapes The polybutadiene chains were generated using Flory's Rotational Isomeric State approach (RIS) [9]. Each chain realization in RIS provides an independent sample.
Thus while the standard Molecular Dynamics and Monte Carlo methods provides sequences of related states, the small changes that occur in each step lead to correlations that must be relaxed to sample an equilibrium distribution. The RIS method offers an advantage of generating a large number of uncorrelated random chain conformations in a computationally cheap manner.
Mark [10,11] and Abe and Flory [12] previously used the RIS method to gen- and by Abe and Flory [12]. Population probability distributions about the bond angle supplements and torsion angles were found to be different for single chain and bulk structures. This difference suggested intermolecular origin conformational differences between single chain and bulk structures. The bulk structure of polybutadiene allowed Mattice and Li to compute cohesive energy of the system.
Cohesive energy is defined as the energy needed to remove a molecule from the bulk system and it was around 4100 cal/mol.
Our focus is on studying size and shape properties of random chain conformations of polybutadiene. We computed characteristic ratios of cis-and trans-1,4 polybutadiene chains at different chain lengths and over a range of temperatures.
Comparing the probability density distribution of the chains at different temperatures has explained the reason behind average swelling of chains with increasing temperature. We also studied chain shapes at different chain lengths and over a temperature range. Finally, we looked at joint probability correlations between chain size and shape and the extents of correlation and anti-correlation for cisand trans-1,4-polybutadiene chains.

Methodology
In the RIS approximation, torsions about bonds are treated as existing in one or more discrete rotational states, with each of these states chosen to coincide with a region of low potential energy. States differ in relative energy and thus in Boltzmann-weighted probability. Discrete states are defined only around bonds that allow torsion. Rotations about the double bond are not allowed.

Chain Generation
Each polybutadiene chain was built in an atom-by-atom manner considering H i+2 ) attached to the C atoms (C i+1 and C i+2 ). It also directly affects positions of the C atoms (C i+2 and C i+3 ). Bond angle supplements and bond lengths used in our computations were obtained from Mark [10,11] and are shown in Table 1.
Abe and Flory [12] used the same values in their calculations.
Transformation matrices are orthogonal matrices which are used to transform bond vectors from one reference system to another [9]. These transformation matrices were used in determining atom positions for each single chain of polybutadiene.
A total of four transformation matrices were used per repeat unit of polybutadiene i.e. three for the C-C single bonds and one for the C=C double bond. For the C=C double bond, the torsional angle (φ) is zero. For the C-C single bond, the torsional angles were chosen based on regions of low potential energy.
Statistical weight matrices [9] were suggested by Mark [10] for 1,4-polybutadiene systems. The same set of matrices and statistical weights were used in our work.
The partition function [9] shows each possible combination of rotational isomeric states of a chain. The pair wise probability of a single conformation equals its contribution to the partition function, divided by the partition function. Please refer to the appendix for a discussion on statistical weights, statistical weight matrices, partition function, and transformation matrices.
Total energy of the system is a summation of the torsional energy [9] and the energy resulting from the dispersion interactions between non-bonded atoms calculated using the Lennard-Jones (6-12) potential [15]. Non-bonded atoms separated by three or more bonds contribute to the non-bonded interaction energy computed using the Lennard-Jones (6-12) potential. Every conformation of polybutadiene generated in our work have fixed bond lengths and bond angles, and thus the bond energies do not affect the overall energy of the system.

