30 2 Structure-Property Relationships Mi+M Mi r=kulkn M; M;+M Mi 一M+M, r2-ka/ka M F=[M VM,] Fig.2.11 Illustration of free-radical copolymerization with the statistical propagation of two The sequences are thus determined by the reactivity ratios(.r)and 2.6.1 Chemical Irregularities This kind of sequence defect occurs in the statistical copolymers,where the species of monomers can crystallize.On the backbone of polyethylene chains,the short branchescan be regarded as the non-rystallizablecomomers In high-density (HDPE) gprobability is about nches/1 000 back one carbon ato nd its talli nity can reach levels as hig吵 s9%:while ir low-density polyethylene(LDPE),the branching probability is about 30 branches/ 1,000 backbone carbon atoms,and its crystallinity reaches only 50 %The most common industry product is actually linear low-density polyethylene (LLDPE),and its branching probability is determined by the copolymerization process of CH2= CH2 and CH2 =CHR (R means side alkane groups for short branches). Assuming an ideal free-radical copolymerization reaction,the reactivity ratios r and r are fixed by the single-site Ziegler-Natta catalyst,as shown in Fig.2.11.The of det es the cha of statis zero for r al ting copolymers a and dibl copolymers.Different feeding styles such as the continuous loop reaction or the intermittent batch reaction also make different characteristic distributions of chair sequences.The continuous reaction keeps the feed composition F constant,and consequently the comonomer distributions are homogeneous among copolymers The products can be referred to homogeneous copolymers.The intermittent reaction shifts the feed composition upon the proceeding of batch reaction,so the comono- mer distributions are heterogeneous among copolymers.The products can be called In extreme cases.the heterogeneous copolymers can be a kind of binary blends 2.6.2 Geometrical Irregularities There may exist asymmetry in the structure of repeating units like-(CH2-CHR)- where RH.Therefore,along a polymer chain,the repeating units can have head-
2.6.1 Chemical Irregularities This kind of sequence defect occurs in the statistical copolymers, where the species of monomers can crystallize. On the backbone of polyethylene chains, the short branches can be regarded as the non-crystallizable comonomers. In high-density polyethylene (HDPE), the branching probability is about 3 branches/1,000 backbone carbon atoms, and its crystallinity can reach levels as high as 90 %; while in low-density polyethylene (LDPE), the branching probability is about 30 branches/ 1,000 backbone carbon atoms, and its crystallinity reaches only 50 %. The most common industry product is actually linear low-density polyethylene (LLDPE), and its branching probability is determined by the copolymerization process of CH2 ¼ CH2 and CH2 ¼ CHR (R means side alkane groups for short branches). Assuming an ideal free-radical copolymerization reaction, the reactivity ratios r1 and r2 are fixed by the single-site Ziegler-Natta catalyst, as shown in Fig. 2.11. The product of r1 r2 determines the sequence characters of statistical copolymers, with the values between zero for alternating copolymers and infinity for diblock copolymers. Different feeding styles such as the continuous loop reaction or the intermittent batch reaction also make different characteristic distributions of chain sequences. The continuous reaction keeps the feed composition F constant, and consequently the comonomer distributions are homogeneous among copolymers. The products can be referred to homogeneous copolymers. The intermittent reaction shifts the feed composition upon the proceeding of batch reaction, so the comonomer distributions are heterogeneous among copolymers. The products can be called heterogeneous copolymers. In extreme cases, the heterogeneous copolymers can be regarded as a kind of binary blends. 2.6.2 Geometrical Irregularities There may exist asymmetry in the structure of repeating units like –(CH2-CHR)-, where R 6¼ H. Therefore, along a polymer chain, the repeating units can have headFig. 2.11 Illustration of free-radical copolymerization with the statistical propagation of two monomers along the chain. The sequences are thus determined by the reactivity ratios (r1, r2) and the feed composition (F) 30 2 Structure–Property Relationships
2.6 Sequence Irregularities 31 c Fig.2.12 Illustration of the structural isomerism of double bondson polymer chains with cisand trans-configurations Fig.2.13 Illustration of sequence regularities of 人人人人人人 Isotactic syndiotactic and atactic sequences 人人入人 Syndiotactic 人人个个人 Atactic to-head and head-to-tail connections,which are called the sequence isomerism.The small amount of head-to-head connections can be regarded as defects if most of connections are head-to-tail.Some unsaturated repeating units such as 1.4.- polybutadine have cis-or trans-configurations in the sequences,as demonstrated isomerism. 2.6.3 Spatial Irregularities Each carbon atom on the backbone chain may contain four substitutes different from each other,two of which are the rest parts of the chain at both sides with different chain lengths.These carbon atoms exhibit chiral asymmetry.giving rise to the optical activity labeled with R for the left-handed direction and with S for the right-handed direction.Irregularities of chain sequences in optical activities are called merism.A central m r exists h middle of the cha (RRRRRRRR. SSS SSSSSsss).Th h are in e meso-form will not y opti ca However,in polypropylene,for example,if we stretch the backbone chain int fully trans-conformations,we can see varying sequence regularities from the orientations of the side methyl groups.as illustrated in Fig.2.13.Three typical cases are the isotactic.syndiotactic and atactic sequences.The examples of regular
