20 2 Structure-Property Relationships less than the sum of two van der Waals radii of the carbon ato s,which means that the carbon ends of the pentane segment will overlap with each other.Therefore,due to their volume-exclusion interactions upon overlapping,such kinds of conformations cannot be accepted. 2.2.4 Characterization of Static Semi-Flexibility of Polymers The static flexibility of a semi-flexible polymer chain is related not only to the potential energy difference e,but also to the temperature T.The following quantities are often used to characterize the conformational states of semi-flexible polymer chains. 1.Persistence length The persistence length is theoretically defined by the projection of the chain end along the direction of the first bond vector(Flory 1969)as ,三e即（告） (2.10) n of each backbone bond on the direction of chain exten sio.me persistence length represents the correlation ln ofeacke orientations along the polymer chain.It is also used to describe the chain rigidity from other sources besides hindered intemal rotation,such as the charge interactions along the polyelectrolyte chain,the double helix formation of DNA,microtubules, and the conjugated covalent bonds in the liquid crystal polymers or conductive 2.Length of Kuhn segment The Kuhn segment is defined by the minimum freely jointed unit along the chair (Kuhn 1936).Assuming that a chain contains n backbone bonds,and each bond contributes bo projection length,the projection length of the whole chain is thus L=nbo nkbk (2.11) where nk and bx are the number and the sequence length of Kuhn segments respectively.They m ake the mean-square end-to-er nd dist ance <R>=nKb候 (2.12) The polymer chain formed by ng and b is also called the equivalently freely jointed chain.Therefore,the Kuhn segment length b is
less than the sum of two van der Waals radii of the carbon atoms, which means that the carbon ends of the pentane segment will overlap with each other. Therefore, due to their volume-exclusion interactions upon overlapping, such kinds of conformations cannot be accepted. 2.2.4 Characterization of Static Semi-Flexibility of Polymers The static flexibility of a semi-flexible polymer chain is related not only to the potential energy difference De, but also to the temperature T. The following quantities are often used to characterize the conformational states of semi-flexible polymer chains. 1. Persistence length The persistence length is theoretically defined by the projection of the chain end along the direction of the first bond vector (Flory 1969) as bp b0 exp De kT (2.10) where b0 is the projection of each backbone bond on the direction of chain extension. The persistence length represents the correlation length of the backbone-bond orientations along the polymer chain. It is also used to describe the chain rigidity from other sources besides hindered internal rotation, such as the charge interactions along the polyelectrolyte chain, the double helix formation of DNA, microtubules, and the conjugated covalent bonds in the liquid crystal polymers or conductive polymers. This quantity originates from the worm-like-chain model describing the semi-rigid polymer chains (Kratky and Porod 1949). 2. Length of Kuhn segment The Kuhn segment is defined by the minimum freely jointed unit along the chain (Kuhn 1936). Assuming that a chain contains n backbone bonds, and each bond contributes b0 projection length, the projection length of the whole chain is thus L ¼ nb0 ¼ nKbK (2.11) where nK and bK are the number and the sequence length of Kuhn segments, respectively. They make the mean-square end-to-end distance <R2 > ¼ nKb2 K (2.12) The polymer chain formed by nK and bK is also called the equivalently freely jointed chain. Therefore, the Kuhn segment length bK is 20 2 Structure–Property Relationships
2.3 Local Inter-Chain Interactions 21 (2.13) 3.Stiffness parameter or steric hindrance The stiffness parameter,or steric hindrance,is defined by () (2.14) which reflects the degree of hindrance in interal rotations 4.Unperturbed dimension The unperturbed dimension of the polymer chain is defined by 4(学) (2.15) where M is the molar mass of the polymer.The unperturbed dimension reflects the ex ansion of the polymer coil relative to that predicted by the freely-jointed-chain model. 5.Characteristic ratio The characteristic ratio C is defined by <R2 Cn三 (2.16 2.3 Local Inter-Chain Interactions Along a long polymer chain,each chain unit may carry specific chemical groups which bring arious kinds of inter-chain interactions,either the common van de Waals inte or cial su olecular inte such as the hyd 9 bonding Coulo ng.hydrophobic tera interacti i讪 netal itions,one kin er-cha interaction may play a dor inant role in deteri ing certain physical behaviorsof polymers.Therefore,we need to introduce multiple molecular energy parameters to characterize the corresponding inter-chain interactions,especially when we study the
