3. Structure Morphology Flow of Polymer

The geometrical arrangement of the atoms in a polymer chain can be divided into two categories. The first category is configurational arrangements which are fixed by the chemical bonding in the molecule. The configuration of a polymer chain cannot be altered unless chemical bonds are broken and reformed. The second category is conformational arrangements which arise from the rotation of single bonds. Examples involving conformations of polymer chains include trans versus gauche arrangement of consecutive carbon–carbon single bonds and the helical arrangements found in some polymer crystal structures. Polymer configurations include head-to-head, tail-to-tail and head-to-tail arrangements in vinyl polymers, several stereoregular arrangements of 1,2- and 1,4-addition cis or trans isomers, or d and l forms, and arrangements around asymmetric carbon atoms.

Stereo-isomerism in polymers arises from different spatial arrangements (configurations) of the atoms or substituents in a molecule. Tacticity is the regularity in the configurations of successive stereo centers that determines the order of the polymer chain, such as (–CH₂–C*HR–) _n where * is stereo center. Figure 3.2 shows the different examples of tacticity. Atactic is that the R group on successive stereo centers are randomly distributed on the two sides of the planar zig–zag polymer chain and thus the polymer chain does not have order. Isotactic is that the stereo center in each repeating unit in the polymer chain has the same configuration. All the R groups are located on one side of the plane of the carbon–carbon polymer chain either all above or all below the plane of the chain. Syndiotactic is that the stereo center alternates from one repeating unit to the next with the R groups located alternately on the opposite sides of the plane of the polymer chain.

In addition to poly(alpha) olefin, polystyrene, poly(methyl methacrylate) and 1,2 addition of polybutadiene can also exhibit either isotactic or syndiotactic structure. Whether a polymer is isotactic or syndiotactic usually determines its crystal structure, and the assignment of an all-isotactic or all-syndiotactic structure to a polymer can be made from its crystal structure. NMR spectroscopy is a powerful technique to determine the stereospecificity of polymer (the principle of NMR spectroscopy will be discussed in Chap.​ 5). It allows the determination of the stereoregular configuration of successive monomers in sequences. For example, the NMR spectrum of methylene protons in poly(methyl methacrylate) allows one to distinguish between and determine the relative numbers of sequences of two monomer units (dyads) that have syndiotactic (racemic, r) and isotactic (meso, m) symmetry. The alpha-methyl proton resonance allows estimation of the numbers of 3-monomer sequences (triads) with configurations mm, mr, and rr. Figure 3.3 shows the configurations of mm, mr, rr of poly(methyl methacrylate).

The synthesized poly(3-(n-hexyl)thiophene) (P3HT) can have three configurations: head-to-head, tail-to-tail and head-to-tail. Only the head-to-tail configuration shows high crystallinity due to the presence of even spacing among the monomers in the head-to-tail configuration as shown in Fig. 3.4. The stereospecificity of P3HT can also be determined by the NMR as shown in Fig. 3.5. The chemical shift of methylene proton of hexyl side chain of thiophene is different resulted from their position, so it can be easily identified and calculated the stereospecificity of the polymer.

Many polymers are capable of rotating the plane of polarization of light and are optical active such as poly(L-propylene oxide). The optical activity of low molecular weight compound is associated with the presence of asymmetric carbon atoms. However, it is not universally true in polymers. The every second chain of substituted vinyl polymer is theoretically asymmetric, yet such polymers are not usually optically active, even when isotactic or syndiotactic, because of intramolecular compensation.

The X-ray patterns of most crystalline polymers show both sharp features associated with ordered regions and diffuse features with molecularly disordered regions. Therefore both crystalline structure and amorphous structure coexist in the crystalline polymer. Additional evidence indicates that the density of crystalline polymer is in between the theoretical calculated value of complete crystalline polymer and amorphous polymer as shown in Table 3.1.