Chain size and shape parameters
An important chain size parameter is the squared end-to-end distance r 2 , which is calculated as where r x , r y , r z are the x, y and z coordinates of the end-to-end distance vector r.
The squared radius of gyration (r 2 g ) is computed using the distance of each atom in the polymer chain to the center of mass, where x j , y j , z j are the x, y, and z coordinates of atom j of a polymer chain, and x com , y com , z com are the x, y, and z coordinates of the center of mass of the polymer chain. The overbar indicates average over all chain atoms. We transformed the radius of gyration matrix to a principal axis system, which diagonalised the radius of gyration matrix in such a manner that the eigenvalues of the matrix were in descending order (λ 1 ≥ λ 2 ≥ λ 3 ). Eigenvalue λ 1 corresponds to the longest principal direction while λ 2 and λ 3 correspond to secondary directions. This effectively represents the size of a polymer chain in each direction, rather than with the radius r g of a hollow sphere having the same mass and moment of inertia as the polymer chain. The squared radius of gyration equals the sum of the three eigenvalues, Computing the radius of gyration matrix (equation 3) enabled quantifying chain shape. The chain shape parameters studied were b (asphericity or deviation from spherical shape), c (acylindricity or deviation from cylindrical shape) and κ 2 (relative shape anisotropy) [16]: Averages of r 2 , r 2 g , b/r 2 g , c/r 2 g , and κ 2 used an equal weighting for each chain at each condition. This is appropriate because relative Boltzmann-weighted probabilities are taken into account while generating the chain conformations.

Chain size
Characteristic ratio (C n ) of unperturbed chains [9,17] is defined as the ratio of mean squared end-to-end distance of a real chain under the theta condition to that of a freely jointed chain with the same number of bonds and bond length, n is the number of backbone bonds along a polymer chain and l is the bond length.
C n quantifies chain expansion due to bond angle and torsion angle correlations.
The subscript 0 of the mean squared end-to-end distance represents unperturbed conditions.
We computed characteristic ratios of cis-and of trans-1,4-polybutadiene chains of different chain lengths at one temperature (T = 343 K) and of a single chain length (n = 50) at multiple temperatures. Figures 3 and 4 [7] and Abe and Fujita [8]; computed values are from Mark [10,11]. The characteristic ratio increased with temperature for both cis and trans chains, as shown in figure 4, and the increase was larger for trans than for cis polybutadiene chains. This indicates swelling of the average chain size upon heating.
In the limit of long chains, the mean squared radius of gyration r 2 g 0 should equal 1/6 of the mean squared end-to-end distance r 2 0 [9]. Figure 3 shows the ratio r 2 0 / r 2 g 0 was higher than 6 for shorter trans chains and decreased to 6 for longer chains. The ratio was slightly higher than 6 for cis chains at all chain lengths. Figure 4 shows that the ratio r 2 0 / r 2 g 0 was almost independent of temperature for cis chains, whereas for trans chains it increased with increase in temperature.  The probability density distribution of the squared end-to-end distance was calculated and compared with a Gaussian probability density distribution [9,17], A Gaussian model assumes each chain behaves like a freely jointed chain. The segments of each chain in such an ensemble can be considered as performing a random walk in three dimensions with the only constraint being that each segment must be joined to its neighbors with a fixed bond length [9,17].  The temperature dependences of the probability density distributions of chain sizes for cis-and trans-1,4-polybutadiene are shown in figure 6. Squared end-toend distance has a much wider distribution than the squared radius of gyration.
Smaller size trans chains were slightly more probable at lower temperatures than at higher ones. Cis chains showed probabilities more independent of temperature. To examine this effect further, the characteristic ratio was calculated using only a subset of the chain size distribution, shown in figure 7. Chains with squared end-to-end distance ranging from 10 to 300Å 2 were considered as smaller chains and chains with squared end-to-end distance greater than 4000Å 2 were considered as larger chains. The characteristic ratios increased more with temperature for larger chain sizes of both cis and trans chains, whereas very little increase in characteristic ratio was observed for smaller chain sizes of cis and trans chains. The increase in charactersitic ratio of larger chain sizes was much more prominent in trans than in cis chains. Increases in characteristic ratio with temperature ( figure   4) can thus be attributed to the size increases of extended and taut chain conformations. Polymer chain swelling with heating can be attributed to a size increase of the relatively few extended and taut conformations, rather than expansion uniformly across conformations of all sizes. The greater increase of characteristic ratio with temperature for larger chains, as shown in figures 4 and 7, indicates that this "taut conformation effect" was more prominent for trans than for cis polybutadiene chains.

