2.4 Molecular Weights and Their Distributions 25 and the weight-average molecular weight by ∑W,-∑NM (2.18) Here the molecular weight commonly means the molar mass of polymers,in units of grams per mole(g/mol).In practice,the molecular weight is also used with referen e to the molar mass of C2 divided by given alon (Da) characterize the width of molec a is defined by d MN (2.19) For a monodisperse polymer sample,d=1.The ranges of d values change drastically with the different mechanisms of polymerization.The values of d are 1.01-1.05 in living polymerization (anionic,cationic.living free radical,etc.) nsation polym erization or n of polyn aroun s-ac olefins. reacti ons on polymerizat 2 5 fo merization.8-30 in coordination polymerization,and 20-50 in branching reactions on polymerization. Another often used characterization of molecular weight is the viscosity-average molecular weight 4=() (2.20) which is obtained from the viscosity measurement of polymer dilute solutions gto Mark-Houwink n th c viscosity [n]=KM"where .See M=（NM MN (2.21) and when=1, WM:Mw Mn=Wi (2.22) Conventionally,=0.5~1,thus MN<M,≤Mw (2.23)
and the weight-average molecular weight by MW P P WiMi Wi ¼ PNiM2 P i NiMi (2.18) Here the molecular weight commonly means the molar mass of polymers, in units of grams per mole (g/mol). In practice, the molecular weight is also used with reference to the molar mass of C12 as divided by 12, given the units of Dalton (Da). The index of polydispersity d can be used to characterize the width of molecular weight distributions, which is defined by d MW MN (2.19) For a monodisperse polymer sample, d ¼ 1. The ranges of d values change drastically with the different mechanisms of polymerization. The values of d are 1.01–1.05 in living polymerization (anionic, cationic, living free radical, etc.), around 1.5 in condensation polymerization or coupling termination of polymerization, around 2 in disproportionation reactions on polymerization, 2–5 for high-conversion olefins, 5–10 in self-acceleration on common free radical polymerization, 8–30 in coordination polymerization, and 20–50 in branching reactions on polymerization. Another often used characterization of molecular weight is the viscosity-average molecular weight M SWiMa i SWi 1=a (2.20) which is obtained from the viscosity measurement of polymer dilute solutions (according to Mark-Houwink equation, the intrinsic viscosity [] ¼ KM a , where K is constant. See (5.18) in Sect. 5.1). When a ¼ 1, M ¼ SNi SNiMi 1 ¼ MN (2.21) and when a ¼ 1, M ¼ SWiMi SWi ¼ MW (2.22) Conventionally, a ¼ 0.5 ~ 1, thus MN < M MW (2.23) 2.4 Molecular Weights and Their Distributions 25
2 Structure-Property Relationships H te or Ve Fig.2.5 Illustration of the principle of GPC.(a)The volume exclusion chromato phy for the lection of chain lengths (b)the efflux curve of a polydisperse polymer sample:(e)the standard curve based on the standard polystyrene samples In the early history of polymer sciences,the measurement of molecular weights was the key experimental evidence proving the presence of macromolecules.In the following decades,many methods have been invented to measure the molecular weights of polym ers.The stoichiome ethods include the end-s The ermo of the colligativ propertie P09 the n of po melting points,the vapor-pha e osmom etry,th othermal distillation,and th osmotic pressure.The scattering methods include the small-angle scattering of visible light (or enhanced with laser beams),X-ray and neutron beams.The microscopy methods include the electron microscopy.The fluid mechanics methods include the viscosity of dilute solutions,the melt index(MI,or melt flow rate MFR,are often used in industry for a fast identification of molecular weights of polymer products,and are defined as the grams of melt mass flowing through a hole within 10 min under a sp nd a ific t mp yea the melt ate MVR is use th the and the GPC metho ome methods me asure the molecular-weight butior as well. In the modern chemistry laboratories,the technology of gel permeation chroma- tography(GPC)has been well commercialized for the characterization of polymer molecular weights and their distributions.In principle,GPC is a kind of volume h matogra In each pore. the low molecular cight fractions ge regio Auid-w g shing while the high mole we fra ctio washing,as dem 1g. 2.5a.Therefore r the detect of pon we obr on curv on the outflow volume.The first signal corresponds to the fraction of the highest molecular weights.The adsorption strength A corresponds to the total weight of the fraction,and the efflux volume Ve corresponds to the molecular weight of that fraction.The latter is calibrated by a standard curve obtained from the standard polystvrene samples.as demonstrated in Fig.2.5b.c.Therefore. ΣH:M Mw =H (2.24)
