2.2 Semi-Flexibility of Polymer Chains 15 ideal-chain model of a single polymer chain.In order for a simple statistical estimation,a long-enough ideal chain has been assumed.and the long-range er the structural units alon the chain have been neglecte Such an idea I polyme is often referred as unperturb mer.In the fol wing,we start with t e simplest freely-join chain model,and then consider the short-range interactions along the chain.The first short-range interactions are the fixed bond angles along the backbone atoms,as described by the freely-rotating-chain model.The second short-range interactions are the hindrances of intemal rotation as described by the hindered-rotating-chain model.In this way.we progressively approach the description to the semi- flexibility of real polymers. 2.2.1 Freely Jointed Chains The freely-jointed-chain model considers only the chain connection of monomers with no restriction on the connection angles.A common method to characterize the semi-flexibility of polymer chains is to measure the size of a random coil consisting of a single polymer chain.The end-to-end distance of a polymer chain is the firs quantity to characterize the coil size.which can be calculated by using the end to the othe end of the chain.Assu the le ngth b of each vect ng to the length of the mai the vector for th d vectors along the chain. R=A+公++6=2a (2.1 For a large number of polymer chains.their random-oriented end-to-end vectors cancel each other,and their summation approaches zero.Therefore,we need to use se the square end-to-end distances.The sum of end dista ces over a of polymer chain characteristic size of polymer coils. arge number R2=mb=b+2∑∑Mb (2.2) Since the dot product of two bond vectors relies on the angle between them (2.3) The angles between any two bond vectors of the freely-jointed chain are uniformly distributed betw en 0 and 2r.leading to a symm tric distribution of ositive and negative ine values between I and-1.Th in the a large num ch epe t dot products,the positive values cancel h negative counterparts.and eventually
ideal-chain model of a single polymer chain. In order for a simple statistical estimation, a long-enough ideal chain has been assumed, and the long-range interactions between the structural units along the chain have been neglected. Such an ideal polymer chain is often referred as a phantom polymer, or an unperturbed polymer. In the following, we start with the simplest freely-jointedchain model, and then consider the short-range interactions along the chain. The first short-range interactions are the fixed bond angles along the backbone atoms, as described by the freely-rotating-chain model. The second short-range interactions are the hindrances of internal rotation as described by the hindered-rotating-chain model. In this way, we progressively approach the description to the semi- flexibility of real polymers. 2.2.1 Freely Jointed Chains The freely-jointed-chain model considers only the chain connection of monomers with no restriction on the connection angles. A common method to characterize the semi-flexibility of polymer chains is to measure the size of a random coil consisting of a single polymer chain. The end-to-end distance of a polymer chain is the first quantity to characterize the coil size, which can be calculated by using the vector R connecting one end to the other end of the chain. Assuming the length b of each bond vector contributing to the contour length of the main chain, the vector for the end-to-end distance is the sum of n bond vectors along the chain, R ¼ b1 þ b2 þþ bn ¼ Xn i¼1 bi (2.1) For a large number of polymer chains, their random-oriented end-to-end vectors cancel each other, and their summation approaches zero. Therefore, we need to use a scalar to avoid zero result, for example, use the square end-to-end distances. The sum of square end-to-end distances over a large number of polymer chains represents the characteristic size of polymer coils. R2 ¼ Xn i¼1 bi Xn j¼1 bj ¼ nb2 þ 2 XX j>i bi bj (2.2) Since the dot product of two bond vectors relies on the angle gij between them, bi bj ¼ b2 cos gij (2.3) The angles between any two bond vectors of the freely-jointed chain are uniformly distributed between 0 and 2p, leading to a symmetric distribution of positive and negative cosine values between 1 and 1. Thus, in the summation over a large number of such independent dot products, the positive values cancel the negative counterparts, and eventually 2.2 Semi-Flexibility of Polymer Chains 15