Chain shape
Ensemble averages of chain shape parameters were obtained in order to quantify shape variations among polybutadiene chains. Since each chain establishes its own principal axes, the analysis uses a different coordinate system for each chain.
The results thus emphasize the deviations of each chain from a symmetric shape.
Rotation differences between the principal axes and the original (x,y,z) coordinates are not important and were not taken into account when combining the results into averages and distributions.
The eigenvalues λ 1 , λ 2 , and λ 3 of the radius of gyration matrix indicate the extents of orthogonal principal axes that span the region occupied by a chain in primary and secondary directions. Ratios of eigenvalues thus indicate if chains are being stretched or compressed. Figures 8 and 9 show the eigenvalue ratios as functions of inverse of chain length and temperature, respectively. These calculations were carried out at 343 K and for 50 repeat units respectively.   suggests a spherical shape and 1 suggests a rod-like shape, while an acylindricity factor (c/r 2 g ) of 0 suggests a round cross section and 0.5 suggests a more flat cross section normal to the longest axis. κ 2 of 0 suggests a rod-like shape whereas 1 suggests structures of tetrahedral or of higher symmetry [16]. Figure 10 shows that both cis and trans chains show similar asphericity of 0.6 at longer chain lengths. An asphericity of 0.6 corresponds to a chain with contribution to the squared radius of gyration that is around 5.5 times larger in the longest direction; it is also consistent with the 12:2.5:1 ratios shown in figures 8 and 9. Cis chains were more spherical at shorter chain lengths and gradually were slightly less spherical with increasing chain length, whereas trans chains were less spherical at smaller chain lengths and were very slightly more spherical with increasing chain length. This change in shape was more subtle for trans chains than cis. This behavior followed the same trend shown in figure 8. Figure 11 shows that cis chains exhibited little or no change in shape with temperature. Trans chains were slightly less spherical with increasing temperature.
This behavior followed the same trend shown in figure 9.
The relative shape anisotropy followed the same trend as asphericity as func-

Joint correlations in size and shape
Joint correlations between chain size and shape were studied to determine if their variations with chain length and temperature were independent or dependent properties. Cis and trans chains showed similar joint correlation behavior, with correlation and anti-correlation between chain size and shape occuring to a greater extent for trans chains as compared to cis chains. Multiple visualizations of these three-dimensional plots are available as supplementary material (chapter 3). Differences P (b/r 2 g , r 2 g ) − P (b/r 2 g )P (r 2 g ) and P (c/r 2 g , r 2 g ) − P (c/r 2 g )P (r 2 g ) of 0 indicate size and shape are completely independent of each other, i.e. they act as mutually exclusive events. A positive difference indicates correlated events, while negative indicates anti-correlation.
For small rod-like chains, which arise less typically than average, figure 12 indicates some anti-correlation between size and shape. Small chains were nearer to spherical in shape, and high correlation between chain size and shape was observed for them. For medium size chains, some correlation was found for chains that are near rod-like, while notable anti-correlation was found for more spherical chains.
Rod-like large chains showed correlation between chain size and shape. In total, different size and shape probability density distributions were found for cis and trans chains over different chain lengths and across a range of temperatures. Probability densities are related to the work required to alter chain size and shape, and thus different probability densities for cis-and trans-1,4-polybutadiene indicate different extents of work that must be done in order to alter chain size and shape. Quantifying this deformation work is the subject of ongoing research.