The crystallinity of polymers is closely related to the chemically and geometrically regular structure of polymer chain. However, atactic polymer can form crystalline as long as the size of repeating unit can fit into the crystal lattices despite of stereochemical irregularity. Polyethylene can exhibit highly ordered arrangement with all of the carbon atoms in one plane when the C–C bonds form a zig–zag. Figure 3.6a shows these zig–zag sections of chains which easily pack together closely to form orthorhombic crystalline. Single crystal of linear polyethylene has been fabricated from a solution in perchloroethylene as shown in Fig. 3.6b. Irregularities such as branching in the polyethylene or copolymerization with other different structure monomer will reduce crystallinity.

The crystal structure of poly(vinyl alcohol) is similar to that of polyethylene, since the CH(OH) group is small enough to fit into the polyethylene structure in place of a CH₂ group. The unit cell is monoclinic. Pairs of chains are linked together by hydrogen bonds and then linked into sheets, as long as the stereochemical irregularity allows. Fully extended planar zig–zag is 3.3 kJ/mole of molecular dynamic energy less than that of the gauche. Thus, the zig-zag conformation is favored to form crystalline structure unless substituents on the chains cause steric hindrance. Syndiotactic polymers such as poly(vinyl chloride), poly(1,2-butadiene), most polyamides and cellulose exhibit the similar crystalline structure. In most aliphatic polyesters and in poly(ethylene terephthalate), the polymer chains are shortened by rotation about the C–O bonds to allow close packing. As a result, the main chains are no longer planar. However the terephthalate unit in poly(ethylene terephthalate) remains planar as required by resonance. That results in higher packaging and thus higher crystallinity for poly(ethylene terephthalate) as compared with aliphatic polyesters.

The substituted vinyl polymer –(CH₂–CHR)– with large bulky R group usually is in amorphous structure through free radical polymerization. For example, poly(methyl methacrylate), polyacrylonitrile, rubber and polystyrene. Amorphous polymers do not scatter light, so they are transparent in visible light. Thus, they can be used as light weight glass such as poly(methyl methacrylate) with a trade name of Flexglas.

The large size substituted polymer can be organized in isotactic structure through coordination polymerization. For example, poly(methyl methacrylate), polystyrene can crystallize with a helical conformation in which alternate chain bonds take trans and gauche positions. For the gauche position, the rotation is always in the direction that relieves steric hindrance by placing R and H groups in juxtaposition, generating either a left-hand or a right-hand helix as shown in Fig. 3.7. If the side group is not too bulky, the helix has exactly three units per turn and the arrangement is similar to that in Fig. 3.7a. This form has been found in isotactic polypropylene, poly(1-butene), polystyrene. More bulky side groups require more space, resulting in the formation of looser helices as shown in Fig. 3.7b–d. Isotactic poly(methyl methacrylate) forms a helix with five units in two turns, while polyisobutylene forms a helix with eight units in five turns. The poly(tetrafluoro ethylene) contains larger size of F atom than that of H atom. Two helical conformations are existed. They are twist ribbons in which the fully extended planar form is distorted to have an 180^o twist in 13 CF₂ units in the more stable form at low temperature. Above 19°C, this form is replaced by a slightly untwisted conformation with 15 CF₂ units per half-twist.

Crystallinity is not expected to find in alternating copolymer and random copolymer. The repeating unit of alternating copolymer is too short to form organized structure. If A and B unit in the alternating copolymer are in similar molecular size, the crystallization behavior will be similar to that of homopolymer. The random copolymer does not have regular and ordered sequence, so the crystalline cannot be formed.

For the block copolymer, various crystalline structures can be formed according to their chemical structure, chemical composition and interactions among blocks. The crystalline behaviors of diblock copolymers have been extensively studied. Figure 3.8a shows the varieties of crystalline structure in diblock copolymers with different volume fractions. A phase diagram can be constructed for various diblock copolymer compositions at different temperature as shown in Fig. 3.8b, c. The formation of crystalline structure in diblock copolymer is due to the phase separation between two blocks. It is called self assembly phenomena. The crystalline domain is in the range of 10–20 nm depends on the size of each block. These nanostructured copolymers have generated many interests in the nanodevice applications, because they can be used as low cost nanolithographic resist defining nanopitch circuits without using expensive electron beam exposure unit.