to-head and head-to-tail connections, which are called the sequence isomerism. The small amount of head-to-head connections can be regarded as defects if most of connections are head-to-tail. Some unsaturated repeating units such as 1,4,- polybutadine have cis- or trans-configurations in the sequences, as demonstrated in Fig. 2.12, referred to the structural isomerism. 2.6.3 Spatial Irregularities Each carbon atom on the backbone chain may contain four substitutes different from each other, two of which are the rest parts of the chain at both sides with different chain lengths. These carbon atoms exhibit chiral asymmetry, giving rise to the optical activity labeled with R for the left-handed direction and with S for the right-handed direction. Irregularities of chain sequences in optical activities are called optical- or stereo-isomerism. A central mirror exists at the middle of the sequence-regular chain (RRRRRRRR. . . . .|. . . . .SSS SSSSSSSS). Therefore, the whole polymers are in the meso-form and will not show any optical activity. However, in polypropylene, for example, if we stretch the backbone chain into fully trans-conformations, we can see varying sequence regularities from the orientations of the side methyl groups, as illustrated in Fig. 2.13. Three typical cases are the isotactic, syndiotactic and atactic sequences. The examples of regular Fig. 2.12 Illustration of the structural isomerism of double bonds on polymer chains with cis- and trans-configurations Fig. 2.13 Illustration of sequence regularities of stereo-isomers along polymer chains with three typical cases for isotactic, syndiotactic and atactic sequences 2.6 Sequence Irregularities 31
32 2 Structure-Property Relationships which all belong to the crystalline polymers.The examples of irregular sequences include aPS,aPP,aPMMA,poly(vinyl acetate).which all belong to the non- crystalline polymers.Some hetero-nuclei polymers may contain different backbone atoms in each repeating unit with intrinsic optical isomerism,such as in poly (propylene oxide)and poly(lactic acid).They have Dor L types,while the optical activity can be completely compensated in the racemate with half-half compositio Question Sets 1.Why are polymer chains still flexible although the angles between backbone bonds are fixed? 2.Which factor is more important in polymer glass transition,the static flexibility or the dynamic flexibility? 3.Try to analyze the usefulness of oriented polymers on account of the local anisotropy of chain structures. 4.Why is the characterization of molecular weights important for polymers? on the olecular i a1 eystalline polymers ofer contain a high conient of seqenc References Flory PI (1969)Statistical mechanics of chain molecules.Interscience.New York Flory PJ(1993)Nobel lecture:spatial configuration of macromolecular chains.In:Nobel lectures y1971- /orld Scient Singapore,p 156 ekularen Statistik.insbesondere bei Kettenmolekilen I Kratky O.Porod G(1949)Rontgenuntersuchung geloster Fadenmolekule.Rec Trav Chim Pays- Kolloid 7 68-2-15 Eigenschaften hochpolymerer Stoffe.KolloidZ76:258-271 in MV (1963)Confi Chem C Int polymer c ience,New York
sequences include isotactic polystyrene (iPS), isotactic polypropylene (iPP), isotactic poly(methyl methacrylate) (iPMMA), and syndiotactic polypropylene (sPP), which all belong to the crystalline polymers. The examples of irregular sequences include aPS, aPP, aPMMA, poly(vinyl acetate), which all belong to the noncrystalline polymers. Some hetero-nuclei polymers may contain different backbone atoms in each repeating unit with intrinsic optical isomerism, such as in poly (propylene oxide) and poly(lactic acid). They have D or L types, while the optical activity can be completely compensated in the racemate with half-half compositions. Question Sets 1. Why are polymer chains still flexible although the angles between backbone bonds are fixed? 2. Which factor is more important in polymer glass transition, the static flexibility or the dynamic flexibility? 3. Try to analyze the usefulness of oriented polymers on account of the local anisotropy of chain structures. 4. Why is the characterization of molecular weights important for polymers? 5. How can we mix two incompatible polymers on the molecular level? 