16 2 Structure-Property Relationships ∑∑b·b=0 (2.4) Accordingly,we obtain <Ri.>=nb2 (2.5) where <.means the ensemble average over many polymer chains.This quantity is referred as the mean-square end-to-end distance of polymer chains. Asan organic chemist.Staudinger regarded initially as rigid rods (Stauding nd mark ger and Nodzu 193).Latero rsical che ts.Gu 14 recogn ized the free inte I rotation of polymer c s(Guth and Mark most at the same time, Kuhn proposed more explicitl the random coi model and made an analogy for the conformation of a freely jointed chain in the trajectory of a random-walking particle in Brownian motion,provided that the chain length corresponds to the walking time (Kuhn 1934).One can see that the formula(2-5)is actually consistent with Einstein's calculation on Brownian motion in 1905 (Einstein 1905).The consistency implies that the random conformations of polymer chains actually result from an integration of random ian motions of ono ers in the chair haracteriz the coil size.probably the mean-square radius o e appears to be mo e mean-square measured directly by using light scattering.The mean-square radius of gyration is defined as the summation of mean-square distances of all the monomers relative to the mass center of the polymer coil.Thus,if we define the square vectors ras the distances radiating from the mass center of the whole polymer chain,their average over n+1 chain monomers(each with the mass m)gives (2.6 When the polymer chain is long enough,we can derive that <R>=石<R之 (2.7 2.2.2 Freely Rotating Chains The carbon-carbon bonds constitute the backbone of a polyolefin chain.In princi- ple,each carbon atom contains four bonds aligning along the tetrahedron sphybrid orbits with the bond angles fixed at 10928.In other words,the connection of
XX j>i bi bj ¼ 0 (2.4) Accordingly, we obtain <R2 f :j: > ¼ nb2 (2.5) where < . > means the ensemble average over many polymer chains. This quantity is referred as the mean-square end-to-end distance of polymer chains. As an organic chemist, Staudinger regarded macromolecules initially as rigid rods (Staudinger and Nodzu 1930). Later on, two excellent physical chemists, Guth and Mark, recognized the free internal rotation of polymer chains (Guth and Mark 1934). Almost at the same time, Kuhn proposed more explicitly the random coil model and made an analogy for the conformation of a freely jointed chain in the trajectory of a random-walking particle in Brownian motion, provided that the chain length corresponds to the walking time (Kuhn 1934). One can see that the formula (2–5) is actually consistent with Einstein’s calculation on Brownian motion in 1905 (Einstein 1905). The consistency implies that the random conformations of polymer chains actually result from an integration of random Brownian motions of monomers in the chain. Another quantity to characterize the coil size, probably a more widely used one, is the mean-square radius of gyration. This size appears to be more practical than the mean-square end-to-end distance, since it can be measured directly by using light scattering. The mean-square radius of gyration is defined as the summation of mean-square distances of all the monomers relative to the mass center of the polymer coil. Thus, if we define the square vectors r 2 as the distances radiating from the mass center of the whole polymer chain, their average over n þ 1 chain monomers (each with the mass m) gives <R2 g> Xn i¼0 mir 2 i Xn i¼0 mi , (2.6) When the polymer chain is long enough, we can derive that <R2 g> ¼ 1 6 <R2 f :j: > (2.7) 2.2.2 Freely Rotating Chains The carbon-carbon bonds constitute the backbone of a polyolefin chain. In principle, each carbon atom contains four bonds aligning along the tetrahedron sp3 hybrid orbits with the bond angles fixed at 10928´. In other words, the connection of 16 2 Structure–Property Relationships
2.2 Semi-Flexibility of Polymer Chains 17 Fig.2.1 Illustration of the fixed bond angle and the monomers on a real polymer chain has a restricted bond angle.However,even if the bond angle keeps fixed,the interal rotation of each bond around the previous bond on the chain is still possible.Therefore,supposing no hindrance in the internal rotation,we obtain the freely-rotating-chain model.As illustrated in Fig.2.1,the hackbo end-to-end distanc of the fre ely-rot g-chain model car be derived by maki ction term t fixe ngles.or distance of th freely-jointed-chain model,giving by <R院>=mb.1+cos0 (2.8) 1-cos0 2.2.3 Hindered Rotating Chains When real polymer chains perform the internal rotation along the backbone bonds. the substituted side groups will interact with each other,causing a hindrance to the internal rotation.Therefore,the hindered-rotating-chain model must be considered. As illustrated in Fig.2.1,along the chain backbone.a bond can perform intemal rotation ar ound the is bond with a fixed bond an gle The /made by the end oft ele.On this cir the sec face forme e previous two b ong the chain,on angle When w hydrogen substitutes on wo separate carbo atom of ehane CHCH locate at the overlapping positions of internal rotation,their distance is 2.26 A. smaller than the sum of van der Waals radius of hydrogen atoms 2.40 A.Thus,the