Factors that influence the phase diagram of diblock copolymers are determined by: I (intrinsically immiscible interaction parameter), χ (the degree of thermodynamic incompatibility), N (degree of polymerization). The narrow molecular weight distribution copolymers have to be used, so one can obtain a clear-cut phase boundary. Therefore, the copolymer usually is synthesized using living polymerization (will be discussed in Chaps.​ 8 and 10). Figure 3.9 shows poly(3-hexyl thiophene)-b-poly(2-vinyl pyridine) can be synthesized by sequential polymerization. The vinyl terminated poly(3-hexyl thiophene) was first synthesized by Grignard metathesis polymerization with one end capped with phenyl group then subsequently reacted with 2-vinyl pyridine to obtain the block copolymer by anionic polymerization [7].

Various compositions of copolymers can be synthesized using anionic macroinitiator of polythiophene to initiate 2-vinylpyridine monomer to obtain P3HT-b-P2VP copolymers as summarized in Table 3.2. Their TEM microstructures are shown in Fig. 3.10. The copolymer exhibits structures of nanofibrils, lamella, cylinder and sphere by changing the volume fraction of P2VP from 20, 29, 68, 87 % respectively [8]. The P3HT segment is more rigid than P2VP segment, so at lower volume fraction of P2VP, less curvature morphology is observed such as fiber, lamella.

The TEM technique can only examine the morphology of copolymer locally. We have to use small angle X-ray scattering (SAXS) technique to study long range order behaviors of copolymers as shown in Fig. 3.11. The instrument set up is described in Fig. 5.​15. The long range order of different copolymers has been observed for all the samples and it is in agreement with TEM observation.

The phase diagram of P3HT-b-P2VP has also been constructed using temperature varied TEM, SAXS and WAXS as shown in Figs. 3.12, 3.13 and 3.14 respectively.

The feature of phase diagram of the self assembly diblock copolymer depends on the chemical structure of each block as shown in Fig. 3.14. Figure 3.15 shows the phase diagrams of different diblock copolymers. The P3HT-b-P2VP is a rod-coil system. The P3HT rod is not flexible enough to obtain gyroid structure easily as observed in coil–coil system. The PPV-b-PI system exhibits simpler phase diagram than the P3HT-b-P2VP system, because the PPV is more rigid (higher aspect ratio of diameter versus length of molecule) as compared with P3HT.

Liquid crystals are neither true liquids nor true solids. Liquid crystallinity (LC) occurs when molecules become aligned in a crystalline array while still in the liquid state that exhibits anisotropic property. The ordered regions in the liquid are called mesophases. Molecules exhibit LC behavior when their structures are relatively rigid, elongated or disc like. The morphology of LC may be influenced by external magnetic or electrical field, shear force, sometimes they change color with temperature. They exhibit fluidity of liquids and the opaqueness of crystalline solids. The LC polymers developed in late 1970. They contain rigid moiety in the polymers. The rigid moiety is called mesogen which is responsible for the mesophases.

There are two major classifications of liquid crystals: lyotropic type and thermotropic type. Lyotropic liquid crystals form under the influence of solvent. A commercially available lyotropic liquid crystalline polymer is the aromatic polyamide (du Pont trade name Kevlar). The structure of Kevlar is shown in below.

The melt viscosity of copolyester was increased initially when the amount of p-hydroxy benzoic acid was increased due to the incorporation of “rigid” p-hydroxy benzoate unit [10]. At levels of about 30 mol %, however, the melt viscosity began to decrease, reaching a minimum at about 60–70 mol %. This is shown in Fig. 3.16 at four different shear rates. The decrease in viscosity and increase in opaqueness arises from the onset of LC polymers which is due to the increased backbone rigidity. The increase in rigidity decreases the entanglement of polymer chain which is an effective way to reduce viscosity.

Major drawbacks to liquid crystalline polymers are that they have a very high melting point and are difficult to dissolve in the common organic solvents. One approach to resolve these difficulties is to separate the rigid back bone groups with flexible spacers such as ethylene units, ethylene oxide units, or silane units (their structures are shown below). Another approach is to attach mesogens with flexible spacers to the polymer backbone. Both types of liquid crystalline polymers are illustrated schematically in Fig. 3.17.