In the early history of polymer sciences, the measurement of molecular weights was the key experimental evidence proving the presence of macromolecules. In the following decades, many methods have been invented to measure the molecular weights of polymers. The stoichiometric methods include the end-group titration. The thermodynamic methods make use of the colligative properties of polymer dilute solutions, such as the rise of solvent boiling points, the depression of polymer melting points, the vapor-phase osmometry, the isothermal distillation, and the osmotic pressure. The scattering methods include the small-angle scattering of visible light (or enhanced with laser beams), X-ray and neutron beams. The microscopy methods include the electron microscopy. The fluid mechanics methods include the viscosity of dilute solutions, the melt index (MI, or melt flow rate MFR, are often used in industry for a fast identification of molecular weights of polymer products, and are defined as the grams of melt mass flowing through a hole within 10 min under a specific pressure and a specific temperature. In recent years, the melt volume flow rate MVR is also used with the unit of cm3 /10 min), the sedimentation equilibrium, the sedimentation diffusion, and the GPC method. Some methods measure the molecular-weight distributions as well. In the modern chemistry laboratories, the technology of gel permeation chromatography (GPC) has been well commercialized for the characterization of polymer molecular weights and their distributions. In principle, GPC is a kind of volumeexclusion chromatography, because the molecular-weight fractionation is based upon volume exclusion. The porous silica beads are filled in the column of chromatography. In each pore, the low molecular weight fractions get into the deeper region and stay longer upon fluid-washing, while the high molecular weight fractions stay shorter upon fluid-washing, as demonstrated in Fig. 2.5a. Therefore, under the detection of the ultraviolet spectroscopy, we obtain the adsorption curve on the outflow volume. The first signal corresponds to the fraction of the highest molecular weights. The adsorption strength H corresponds to the total weight of the fraction, and the efflux volume Ve corresponds to the molecular weight of that fraction. The latter is calibrated by a standard curve obtained from the standard polystyrene samples, as demonstrated in Fig. 2.5b, c. Therefore, MW ¼ SHiMi SHi (2.24) Fig. 2.5 Illustration of the principle of GPC. (a) The volume exclusion chromatography for the selection of chain lengths; (b) the efflux curve of a polydisperse polymer sample; (c) the standard curve based on the standard polystyrene samples 26 2 Structure–Property Relationships
25 Topological Architectures 2 Fig.2.6 Illustration of(a)linear polymers and(b)ring polymers CH MN=Hi/Mi (2.25) The m (MALDI-TOF MS)can be applied to measure the molecular weights of macromolecules (up to one million Dalton)due to its high sensitivity and its wide response range.In combination with the distribution of fragment lengths,it can be used to characterize those bio-and synthetic macromolecules with complicated molecular architectures. 2.5 Topological Architectures Polymers are not only simply linear chains,but also the building blocks to construct topologically more complicated three-dimensional macromolecules.Some typical cases are listed below. 1.Linear polymers.Linear structure is the basic topological shape of poly ne in Fig 2.6a.The assem of linear polymers normally of chain lengths 2.Ring polymers.The rin contains no chain end.a shown in Fig. 2.6b.Since ring polymers cannot make entanglement with each other,their mobility is much higher than the linear polymers with the same chain lengths. 3.Branched polymers.There can be multiple branches on the linear chains, characterized by the degree of branching.There are several typical cases.The comb-like polymers contain all the branches derived from the same backbone chain as sho yn in fig.2 7a if the branches of co h like olymers are chemi cally diffe h have the cop s exa mple f the random branching is the tin(branched starch) hed polymers (more often c several levels of branching at the chain ends,like a Cayley tree,as shown in Fig.2.7b.Chain branching destroys the sequence regularity of polymer chains. hindering crystallization and thus depressing the mechanical performance