monomers on a real polymer chain has a restricted bond angle. However, even if the bond angle keeps fixed, the internal rotation of each bond around the previous bond on the chain is still possible. Therefore, supposing no hindrance in the internal rotation, we obtain the freely-rotating-chain model. As illustrated in Fig. 2.1, the angle between the bond vector and its preceding is defined as y, and for the backbone carbon chains, y ¼ 180–10928´. The mean-square end-to-end distance of the freely-rotating-chain model can be derived by making a correction term for the fixed bond angles, on the basis of the mean-square end-to-end distance of the freely-jointed-chain model, giving by <R2 f :r: > ¼ nb2 1 þ cos y 1 cos y (2.8) 2.2.3 Hindered Rotating Chains When real polymer chains perform the internal rotation along the backbone bonds, the substituted side groups will interact with each other, causing a hindrance to the internal rotation. Therefore, the hindered-rotating-chain model must be considered. As illustrated in Fig. 2.1, along the chain backbone, a bond can perform internal rotation around the previous bond with a fixed bond angle. The trajectory made by the end of this rotating bond forms a circle. On this circle, by making a reference to the sectional line of the face formed by the previous two bonds along the chain, one can define the angle of the projected line of the rotating bond as the rotation angle f. When two hydrogen substitutes on two separate carbon atoms of ethane CH3–CH3 locate at the overlapping positions of internal rotation, their distance is 2.26 A˚ , smaller than the sum of van der Waals radius of hydrogen atoms 2.40 A˚ . Thus, the Fig. 2.1 Illustration of the fixed bond angle y and the internal rotating angle f along the chain backbone 2.2 Semi-Flexibility of Polymer Chains 17
18 2 Structure-Property Relationships 水 180°240° Fig.2.2 (a)Illustration of the overlapping position (=)and the interleaving position (60)of the hydrogen atoms substituted on two carbon atoms of ethane.(b)The potential energy curve of the internal rotation for ethane potential energy of internal rotation will increase due to the strong volume exclusior of two overlapping substitutes,which is unfavorable for the stability of the confor g22 On each middle carbon-carbon bond of the backbone polyethylene,the rest chains at two ends can be regarded separately as two big groups.replacing the two hydrogen atoms of an ethane discussed above.Thus,the strong interaction between two rest chains will greatly raise the potential energy at the overlapping positions,and make a big difference from their interleaving positions.As illustrated in Fig.2.3.the overlapping position exhibits the highest potential energy in the internal rotation.the trans (denoted as t)conformation shows the lowest.and the ations (s .three relatively stable states ie.one d tw ns.can be regarded in th polymer chain conformations.Such an idea chain model called rotational-isomerism-state model (RISM,see Ref.(Volkenstein 1963;Birshtein and Ptitsyn 1966)).This model can characterize well the semi-flexibility of real polymer chains.Several examples for semi-flexibility of real polymer chains can be found in Flory's specialized text (Flory 1969). The semi-flexibility of polymer chains due to the hindered internal rotation is revealed by the correction from the contribution of the internal rotation in the mean square end-to. -end distance,as <Ri,>=mb2.1+cos0.1+<cos> (2.9 1-cos0 1-<cos> where <cos= cosd and k is the Boltzmamn's constant with T the absolute temperature. From the potential energy curve of the internal rotation of polyethylene shown in Fig.2.3.one can recognize two potential energy differences with important
potential energy of internal rotation will increase due to the strong volume exclusion of two overlapping substitutes, which is unfavorable for the stability of the conformation. When the hydrogen atoms locate at the interleaving positions with a rotation angle of 60, the potential energy of internal rotation can be effectively lowered, as illustrated in Fig. 2.2. On each middle carbon-carbon bond of the backbone polyethylene, the rest chains at two ends can be regarded separately as two big groups, replacing the two hydrogen atoms of an ethane discussed above. Thus, the strong interaction between two rest chains will greatly raise the potential energy at the overlapping positions, and make a big difference from their interleaving positions. As illustrated in Fig. 2.3, the overlapping position exhibits the highest potential energy in the internal rotation, the trans (denoted as t) conformation shows the lowest, and the gauche conformations (separated into the left g+ and the right g- ) are two metastable states. Therefore, three relatively stable states, i.e. one trans and two gauche conformations, can be regarded as the representative states in the statistics of polymer chain conformations. Such an ideal chain model is often called the rotational-isomerism-state model (RISM, see Ref. (Volkenstein 1963; Birshtein and Ptitsyn 1966)). This model can characterize well the semi-flexibility of real polymer chains. Several examples for semi-flexibility of real polymer chains can be found in Flory’s specialized text (Flory 1969). The semi-flexibility of polymer chains due to the hindered internal rotation is revealed by the correction from the contribution of the internal rotation in the meansquare end-to-end distance, as <R2 h:r: > ¼ nb2 1 þ cos y 1 cos y 1 þ < cos f> 1 < cos f> (2.9) where < cos f> ¼ Ð 2p 0 eEðfÞ=kT cos fdf Ð 2p 0 eEðfÞ=kTdf and k is the Boltzmann’s constant with T the absolute temperature. From the potential energy curve of the internal rotation of polyethylene shown in Fig. 2.3, one can recognize two potential energy differences with important Fig. 2.2 (a) Illustration of the overlapping position (f ¼ 0) and the interleaving positions (f ¼ 60) of the hydrogen atoms substituted on two carbon atoms of ethane. (b) The potential energy curve of the internal rotation for ethane 18 2 Structure–Property Relationships