The crosslink density \(\varGamma \) is determined by the number average molecular weight between crosslinks \( (\bar{M}_{n} )_{c} \) dividing the number average molecular weight of uncrosslinked polymer \( (\bar{M}_{n} )_{o} \) as shown below.

Crosslink density is the number of crosslinked monomer units per main chain (theoretical value). For example, as a polymer where one of every 20 molecules is a trifunctional isocyanate, nine of every 20 molecules is a bifunctional isocyanate, and 10 of every 20 molecules is a bifunctional alcohol, we could say the crosslink density of this polymer is 0.05 [1/20]. In practice for actual value, the crosslinking density is measured by swelling ratio and extract ratio test (ASTM D2765-95), and their ratios are equal to:

By introducing strong secondary bonding attraction between polymer chains, the polymer can exhibit properties of a thermosetting material while remaining thermoplastic behavior. Crystalline polymers fit into this category. Because of the very strong secondary forces arising from close chain packing, many of the mechanical and solution properties of crystalline polymers are similar to those of crosslinked amorphous polymers. For example, thermoplastic elastomer can be crosslinked through hydrogen bonds. A common type of thermoplastic elastomers (TPE) has polyurethane structures. This type of polyurethane can be prepared by reacting excess amount of methylene 4,4′-diphenyl diisocyanate with polyols (hydroxyl terminated long chain aliphatic polyester or polyether) to form isocyanate terminated oligomer. Then the oligomer is reacted with ethylene diamine to yield polyurethane with the structure shown in Fig. 3.18 [2]. The polymer contains soft segment and hard segment from the polyols domain and urethane urea domain respectively.

Polystyrene-polybutadiene-polystyrene (-ABA-copolymer, rigid-flexible-rigid) is another example. A and B are immiscible, B block aggregates into microdomain within the polymer matrix, as shown in Fig. 3.19 to have physical crosslinking with improved mechanical properties.

Polymer blend is a mixture of two or more polymers without chemical bonding. It is also called polymer alloy. Miscible blend is clear and exhibits one phase transition, for instance, GE Noryl is a polymer blend of polystyrene and poly (oxy-2,6-dimethyl-1,4-phenylene). Immiscible blend is opaque and exhibits more than one phase transition, such as ABS (acrylonitrile–butadiene–styrene) plastics. It is prepared by an amorphous styrene-butadiene copolymer dissolving in styrene and acrylonitrile first and then undergoing further polymerization. Chain transfer reactions may occur to graft one copolymer to the other [11].

The property (P) of miscible binary polymer blend can be quantified by the semi empirical relationship,where \( \emptyset \) is the volume fraction in the blend and I is the interaction factor that may be negative, zero, or positive. If the property is entirely additive, I = 0. If I is positive, every component has synergistic effect to each other which results in better property than the weighted average. If I is negative, every component is nonsynergistic, then the property is worse than the weighted average. Figure 3.20 shows these three possible phenomena in the plot of property versus concentration.

When the two polymers are not miscible, by incorporate compatibilizers into the blend, the adhesion between phases can be improved. For example, an incompatible blend of homopolymers of poly(A) and poly(B), by adding the block copolymer made from A and B monomers, the natural affinities of the blocks for their respective homopolymers will localize the copolymer at the phase boundary and help “glue” the two immiscible phases together, as depicted in Fig. 3.21.

A variety of forces are applicable to polymer deformation, and the most important one is shear force (or tangential stress). Shear is a force applied to one side of a surface in a direction parallel to the surface. If a rectangle is subjected to shear, for example, it becomes a parallelogram as illustrated in Fig. 3.22. Shear stress \( (\tau ) \) is defined as the force \( (F) \) in dynes (or newtons) per unit of surface area \( (A) \) in square centimeters (or square meters); that is,Shear causes polymer molecule to flow past one another, when a polymer is in the liquid or molten state.