bK ¼ <R2> L (2.13) 3. Stiffness parameter or steric hindrance The stiffness parameter, or steric hindrance, is defined by s <R2> <R2 f :j: > 1=2 (2.14) which reflects the degree of hindrance in internal rotations. 4. Unperturbed dimension The unperturbed dimension of the polymer chain is defined by A <R2> M 1=2 (2.15) where M is the molar mass of the polymer. The unperturbed dimension reflects the expansion of the polymer coil relative to that predicted by the freely-jointed-chain model. 5. Characteristic ratio The characteristic ratio Cn is defined by Cn <R2> nb2 (2.16) and when the number of chain units n approaches infinity, one can obtain the limiting characteristic ratio C1. 2.3 Local Inter-Chain Interactions Along a long polymer chain, each chain unit may carry specific chemical groups, which bring various kinds of inter-chain interactions, either the common van der Waals interactions, or some special supermolecular interactions, such as the hydrogen bonding, Coulomb forces, p-p stacking, hydrophobic interactions, and coordination interactions with metal atoms. Under proper conditions, one kind of inter-chain interaction may play a dominant role in determining certain physical behaviors of polymers. Therefore, we need to introduce multiple molecular energy parameters to characterize the corresponding inter-chain interactions, especially when we study the 2.3 Local Inter-Chain Interactions 21
22 2 Structure-Property Relationships hierarchical functions at different levels of polyme assembly.Protein folding can be regarded as a typical example of this case The chain simult aneously contains various specific interactions,and adjusts its complex hierarchical self-assembly structure with the change of the local environment for maintaining its living functions.Such a function is also known as the"robust stability". One kind of inter-chain interaction can even play multiple roles in determining y of chain-like structures endo each chain unit conn s its two o neighbors with strong che onds.H er if we look at the dire to the chain each cha mnit inte eracts w with ely weake ence b For example in the tensile strength along the chain direction originating from the covalent bonds is as high as 350 GPa: while the theoretical tensile strength normal to the chain coming from the van der Waals interactions is as low as 10 GPa.Such anisotropy also makes the thermal conductivity of polymer crystals along the chain direction much higher than that normal to the chain.Therefore,the local anisotropy of chain-like structures makes the local inter-chain interactions behave as the interactions between rigid-rod cules.If the c van Wa s can be separa ted i hort-range strong repu nteracti and the long-range r split each kind of interaction into isotropic and aniso tropic parts for such rod-like molecules. The van der Waals interactions are one of the important driving forces for the physical behavior in polymer assembly states.On the one hand,the packing structure of molecules in the liquid phase is dominated by the isotropic part of volume repulsive interactions between polymer chains,especially the combinato- rial entropy for liquid mixtures.Such kind of inter-molecular spatial combination can be ell ted by the lattice nodel.This is the on why the lattice odel can st cessfully desc e the statistical thern dynamic s of multi componen systems containing polymers.The hydrodynamic volume-exclusion interactionso rigid-rod molecules can be regarded as effective anisotropic interactions,which result in an entropy change driving the lyotropic liquid crystal ordering.In addition the combinatorial entropy of bond orientations at the neighboring positions of each chain unit can be employed to explain the screening effect of the repulsive interactions along a polymer chain due to the interpenetration of other polymer chains.The screening effect makes polymer chains exhibit the scaling behavior of unperturbed chain confor rmations in the melt phase.On the other hand,the isotropic s of attra activ s play a deter nt role in driving mixing demixing in multi-component polymer systems. h anisotrop interactionsill drive the thermotropic liquid crystal orderin i the uphas via the enthalpy change.In addition,the local anisotropic attractive interactions between chains can be utilized to describe molecular driving forces for spontaneous crystallization of polymers
hierarchical functions at different levels of polymer assembly. Protein folding can be regarded as a typical example of this case. The chain simultaneously contains various specific interactions, and adjusts its complex hierarchical self-assembly structure with the change of the local environment for maintaining its living functions. Such a function is also known as the “robust stability”. One kind of inter-chain interaction can even play multiple roles in determining the physical behavior of a polymer. The local anisotropy of chain-like structures endows polymers a typical character. Along the backbone of the polymer chain, each chain unit connects its two neighbors with strong chemical bonds. However, if we look at the direction normal to the chain, each chain unit interacts with other neighboring units with relatively