2.2 Semi-Flexibility of Polymer Chains 人水水 ig.2.3 (a)Illus of polyethylene.(b)Th ntial overla physical meanings:one is the potential energy differene gauche conton ations,and the other is the potential energy barrier 4E for the transition from the trans to gauche conformations.The thermodynamic equilibrium based on the energy potential Ae defines the static flexibility of polymer chains.We know that the thermodynamic distributions in three conformation states are related to the capability of local thermal fluctuations,which is at the energy level of 1 kT. When AT>>4e.the states t.g'.g'will occur with almost the same probabilities and polymer chains will exhibit random coils with a high flexibility.When N the tra cMeo欧牌 tion will be dominar and p will exhibit the onformation with a high eextendcedchainsmainlyexi the ordering of polyme chains.On the other hand,the transition kinetics based on the activation energy 4E defines the dynamic flexibility of polymer chains.When kT>>AE,it is easy for polymer chains to change their conformation,so they are in the fluid state.When AT<<E,polymer chains are unable to change their conformation.so they are in the solid state.either in the glass states or in the crystalline state.Therefore.the chain semi-flexibility provides an intra-molecular source of the activation energy to the glass transition chain h ch bonds on one chuin imply that thedso oul ve ≈107 ways to arrange all the micro-conformations.Although compared to the real polymer chain this chain is not very long.we could not count out one-by-one the astronomical figures of conformations.Therefore,if we want to leamn the conformational properties and their variation laws.we have to employ the statistical method introduced in the next chapter. In practice,not all the combinations of three representative interal rotation n he ins.the exists the s w van der s that the summation A.Two consecutive gche conformations,g'g*or g'g,bring the end-to-end distance of the pentane segment to 3.6 A.Thus,these two chain ends interleave with each other,which can be acceptable.However,the conformation g'gor g'g*bring the end distance to 2.5 A
physical meanings: one is the potential energy difference De between trans and gauche conformations, and the other is the potential energy barrier DE for the transition from the trans to gauche conformations. The thermodynamic equilibrium based on the energy potential De defines the static flexibility of polymer chains. We know that the thermodynamic distributions in three conformation states are related to the capability of local thermal fluctuations, which is at the energy level of 1 kT. When kT >> De, the states t, g+ , g- will occur with almost the same probabilities, and polymer chains will exhibit random coils with a high flexibility. When kT << De, the trans conformation will be dominant, and polymer chains will exhibit the fully extended conformation with a high rigidity. The extended chains mainly exist in the ordered states, so the static semi-flexibility facilitates the ordering of polymer chains. On the other hand, the transition kinetics based on the activation energy DE defines the dynamic flexibility of polymer chains. When kT >> DE, it is easy for polymer chains to change their conformation, so they are in the fluid state. When kT << DE, polymer chains are unable to change their conformation, so they are in the solid state, either in the glass states or in the crystalline state. Therefore, the chain semi-flexibility provides an intra-molecular source of the activation energy to the glass transition of polymers. For a flexible polymer chain, if the internal rotation of each bond along the backbone chain has three possible rotational isomerism states, 1,000 such bonds on one chain imply that the random coil could have as many as 31,000 10477 ways to arrange all the micro-conformations. Although compared to the real polymer chain this chain is not very long, we could not count out one-by-one the astronomical figures of conformations. Therefore, if we want to learn the conformational properties and their variation laws, we have to employ the statistical method introduced in the next chapter. In practice, not all the combinations of three representative internal rotation states can be accepted along the real polymer chain. For polyethylene chains, there exists the so-called “pentane effect” (Flory 1993). We know that the summation of two van der Waals radii of carbon atoms is 3.0 A˚ . Two consecutive gauche conformations, g+ g+ or gg- , bring the end-to-end distance of the pentane segment to 3.6 A˚ . Thus, these two chain ends interleave with each other, which can be acceptable. However, the conformation g+ g- or gg+ bring the end distance to 2.5 A˚ , Fig. 2.3 (a) Illustration of the overlapping, gauche and trans positions of polyethylene. (b) The potential energy curve of the internal rotation of polyethylene 2.2 Semi-Flexibility of Polymer Chains 19