Viscosity is a measure of resistance to flow which has been used in the discussion of polymer solution in Chap.​ 2. In this chapter, we concentrate on the flow properties of neat polymers. When the polymer is under the shear force \( \tau = F/A, \) shear strain, \( \gamma \) is the amount of deformation can be expressed byThe resistance to shear is the shear modulus, G, which is the ratio of stress to shear strain and expressed byShear rate, \( (\dot{r}) \), also called the velocity gradient, is the rate at which the planes (see Fig. 3.22) or molecules (in an amorphous liquid) flow relative to one another:If the shear stress increases in proportion to the shear rate, the liquid is called Newtonian (or ideal) liquid. It follows the Newton’s law of viscosity:where the proportionality constant, \( \eta \), is the viscosity. Units of viscosity are poises (dyne s/cm²), or in the SI system, Pascal-seconds (Pa s = newton s/m²). Table 3.3 is the list of viscosity of different matters as compared to glycerine. The polymer exhibits at least two orders higher viscosity than glycerine.

The viscosity of neat polymer can be described by the Williams-Landel-Ferry (WLF) equation for temperature ranged from \( T_{g} \) to \( T_{g} + { 1}00\;{\text{K}} \) as shown in Eq. 3.10. \( T_{g} \) is the glass transition temperature of polymer which indicates the transition temperature of polymer from rigid solid state to rubber state (the effect of chemical structure on the \( T_{g} \) of polymer will be discussed in Sect. 3.8).where A, B are adjustable constant, R is gas constant, E _v is apparent activation energy for viscous flow which is related to free volume, \( T_{\rm{0}}\,<\,T_{g} \) (\( T_{\rm{0}} \) is the temperature of free volume being equal to zero). The Arrhenius equation (Arrhenius-Frenkel-Eyring AFE formula) can be used for temperature above \( T_{g} + 100\;{\text{K}} \) The shear rate dependence of viscosity can be expressed by Carrean A model as below which is not temperature dependent, so called compensation effects for polymer melts.where \( \eta \) is the viscosity at a shear rate of \( \dot{r};\,\eta_{\rm{0}} ,k, \) and s are the model parameters. This relation is known to fit most polymer viscosity data obtained from simple shear flow experiment quite well [12]. The viscosity of the polymer depends on the chemical structure of the polymer and temperature. At the same amount of repeating unit of polymer, the bulky structured polymers from bulky branch or substituent have higher \( E_{\nu } \) than that of linear polymers. At a specific temperature \( (T_{1} ) \), the polymer with more bulky chain branch or substituent exhibits higher viscosity because the polymer has less free volume. When the temperature increases from \( T_{1} \) to \( T_{2} \), its effect on the viscosity will be less for linear polymer but more for bulky structured polymer. Table 3.4 shows the extent of temperature effect on the polymer viscosity for different chain structures and different activation energy of polymer flow.

Deviations from ordinary Newtonian liquid behavior are common in polymer flow. One type is called a Bingham Newtonian fluid which is defined aswhere \( \tau_{c} \) is the critical shear stress, or threshold stress. It is the stress needed to initiate flow. Other deviations are non-Newtonian. They occur where shear stress does not increase in direct proportion to shear rate. The deviation may be in the direction of thinning or thickening. These various possibilities are illustrated in Fig. 3.23.

Thixotropic liquid has gel like high viscosity at low stress, but thin out at high shear force such as stirring. The difference between shear thinning and thixotropic is that the former dependent on shear rate, the later is independent of shear rate but dependent on time at a fixed shear rate. Flow behavior may also be expressed by a power law equation:where \( A \) is a constant and B an index of flow. For a Newtonian fluid, the relationships of \( B = 1 \) and \( A = \eta \,\left( {\tau = \eta \dot{r}} \right) \) are held. A plot of log \( \tau \) versus log \( \dot{r} \) is linear with slope equal to B and intercept equal to log A (Fig. 3.24).

The viscosity of polymer usually decreases at high shear rate due to the decrease in molecular entanglement as shown in Fig. 3.25. In the amorphous state there is considerable entanglement of the chains, and while low shear rates may disrupt this to a degree, the mass retains its entangled character. As shear rate increases, disruption may occur faster than the chains can re-entangle, and the resultant decreasing entanglement allows the molecules to flow with less resistance, hence the decrease in viscosity.