MN ¼ SHi SHi=Mi (2.25) In recent years, the technology of mass spectroscopy has been well developed. The matrix-assisted Laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS) can be applied to measure the molecular weights of macromolecules (up to one million Dalton) due to its high sensitivity and its wide response range. In combination with the distribution of fragment lengths, it can be used to characterize those bio- and synthetic macromolecules with complicated molecular architectures. 2.5 Topological Architectures Polymers are not only simply linear chains, but also the building blocks to construct topologically more complicated three-dimensional macromolecules. Some typical cases are listed below. 1. Linear polymers. Linear structure is the basic topological shape of polymer chains, as shown in Fig. 2.6a. The assembly of linear polymers normally contains a specific distribution of chain lengths. 2. Ring polymers. The ring contains no chain end, as shown in Fig. 2.6b. Since ring polymers cannot make entanglement with each other, their mobility is much higher than the linear polymers with the same chain lengths. 3. Branched polymers. There can be multiple branches on the linear chains, characterized by the degree of branching. There are several typical cases. The comb-like polymers contain all the branches derived from the same backbone chain, as shown in Fig. 2.7a. If the branches of comb-like polymers are chemically different from the backbone chain, we have the graft copolymers. A famous example for the random branching is the amylopectin (branched starch). The hyper-branched polymers (more often called dendrimers) may contain several levels of branching at the chain ends, like a Cayley tree, as shown in Fig. 2.7b. Chain branching destroys the sequence regularity of polymer chains, hindering crystallization and thus depressing the mechanical performance. Fig. 2.6 Illustration of (a) linear polymers and (b) ring polymers 2.5 Topological Architectures 27
2 Structure-Property Relationships Fig.2.7 Illustration of(a)comb-like polymers and(b)hyper-branched polymer Fig.2.8 Illustration of (a)diblock copolymers and(b)star polymers Long-chain branching and tree-like polymers exhibit quite different flow The can be a the pomheooicaerfomnc of hehaviors bulk linea processing. 4.Block copolymers.The multi-component systems are intramolecular,with each component occupying a certain length of chain sequences,as shown in Fig.2.8a. They can be diblock,triblock or even multi-block copolymers.Upon the change of composition,the microphase separation in block copolymers can fabricate various geometries of regularly packed microdomain patterns with nano-scale resolution,as will be introduced in Sect.9.3. 5.Star polymers.Derived from the same c nter,the star arms may belong to the .as shown in Fig.2.8b, the diffe rent species.The e arm e nano-scale microd 6.Polymer b or ea ch polymer cl e same solid surfaces of a rod or a flat-plate.When the graft density becomes high,polymer chains will stretch out due to the overcrowding on the surfaces,as shown in Fig.2.9.Polymer brushes can change the surface properties and thus give solid surfaces a responsive function. 7.Cross-linking networks.The degree of cross-linking is characterized by the density of cross-linking es of 6 -de cross-lin points in the networkas shown in Fig.10a.Examp the vulca ized rubber which e the er city of po er chain .and 0 inking are the phenol-formaldehyderesin.th