Conclusions
Ensemble averages and probability density distributions of sizes and shapes of cis-and trans-1,4-polybutadiene chains have been quantified for isolated single chains under undeformed theta conditions. Such conformations are considered to be representative for a chain in its own melt.
Characteristic ratios were larger with increasing chain length for both cis and trans chains, and these were in good agreement with experimental and prior At longer chain lengths, both cis and trans chains showed similar asphericity.
Little or no variation was computed in acylindricity for either cis or trans polybutadiene chains. Relative shape anisotropy followed the same trend as asphericity as functions of both chain length and temperature for cis and trans polybutadiene chains.
Joint correlation studies revealed that size and shape parameters are mutually dependent properties of chains. For asphericity, small size rod-like cis chains indicated anti-correlation between size and shape. Small size spherical chains showed high amount of correlation between size and shape. For medium size chains, notable anti-correlation between size and shape was observed for spherical chains whereas some correlation between size and shape was observed for near rod-like chains. Large rod-like chains showed correlation between size and shape. For acylindricity, round cross section small size chains showed good correlation between size and shape, whereas medium size chains showed correlation between size and shape for flattened cross section chains. Round cross section medium size chains showed anti-correlation between chain size and shape. Large chains showed minor correlation between size and shape with being nearly round in cross section. Trans chains showed similar correlation and anti-correlation between size and shape as cis chains, yet to a greater extent.
Cis-and trans-1,4-polybutadiene show different size and shape probability density distributions, which imply different amounts of deformation work to alter chain shape and size. Quantifying this deformation work and its implications for mechanical properties, viscoelastic properties, and rolling resistance are the subject of ongoing work.

Acknowledgements
We

Conclusions
Characteristic ratios were in good agreement with experimental [1,2] and prior computed values [3] (cis-1,4-polybutadiene), and slightly higher than prior computed values [4] (trans-1,4-polybutadiene). Cis and trans chains characteristic ratios were larger with increasing chain length. Higher characteristic ratios for trans chains than cis chains indicated greater chain extension, which could potentially be a result of greater distance spanned between the carbon atoms bonded to the double bonded carbons. Characteristic ratios computed here increased with increasing temperature, with the increase being more prominent for trans chains than cis polybutadiene chains. Small absolute changes in chain size probability densities with temperature were observed. The increase in characteristic ratio can be attributed to a larger relative increase in probability density of larger size chains as compared to a smaller relative decrease in probability density of the smaller size chains with increasing temperature. This resulted in an increase in the average size of the chains with increasing temperature. The larger chains showed a much higher increase in characteristic ratios with temperature than smaller chains, and this effect was stronger for trans than for cis chains. Increase in characteristic ratios can be attributed to the size increase of the extended and taut chain conformations; hence we have named this effect as the "taut conformation effect". Swelling of these polymer chains upon heating can thus be attributed to a size increase of the relatively few extended and taut conformations, rather than expansion uniformly across conformations of all sizes.
For limit of long chains, the mean squared radius of gyration r 2 g 0 should equal 1/6 of the mean squared end-to-end distance r 2 0 [5]. The ratio r 2 0 / r 2 g 0 was higher than 6 for shorter trans chains and decreased to 6 at longer lengths.
For cis chains, the ratio was slightly higher than 6 for all chain lengths.
The chain size probability density distributions of the cis-and trans-1,4polybutadiene chains were compared to the Gaussian model [5,6]. Gaussian model predicted higher probability than simulation results at shorter and longer chain sizes for both cis and trans chains. Simulation results predicted higher probability than the Gaussian model at certain regions of medium size chains for cis and trans chains while at other regions of medium size chains, simulation and Gaussian results were in agreement.
The eigenvalues λ 1 , λ 2 , and λ 3 of the radius of gyration matrix used by Theodorou and Suter [7] indicate the extents of orthogonal principal axes that span the region occupied by a chain in primary and secondary directions (eigenvectors Ensemble averages of chain shape parameters such as asphericity (deviation from spherical shape), acylindricity (deviation from cylindrical shape) and relative shape anisotropy were studied based on the radius of gyration matrix for both cis and trans chains of different chain lengths and over different temperature ranges.
Cis and trans chains showed similar asphericity behavior at longer chain lengths i.e. an asphericity value of 0.6. An asphericity of 0.6 corresponds to a chain with a contribution to the squared radius of gyration that is around 5.5 times larger in the longest direction than the secondary directions. At the longer chain lengths, the averaged ratio of the eigenvalues along the longest to the shortest direction was around 12, while at the same chain length, the averaged ratio of the eigenvalues along the secondary directions was around 2.5; this corroborates that the contribution to the radius of gyration was around 5.5 times larger in the longest direction than the secondary directions.
The relative shape anisotropy followed the same trends as the asphericity as functions of both chain length and temperature. The acylindricity factor did not show much observable deviation with chain length and temperature for both cis and trans chains.
Joint correlation studies between chain size and shape showed that they are mutually dependent properties. For asphericity, rod-like small size and spherical medium size cis chains showed anti-correlation between chain size and shape.
Spherical small size, near rod-like medium and large size chains showed correlation between chain size and shape.
For acylindricity, medium size chains of flattened cross section, and small and large size chains of round cross section showed correlation between chain size and shape. Round cross section medium size chains showed anti-correlation between chain size and shape. Trans chains showed similar behavior as cis chains with correlation and anti-correlation between chain size and shape occuring to a greater extent.
Probability densities are related to the work required to alter chain size and shape. Cis-and trans-1,4-polybutadiene chains showed different probability density distributions, and thus different amounts of work would be needed to be done on them to bring about a change in their chain conformations. This deformation work can be quantified to determine mechanical properties, viscoelastic properties and rolling resistance. Thus it can be seen that changes in chain conformations directly impacts rolling resistance of vehicle tires.