The molecular entanglement is increased as the molecular weight increases. Therefore, molecular weight is a critical variable in rheology. Studies have shown that there exists a critical molecular weight \( (\bar{M}_{c} ) \) for entanglement to begin for flexible chain polymers as illustrated in Fig. 3.26. The slope of log of Newtonian viscosity \( (\eta_{\rm{0}} ) \) against the log of weight average molecular weight \( (\bar{M}_{w} ) \) becomes steep (very large) after the \( \bar{M}_{c} \). For most polymers, the critical molecular weight falls in the range of 4,000–15,000. The critical molecular weight is a function of chain length and equal to average \( \overline{DP} \) of 600 (e.g. polyethylene –(CH₂CH₂)–, 600 × 14 = 8,400). A critical chain length, rather than a critical molecular weight is necessary for entanglement.

Polymer viscosity is increasing logarithmically with molecular weight (Fig. 3.26). The broader the molecular weight distribution, the lower the shear rate at which shear thinning develops (Fig. 3.27). At a given molecular weight, shear rate and temperature, the more highly branched polymer, the lower will be its hydrodynamic volume and the lower its degree of entanglement. Although the branched polymer has low viscosity for ease of processing, it is usually mechanically weaker than the linear polymer due to lack of intermolecular secondary bond. The flow also depends on polymer conformation (shape), more rigid polymers are significantly more viscous according to Mark-Houwink-Sakurada (M–H–S) equation (Eq.​ 2.​22) except liquid crystalline polymers as discussed before. We observed that \( a \) varies from 0.5 for random coil to about one for more rodlike extended shape.

The polymer starts to flow at a specific temperature which is called glass transition temperature \( (T_{g} ) \). The solid polymer changes from a hard glassy solid to a rubbery (elastomers) or flexible (for themoplastics) state which involves the translational movements of the polymer main chains and the rotational movements of the segments (Fig. 3.28).

The translational movements are related to the molecular weight of the molecule and free volume of the polymers. High crystalline and high density polymers exhibit high \( T_{g} \) due to less free volume. Any structural feature which hinders the rotation of segments of the molecular chain should increase the value of \( T_{g} \) of the polymer. Tables 3.5 and 3.6 list the \( T_{g} \) of some common thermoplastics and elastomers respectively. For the purpose of comparison of thermoplastic with general CRU formula –CH₂–CH(X)–, the \( T_{g} \) of polyethylene is taken as a reference point. Any substituting for the hydrogen atom on the ethylene chain, the \( T_{g} \) will be increased due to increased steric hinderance of large size substituent; the larger the substituent, the higher the \( T_{g} \) becomes. For instance, the \( T_{g} \) of polystyrene (381 K) is higher than that of polypropylene (267 K) and polyethylene (253 K). When the substituent is more polar, the \( T_{g} \) will increase as well, such as poly(vinyl chloride) (354 K) and polyacrylonitrile (378 K). The –Cl and –CN are also larger then –H. This point also can be observed in elastomers with general CRU formula –CH–CH=C(X)–CH–. The –C–O– bond has more free volume than –C–C– bond due to the divalent of –O–, so the \( T_{g} \) of polyoxymethylene is lower than polyethylene (198 K vs. 253 K).

Polysiloxanes are known commonly as ‘silicones’ and have remarkable properties, including chemical inertness and very low values of \( T_{g} \). Figure 3.29 compares the geometries of the extended chain structures of isotactic polypropylene \( (T_{g} = 255\;{\text{K}}) \) and one of the common silicones, poly (dimethyl siloxane) with a \( T_{g} \) of 155 K. It is the relatively large separation of the backbone silicon atoms and their substituents which accounts for the ease of internal rotations and hence the low \( T_{g} \), which is even over 40 K lower than that of polyoxymethylene (198 K).

The \( T_{m} \) refers to a phase transition in which converts crystalline solid to liquid polymer. The \( T_{m} \) can be expressed by the corresponding changes of enthalpy \( \Updelta H_{m} \) and entropy \( \Updelta S_{m} \). The corresponding Gibbs energy change \( \left( {\Updelta {\text{G}}_{m} } \right) \) is zero when the process is conducted under equilibrium conditions so that \( \Updelta H_{m} = T_{m} \Updelta S_{m} \) and \( T_{m} = \Updelta H_{m} /\Updelta S_{m} \). Table 3.7 shows representative values of these parameters for various common thermoplastics in which substantial fractions of bulk samples are usually crystalline.