Long-chain branching and tree-like polymers exhibit quite different flow behaviors compared to linear polymers. They can be a good additive in the bulk linear polymers to improve the rheological performance of polymers upon processing. 4. Block copolymers. The multi-component systems are intramolecular, with each component occupying a certain length of chain sequences, as shown in Fig. 2.8a. They can be diblock, triblock or even multi-block copolymers. Upon the change of composition, the microphase separation in block copolymers can fabricate various geometries of regularly packed microdomain patterns with nano-scale resolution, as will be introduced in Sect. 9.3. 5. Star polymers. Derived from the same center, the star arms may belong to the same species, as shown in Fig. 2.8b, or to the different species. The arms with different components can form the nano-scale microdomain patterns as well. 6. Polymer brushes. One end of each polymer chain is anchored on the same solid surfaces of a rod or a flat-plate. When the graft density becomes high, polymer chains will stretch out due to the overcrowding on the surfaces, as shown in Fig. 2.9. Polymer brushes can change the surface properties and thus give solid surfaces a responsive function. 7. Cross-linking networks. The degree of cross-linking is characterized by the density of cross-linking points in the network, as shown in Fig. 2.10a. Examples of low-density cross-linking are the vulcanized rubber and chewing gum, which release the entropy elasticity of polymer chains, and only swell when in solvents. Examples of high-density cross-linking are the phenol-formaldehyde resin, the Fig. 2.7 Illustration of (a) comb-like polymers and (b) hyper-branched polymers Fig. 2.8 Illustration of (a) diblock copolymers and (b) star polymers 28 2 Structure–Property Relationships
2.6 Sequence Irregularities 29 Fig.2.9 Illustration of molecular structures of polymer brushes Fig.2.10 Ilus stnionoflby interpenetrated network (IPN) Cross-linking networ epolyeern(comonrdb the glas nd do 8.Interpenetrated netw d polymer swel the monomer solvent.and then initiate polymerization of the s olvent molecules t form another cross-linking network,as shown in Fig.2.10b.By this way.the interpenetrated network (IPN)allows two thermodynamically incompatible polymers to be mixed on the molecular level,and thus integrates their favorable properties.The case for the solvent monomers to perform only polymerization without further cross-linking,is called the semi-interpenetrated network. The complexity of macromolecular architectures is limited only by our imagi and our chemical As in the buil ks th mp mt e phys ar polymers are of esse rmining the physical 2.6 Sequence Irregularities Crystallization is a cor mpact-packing process of polymer chains,which is ver se itive to any in th geom chain ences.The f ers is thus restric d by the high conten irregularities.The I facto to characterize polymer microstructures.There are commonly three kinds of irregularities in chain sequences,i.e.chemical irregularities,geometrical irregularities and spatial irregularities
epoxy resin and the unsaturated polyester resin (commonly reinforced by the glass fibers), which exhibit a good thermal resistance, and do not even swell in solvents. 8. Interpenetrated networks. We can first make the cross-linked polymer swell in the monomer solvent, and then initiate polymerization of the solvent molecules to form another cross-linking network, as shown in Fig. 2.10b. By this way, the interpenetrated network (IPN) allows two thermodynamically incompatible polymers to be mixed on the molecular level, and thus integrates their favorable properties. The case for the solvent monomers to perform only polymerization without further cross-linking, is called the semi-interpenetrated network. The complexity of macromolecular architectures is limited only by our imagination and our chemical synthesis skills. As in the building blocks, the physical behaviors of linear polymers are of essential importance in determining the physical behaviors of high-level complex structures of macromolecules. 2.6 Sequence Irregularities Crystallization is a compact-packing process of polymer chains, which is very sensitive to any mismatch in the geometries of chain sequences. The crystallizability of polymers is thus restricted by the high content of sequence irregularities. Therefore, the sequence regularity is a very important chemical factor to characterize polymer microstructures. There are commonly three kinds of irregularities in chain sequences, i.e., chemical irregularities, geometrical irregularities and spatial irregularities. Fig. 2.9 Illustration of molecular structures of polymer brushes Fig. 2.10 Illustration of (a) cross-linking network and (b) interpenetrated network (IPN) 2.6 Sequence Irregularities 29