6. Why do non-crystalline polymers often contain a high content of sequence irregularities? References Birshtein TM, Ptitsyn OB (1966) Conformations of macromolecules. Interscience, New York Einstein A (1905) Investigations on the theory of the Brownian movement. Ann Phys (Leipzig) 17:549–560 Flory PJ (1969) Statistical mechanics of chain molecules. Interscience, New York Flory PJ (1993) Nobel lecture: spatial configuration of macromolecular chains. In: Nobel lectures chemistry 1971–1980. World Scientific, Singapore, p 156 Guth E, Mark H (1934) Zur innermolekularen Statistik, insbesondere bei Kettenmoleku¨len I, Monatshefte f. Chemie 65:93–121 Kratky O, Porod G (1949) Ro¨ntgenuntersuchung gelo¨ster Fadenmoleku¨le. Rec Trav Chim PaysBas 68:1106–1123 Kuhn W (1934) U¨ ber die Gestalt fadenfo¨rmiger Moleku¨le in Lo¨sung. Kolloid Z 68:2–15 Kuhn W (1936) Beziehungen zwischen Moleku¨lgro¨sse, statistischer Moleku¨lgestalt und elastische Eigenschaften hochpolymerer Stoffe. Kolloid Z 76:258–271 Staudinger H, Nodzu R (1930) U¨ ber hochpolymere Verbindungen, 36. Mitteil: viscosita¨tsUntersuchungen an Paraffin-Lo¨sungen. Berich Deut Chem Ges 63:721–724 Volkenstein MV (1963) Configurational statistics of polymer chains. Interscience, New York 32 2 Structure–Property Relationships
Chapter 3 Conformation Statistics and Entropic Elasticity 3.1 Gaussian Distribution of End-to-End Distances of Polymer Coils If the intemal rotation surrounding each backbone bond contains three possible conformation states,the internal rotation of a long chain will generate an astronom- ical amount of possible conformation states.In such a case.we could not count them ne hyon and thus have to make conformation statistics on the basis of a n model. A real polymer c in can be modeled by the freely jointed chain that is consisted of Kuhn segments.A freely jointed chain is analogous to the trajectories of random walks.A random walk of a man who gets lost in the forest often turns back to the starting point.Therefore for a polymer coil,one chain end exhibits a rather stochastic location near another chain end.In mathematics,the stochastic distributions of random events follow the central-limit theorem,i.e.,the distributions of large-enough amount of independent random events exhibit a characteristics of Gaussian function around their mean value.as demonstrated in Fig 3 1 The one-din onal distribution of the -end locations for coil follows the Gaussiar nction around another chain endas given by (3.1) where 3 (3.2) Herex is the end-to-end distance,n is the total bond number,and b is the bond e fractions of three dimensions are independent to each other,we W.Hu,Polymer Physics,.D0I10.1007/978-3-7091-0670-9_3. 33 C Springer-Verlag Wien 2013
Chapter 3 Conformation Statistics and Entropic Elasticity 3.1 Gaussian Distribution of End-to-End Distances of Polymer Coils If the internal rotation surrounding each backbone bond contains three possible conformation states, the internal rotation of a long chain will generate an astronomical amount of possible conformation states. In such a case, we could not count them one-by-one, and thus have to make conformation statistics on the basis of a simplified ideal chain model. A real polymer chain can be modeled by the freely jointed chain that is consisted of Kuhn segments. A freely jointed chain is analogous to the trajectories of random walks. A random walk of a man who gets lost in the forest often turns back to the starting point. Therefore for a polymer coil, one chain end exhibits a rather stochastic location near another chain end. In mathematics, the stochastic distributions of random events follow the central-limit theorem, i.e., the distributions of large-enough amount of independent random events exhibit a characteristics of Gaussian function around their mean value, as demonstrated in Fig. 3.1. The one-dimensional distribution of the one-end locations for a polymer coil follows the Gaussian function around another chain end, as given by WðxÞ¼ðb2 p Þ 1 2= eb2 x2 (3.1) where b2 3 2nb2 (3.2) Here x is the end-to-end distance, n is the total bond number, and b is the bond length. Since the fractions of three dimensions are independent to each other, we obtain W. Hu, Polymer Physics, DOI 10.1007/978-3-7091-0670-9_3, # Springer-Verlag Wien 2013 33
34 3 Conformation Statistics and Entropic Elasticity itiomofd Fig.3.1 Illus W=ae-bx2a.b>0 function the dots (xy.z)with fixed distancesR from the sam starting point at the central Wkx=wew6)w日=fRe9 (3.3) As illustrated in Fig.3.2,the radial distribution of the end-to-end distances can be expressed as W(R)=Wxy,z)·4πR (3.4) Here.R2=+y+2.Accordingly. wR=昏re.4aR (3.5) As demonstrated in Fig.3.3.with the increase of R from zero.w(R)reaches a maximum value,which is called the most probable end-to-end distance R.From aw(R)-0 (3.6 R we obtain R*=B-1 (3.7
Wðx; y;zÞ ¼ WðxÞWðyÞWðzÞ¼ðb2 p Þ 3 2= eb2ðx2þy2þz2Þ (3.3) As illustrated in Fig. 3.2, the radial distribution of the end-to-end distances can be expressed as WðRÞ ¼ Wðx; y;zÞ 4pR2 (3.4) Here, R2 ¼ x2 þ y2 þ z2. Accordingly, WðRÞ¼ðb2 p Þ 3 2= eb2R2 4pR2 (3.5) As demonstrated in Fig. 3.3, with the increase of R from zero, W(R) reaches a maximum value, which is called the most probable end-to-end distance R*. From @WðRÞ @R ¼ 0 (3.6) we obtain R ¼ b1 (3.7) Fig. 3.1 Illustration of a Gaussian distribution function Fig. 3.2 Illustration of a spherical surface formed by the dots (x, y, z) with fixed distances R from the same starting point at the central 34 3 Conformation Statistics and Entropic Elasticity