weaker sub-valence bonds. For example, in the fully-extended-chain polyethylene crystal, the theoretical tensile strength along the chain direction originating from the covalent bonds is as high as 350 GPa; while the theoretical tensile strength normal to the chain coming from the van der Waals interactions is as low as 10 GPa. Such anisotropy also makes the thermal conductivity of polymer crystals along the chain direction much higher than that normal to the chain. Therefore, the local anisotropy of chain-like structures makes the local inter-chain interactions behave as the interactions between rigid-rod molecules. If the common van der Waals interactions can be separated into the short-range strong repulsive interactions and the long-range weak attractive interactions, we can further split each kind of interaction into isotropic and anisotropic parts for such rod-like molecules. The van der Waals interactions are one of the important driving forces for the physical behavior in polymer assembly states. On the one hand, the packing structure of molecules in the liquid phase is dominated by the isotropic part of volume repulsive interactions between polymer chains, especially the combinatorial entropy for liquid mixtures. Such kind of inter-molecular spatial combination can be well represented by the lattice model. This is the reason why the lattice model can successfully describe the statistical thermodynamics of multi-component systems containing polymers. The hydrodynamic volume-exclusion interactions of rigid-rod molecules can be regarded as effective anisotropic interactions, which result in an entropy change driving the lyotropic liquid crystal ordering. In addition, the combinatorial entropy of bond orientations at the neighboring positions of each chain unit can be employed to explain the screening effect of the repulsive interactions along a polymer chain due to the interpenetration of other polymer chains. The screening effect makes polymer chains exhibit the scaling behavior of unperturbed chain conformations in the melt phase. On the other hand, the isotropic contributions of attractive interactions play a determinant role in driving mixing or demixing in multi-component polymer systems. The anisotropic attractive interactions will drive the thermotropic liquid crystal ordering in the bulk phase via the enthalpy change. In addition, the local anisotropic attractive interactions between chains can be utilized to describe molecular driving forces for spontaneous crystallization of polymers. 22 2 Structure–Property Relationships
2.4 Molecular Weights and Their Distributions Fig.2.4 Illustration for some increasing as well 2.4 Molecular Weights and Their Distributions 2.4.1 Molecular Weight Effects with the increase of molar masses from the lower end.some common properties arg enhanced,such as the melting oints.the mechanical properties.the ans cement of te with further inc asing of ver,the enhar ar ma es,as illus tra d in Fig.2.4.A higher molar mas leads to a larger viscosity.and as a result.the fluid processing becomes more difficult.In most cases,the higher molar mass does not bring the better performance of the product.Therefore,from the practical point of view,it is important to control the molar mass in a proper range during polymer processing. High or low molecular weights are also reflected in the concentration of chain ing are 1.High mobility Short chains are viable to disentangle.The addition of plasticizer is actually adding small compatible molecules for enhancing the high-mobility effect of chain ends. The glass transition temperature will be lowered in this way. 2.Defects in the crystal Short-chain crystals have lower melting points,which can be regarded as a result of tent of cha nd defects in t arge-size crystal.The defects bring ar
2.4 Molecular Weights and Their Distributions 2.4.1 Molecular Weight Effects With the increase of molar masses from the lower end, some common properties are enhanced, such as the melting points, the mechanical properties, the glass transition temperatures, etc. However, the enhancement of properties will soon saturate with further increasing of molar masses, as illustrated in Fig. 2.4. A higher molar mass leads to a larger viscosity, and as a result, the fluid processing becomes more difficult. In most cases, the higher molar mass does not bring the better performance of the product. Therefore, from the practical point of view, it is important to control the molar mass in a proper range during polymer processing. High or low molecular weights are also reflected in the concentration of chain ends in the bulk phase. The chain ends often exhibit quite different behaviors from those monomers locating in the middle of polymer chains. In the following are listed the so-called chain-end effects. 1. High mobility Short chains are viable to disentangle. The addition of plasticizer is actually adding small compatible molecules for enhancing the high-mobility effect of chain ends. The glass transition temperature will be lowered in this way. 2. Defects in the crystal Short-chain crystals have lower melting points, which can be regarded as a result of high content of chain-end defects in the large-size crystal. The defects bring an effective depression in the melting points. Fig. 2.4 Illustration for some common properties P increasing with the molecular weight M, but the viscosity increasing as well 2.4 Molecular Weights and Their Distributions 23