Current Work
Currently I am looking at how cis-and trans-1,4-polybutadiene chain size and shape are affected under deformation. I am using the same ensemble of single isolated chains (100,000) under the same range of chain length and temperature, and applying deformation on them. Instead of squared end-to-end distance (r 2 ), I am using end-to-end distance vectors in the x, y, and z directions (r x , r y , and r z ) to study the extent of deformation in each of those directions of the chains.
Probability density distributions of the end-to-end distance vectors help quantify the deformation force acting on the chain ensembles. Deformation leads to changes in chain conformations which results in entropy losses of the chains (since entropy is related logarithmically to chain conformations [8]). These entropy losses lead to computing irreversible work, viscoelastic losses and ultimately rolling resistance. As mentioned in chapter 2, Mark chose to use six discrete rotational isomeric states (based on location of potential energy minima) for each torsional bond [2]. These correspond to φ = -120 • ,-60 • , 0 • ,60 • ,120 • ,180 • . The angles -120 • ,-60 • ,60 • ,120 • correspond to gauche states and the angle 0 • corresponds to trans state. The bond pairs ±60 • ,±60 • or ∓60 • ,±60 • have the same probability and were assigned the statistical weight γ. These were gauche-gauche bond pairs.
Since there are six discrete rotational isomeric states, the statistical weights were arranged in 6×6 statistical weight matrices. Since each repeat unit of polybutadiene has three torsional bonds, three statistical weight matrices were used in our work (as suggested by Mark). These were These statistical weight matrices (U i , U i+1 , U i+3 ) were the same as (U c , U a , U b ) as used by Mark [2].

A.2 Partition function
Partition function (z) can be defined as the sum of the unnormalized probabilities for all possible discrete rotational isomeric states. It is used in the computation of bond pair probability (as stated in chapter 2) as n is the degree of polymerization or number of repeat units in a polymer chain.
J and J are row and column matrices for chain start and end respectively [2].
Carrying out the matrix multiplication in equation A.7 provides one term in z for each possible combination of rotational isomeric states. The probability of a single conformation equals its contribution to z, divided by z.

A.3 Transformation matrices
Transformation matrices are used to transform bond vectors from one reference state to another one. According to Flory [1], the transformation matrix used to transform bond vectors from i + 1 frame to i frame (refer figure 2 from chapter 2) can be given as 10) where θ i = bond angle supplements (as given in Table 1) and φ i = torsional angles.