It is interest to observe that a large difference presents in the \( T_{m} \) between polyethylene (–CH₂–CH₂–) _n and poly(tetrafluoro ethylene) (PTFE) (–CF₂CF₂–) _n . As shown in Table 3.7, the \( \Updelta S_{m} \) of PTFE is much lower than that of polyethylene which is mainly responsible for the higher thermal stability of PTFE crystallites. What is obvious is that the fluorine atoms, much larger than hydrogen atoms, form an interlocking sheath along the PTFE chain. This difference is of particular significance for the molten phases, in which PTFE chains will be much stiffer than polyethylene chains, the latter being able to undergo almost free internal rotations to explore a much larger range of conformations. Both \( T_{g} \) and \( T_{m} \) increase as internal rotations in the bond and the backbones of polymer chains become more hindered as shown in Table 3.8. In fact there is a useful approximate rule of thumb which states that \( T_{g} ({\rm{K}}) \) is approximately equal to two-thirds (0.67) of \( T_{m} ({\rm{K}}) \). The rule is present in the actual case of the common polymers as shown in Fig. 3.30.

For the polymer blends, one T _g is observed for miscible blends such as the blends of polystyrene and poly(oxy-2,6-dimethyl-1,4-phylene) (GE Noryl polymer). Immiscibe blends exhibit more than one T _g such as thermoplastic elastomer of poly(styrene–butadiene–styrene). At room temperature, the polymer behaviors as elastomer with low T _g but the polymer is physical interlocked at the end terminated polymer styrene. At the T _g of polystyrene, the polymer can be soften for ease of processing.

The values of \( T_{m} \) and \( T_{g} \) of copolymers depend on the type of copolymers. For statistical copolymer, the crystallinity of statistical copolymer is lower than that of either of the respective homopolymer because of the decrease in structural regularity. The \( T_{m} \) of any crystalline material formed is usually lower than that of either homopolymer. The \( T_{g} \) will be in between those for the two homopolymers. For block copolymer, each type of block in block copolymer shows its corresponding homopolymer as long as the block lengths are not too short (\( {\bar{M}}_{w} \) > 10,000). This offers the ability to combine the properties of two different polymers into one block copolymer. For alternating copolymer has a regular structure, their crystallinity may not be affected significantly unless one of repeating unit containing rigid, bulky or excessively flexible chain segment. The \( T_{g} \) and \( T_{m} \) of alternating copolymer are in between the two corresponding homopolymers.

Accurate \( T_{g} \) and \( T_{m} \) can be determined by the established methods of DSC, DTA, etc. Their principles and instruments will be discussed in Chap.​ 5. Figure 3.31 shows typical DSC profiles which have been obtained for samples of poly (ethylene terephthalate) (PET) treated in different ways. The top-most profile is for a totally amorphous sample obtained by rapid quenching of melted PET by dropping into liquid nitrogen at 77 K, which allows no time for the development of microcrystallinity. The middle profile is for a PET sample which has been cooled at a moderate rate from the melt and hence has a substantial degree of crystallinity. The bottom-most profile is for a PET sample which has been cooled so slowly that it can be considered to have been annealed. The annealing process is the processing temperature remained near to \( T_{m} \) long enough to develop the maximum possible degree of crystallinity.

The phase transition of the block copolymer, P3HT-b-P2VP has been studied by DSC and shown in Fig. 3.32. The \( T_{g} \) of copolymer is from P2VP homopolymer which is decreased with decreasing the amount of P2VP due to the decreased molecular weight of P2VP. The P3HT homopolymer shows a relative low \( T_{g} \) as compared with the copolymer, but have a \( T_{m} \) at 200°C due to the crystallization of P3HT. However, the crystallization is suppressed by incorporating P2VP segment larger than 20 % by volume, no \( T_{m} \) is observed in copolymer. Figure 3.33 shows the phase transition of pure P3HT at (100) plane using X-ray diffraction measurement. At 425 K, the P3HT is fully crystalline, but the crystallinity decreases with increasing temperature. At 468 K the P3HT is melted which is close to the \( T_{m} \) of P3HT.