Thermal Analysis of Polymeric Materials Part 4 pdf

The empirical rule of change of the heat capacity when goingthrough a glass transition, is described in Sect.. The plastic crystal consist,however, of molecules their mesogens that are a

Fig 2.102

Fig 2.101

changes in entropy which are coupled with the disordering are listed Details aboutthese data are presented in Sect 5.4 Usually these transitions are first-ordertransitions (see Sect 2.5.7) All mesophases, when kept for structural or kineticreasons from full crystallization on cooling, display in addition a glass transition, as

is indicated on the left-hand side of Fig 2.103 The glass transition leads to a glass

2.5 Phases and Their Transitions 167

Fig 2.103

with mesophase order, but without the large-amplitude motion The recognition ofmesophases and mesophase glasses as states of intermediate order and their study bythermal analysis has become of great importance for the understanding of themultitude of materials The empirical rule of change of the heat capacity when goingthrough a glass transition, is described in Sect 2.5.6

2.5.2 Phases of Different Sizes

It was noted about 150 years ago that the properties of phases change when theirdimensions decrease to the micrometer scale This observation was first made afterthe discovery of colloids Today, such small phases are better called microphases[27] In microphases, the effect of the surfaces cannot be neglected Surface freeenergies and charges (surface potentials) govern the properties and stability (or meta-stability) of the microphases Similarly, in Fig 1.6 it is shown that molecules may

be classified into three types, small, large and flexible, and large and rigid In thissection the changes of phase size and molecular size are analyzed briefly

When forming a crystal via nucleation and growth, as described in Sects 3.5 and3.6, the linear growth rates in the various crystallographic directions determine theinitial, usually metastable, crystal shape Linear, flexible macromolecules, forexample, chain-fold to lamellar crystals with much faster growth in the crystallo-graphic a- and b-directions than in direction c, along the chain direction, as outlined

in Fig 2.104 The folds accumulate in the ab-surface (see Sect 5.2) By limiting theavailable material to a few macromolecules, the initial liquid takes the shape of asmall droplet with a micrometer diameter, and the crystal is also a microphase, asassumed in the figure On annealing, the initial, kinetically-determined morphology

Fig 2.104

(shape) may approach equilibrium, as shown in the sketch at the bottom The majorrearrangement of the molecules within the crystal requires sufficient mobility, as hasbeen observed, for example, in the condis mesophase of polyethylene Theequilibrium crystal must have a minimum of the positive surface free energy Based

on this principle, Wulff devised in 1901 a construction to establish the equilibriumshape (Wulff construction) By drawing normals from a point within the crystal tothe various possible surfaces with lengths proportional to the surface free energy, theinnermost complete body of these surfaces is the equilibrium shape The propervolume can be achieved by adjusting the lengths without changing the proportions.The equilibrium polymer crystal should have, thus, its large dimension at right angles

to the high energy fold surface ab, as indicated in Fig 2.104

New properties arise when the molecular sizes increase to the dimensions of thephase, as for example in the case of a typical polyethylene of 500,000 Da It has acontour length of 2.8m and can easily cross microphase boundaries This traversing

of the surface makes neighboring phases interact strongly, in contrast to the weakinteractions by intermolecular forces (see Fig 1.5)

The lower limit of the size of crystals of macromolecules may be as small as 2 nm

and must then be called a nanophase as displayed in Fig 2.98 The noncrystalline

material surrounding the macromolecular crystals, containing folds, chain ends,noncrystallizable repeating units, and tie molecules, has similarly small phasedimensions In nanophases the opposing surfaces of a phase area are sufficientlyclose to interact so that no bulk phase exists anymore Macromolecules traversing

a nanophase will strongly link the surfaces; but even weaker forces, such as caused

by polarity at the interface, or by charges in the surface, may bridge nanophases andmay give rise to special properties of the nanophases For characterization,nanophases need a detailed study of composition, physical state, molecular structure,

2.5 Phases and Their Transitions 169

and molecular motion The macromolecules can now traverse not only one, butseveral or many phase domains and influence the macroscopic properties

The limit of nanophases towards smaller sizes occurs at perhaps one nanometer.Two causes exist for this limit: The increasing loss in homogeneity due to theinherent molecular structure and their fluctuations in energy makes the recognition

of a homogeneous phase impossible; and for copolymers, the decreasing gain inenthalpy on mixing due to the larger surface contribution relative to the entropy loss

on phase separation makes the nanophase unstable compared to a solution

This short summary illustrates that three size-ranges must be considered for eachtype of phase (macrophases, microphases, and nanophases) In addition, the phasestructure may be different for each of the three types of molecules This leads to atotal of nine possible situations for each of the nine condensed phases shown inFig 2.103 Of the 81 combinations, 57 have been suggested to be possible at thebeginning of Sect 2.5 [27] Thermal analysis is the main technique to distinguish andidentify this multitude of different systems

2.5.3 Mesophases

The term mesophase, introduced in Sect 2.5.1, was first coined in 1922 by Friedel

to describe mainly liquid crystals which are known since 1888 They are related toliquids, but maintain a certain degree of orientational order, as shown schematically

in Fig 2.105 A list of characteristic properties of liquid crystals is given in the leftcolumn of Fig 2.107, below In the examples shown in Fig 2.105, the liquidcrystalline order is due to an elongated, rod-like or flat, board or disk-like segment

of the molecules, the mesogen The left example in Fig 2.105 is a two-dimensionalrepresentation of a nematic liquid crystal (Gk.=, thread, from the thread-likeinterference patterns of nematic liquid crystals under the polarizing microscope) Thenematic phase shows orientation of the mesogens in only one direction The rightexample is a macromolecular smectic liquid crystal (Gk.);, soap) In this casethe limited orientational order is in two dimensions The example is of a polymer thathas a mesogen included in the otherwise flexible chain For example, CH2-sequencescan link the mesogens and providing the mobility needed to give a liquid-crystallinecharacter A typical example of a mesogen is shown at the bottom of Fig 2.105 Theorientation in the liquid crystals gives rise to the birefringence and its high mobilityallows the use of liquid crystals in display devices