24 2 Structure-Property Relationships 3.Chemical activity Polyoxymethylene (POM)and polycarbonate(PC)often start thermal degradation from the chain ends.If their plastic products contain too many short chains,the products are liable to tur yellow in color,losing their good quality. 4.Specific interactions Poly(ethylene oxide)(PEO)contains-OH groups at its chain ends,which are able to ociation clusters via hydrogen bon ding.leading to high apparent molecula weights in the measurements 5.Fixed chain ends Vulcanization effectively fixes the ends of flexible polymer chains to form three dimensional networks.By this way,the stress relaxation of stretched polymer chains can be avoided.and the high entropy elasticity of the rubber can be produced.Moreover,the fixed chain ends also increase both melting points and glass transition temperatures of short flexible chains. In the synthesis ss of polyme er chains,polymerization is not able to synchror ini th and the in all mers, d mole ula eight distribution in the prod et.Such a ch aracter is calle the polydispersity of polymers The production of polymeric materials has to pay close attention to both highe and lower ends of molecular-weight distributions.which often play important roles in determining the processing technology as well as the product quality of polymers. For examples,some low molecular weight fractions of poly(vinyl chloride)can help the fluid processing of high molecular weight fractions,like the additives of plasticizers.The low molecular weight fractions of PC could speed up chemical degradation, ausing air bubbles an in the oduct Addi f ultra olecular w of iPP improve the crystal Due to th eous character of Ziegler-Na ow molecular weight frac tion of the low-density polyethylene (LDPE)usually contains more short branches, and thus exhibits a difficulty to crystallize with high molecular weight fractions. 2.4.2 Characterization of Molecular Weights A polydisperse polymer exhibits a certain width in its molecular weight distribu- tion We the number o polyer in the fction of the molecular weightMand the total W=N×M.then we define the number-average molecular weight by ENi (2.17
3. Chemical activity Polyoxymethylene (POM) and polycarbonate (PC) often start thermal degradation from the chain ends. If their plastic products contain too many short chains, the products are liable to turn yellow in color, losing their good quality. 4. Specific interactions Poly(ethylene oxide) (PEO) contains -OH groups at its chain ends, which are able to form association clusters via hydrogen bonding, leading to high apparent molecular weights in the measurements. 5. Fixed chain ends Vulcanization effectively fixes the ends of flexible polymer chains to form threedimensional networks. By this way, the stress relaxation of stretched polymer chains can be avoided, and the high entropy elasticity of the rubber can be produced. Moreover, the fixed chain ends also increase both melting points and glass transition temperatures of short flexible chains. In the synthesis process of polymer chains, polymerization is not able to synchronize the initiation, the propagation and the termination steps in all polymers, and therefore leads to a certain width of molecular weight distributions in the product. Such a character is called the polydispersity of polymers. The production of polymeric materials has to pay close attention to both higher and lower ends of molecular-weight distributions, which often play important roles in determining the processing technology as well as the product quality of polymers. For examples, some low molecular weight fractions of poly(vinyl chloride) can help the fluid processing of high molecular weight fractions, like the additives of plasticizers. The low molecular weight fractions of PC could speed up chemical degradation, causing air bubbles and darkening in the plastic product. Adding a minute amount of ultra-high molecular weight fraction of iPP could significantly improve the crystal nucleation behavior of polypropylene product. Due to the heterogeneous character of Ziegler-Natta catalysts, the low molecular weight fraction of the low-density polyethylene (LDPE) usually contains more short branches, and thus exhibits a difficulty to crystallize with high molecular weight fractions. 2.4.2 Characterization of Molecular Weights A polydisperse polymer exhibits a certain width in its molecular weight distribution. We assume the number of polymer molecules Ni in the fraction of the molecular weight Mi, and the total weight of this fraction Wi ¼ Ni Mi, then we define the number-average molecular weight by MN SNiMi SNi (2.17) 24 2 Structure–Property Relationships