Soaps and lipids are also liquid crystals These molecules consist of two parts thatwould be phase-separated, if not connected by strong bonds In the pure state thesemolecules arrange such that the junction between their two incompatible segmentsform an interface ( | ) between two phase areas, for example, in sodium stearate, asoap: Na+ OOC( | )(CH2)16CH3, and in lipids as seen in Fig 2.106 The domainsize of these phases is about one nanometer in the direction of the molecule, i.e., theyare nanophases On crystallization, the molecules of the liquid crystals have to packclosely, which is not always possible for more complicated structures, so that glassesare common low-temperature phases for such molecules For the soaps and lipids,

a nanophase-separation remains in the crystals Sometimes the two types ofnanophases within the overall crystal undergo separate phase transitions

Fig 2.106

Fig 2.105

Plastic crystals are more closely related to the classical crystals They have fullpositional order as shown in the sketch in Fig 2.105 The plastic crystal consist,however, of molecules (their mesogens) that are almost spherical and can start torotate within the crystal at a given transition temperature Figure 2.107 contains a list

of typical properties of plastic crystals and allows a comparison to liquid crystals.The plastic crystalline state was first recognized in the 1930's Most plastic crystals

2.5 Phases and Their Transitions 171

Fig 2.107

have a cubic crystal structure Due to the rotation of the molecules in the crystal,their actual symmetry is averaged to a sphere, eliminating both the birefringence andcausing an entropy of fusion similar to the metals and noble gases which havespherical motifs (see Sects 5.4 and 5.5) Cubic crystals have also many slip planesand with rotating, weakly bound molecules, the crystals easily deform, they areplastic Metals with similar crystal structures and spherical motifs, but strongerbonding, still show ductility, but no plastic-crystal behavior Both liquid and plasticcrystals show conformational mobility and disorder if the basic molecule is flexible,i.e., can change to different conformational isomers

Conformationally disordered crystals (condis crystals) were discovered in the1980’s They show positional and orientational order, but are partially or fullyconformationally mobile The condis crystals complete the comparison ofmesophases in Figs 2.103 and 2.107 Linear, flexible molecules can show chainmobility that leaves the position and orientation of the molecule unchanged, butintroduces large-amplitude conformational motion about the chain axis Again, thesymmetry of the molecule is in this case increased Condis crystals have often ahexagonal, columnar crystal structure Typical examples of condis crystals are the

high-temperature phase of polyethylene, polytetrafluoroethylene,

trans-1,4-polybutadiene, and the low-temperature phases of soaps, lipids and other crystal forming, flexible molecules

liquid-Figure 2.108 illustrates the chemical structure and a thermal analysis curve of atypical small molecule with liquid-crystal and condis-crystal phases, OOBPD Themesogen is the rigid bisoxybenzalphenylenediamine Two flexible octyl groupsenable conformational disorder by rotation about the CC and OC bonds The letter

N represents the nematic phase, letters C, I, G’, and H’ the increasingly better orderedsmectic phases, and K designates condis phases Note that phase K has still not

Fig 2.108

Fig 2.109

reached the heat capacity expected for the solid crystalline or glassy state indicated

by the thin, bottom line (vibration-only crystalline heat capacity, see Sect 2.3) Since

no further crystallization occurs, a glass transition is expected and is seen in thermalanalysis at about 350 K More details about this compound are given in Sect 5.5.4.Fullerene, C60, is an example of a molecule with a plastic-crystal phase Itsstructure is given in Fig 2.109 together with the other two allotropes of carbon,

2.5 Phases and Their Transitions 173

Fig 2.110

Fig 2.111

diamond and graphite Its calorimetry is discussed in Sect 4.2.7 (see also Figs.2.39–41) Figure 2.110 is a thermal analysis trace (DSC, see Sect 4.3) Thetransition starts rather broad and then becomes sharp as full rotation becomespossible More details about this transition are available through 13

C nuclearmagnetic resonance experiments Figure 2.111 is a recording of the spin-latticerelaxation time T1 It is a measure of the rotation of the molecule Three models

Fig 2.112

Polytetrafluoroethylene and trans-1,4-polybutadiene are two examples of

macro-molecular condis crystals The heat capacity of polytetrafluoroethylene is shown in

Fig 2.63, that of trans-1,4-polybutadiene is illustrated in Fig 2.112 Both polymers

have two endothermic transitions At low temperature they show an endotherm at thedisordering temperature, Td, on going from the crystal to the condis phase, then theyultimately melt at Ti(isotropization temperature) In Fig 2.112 the motion of trans-

1,4-polybutadiene is characterized with1

H solid-state NMR, using the line width ofthe signal as a measure of mobility The line width can be modeled quantitatively interms of its second moment (G2

) The first narrowing is due to the glass transition ofthe not-crystallized, amorphous fraction of the semicrystalline polymer, the second,due to the conformational mobility gained at Td, the third to final melting (isotro-pization) at Ti

Figure 2.113 indicates with an entropy plot that only the trans isomer of the butadienes displays a stable mesophase The crystals of the cis isomer have a more

1,4-helical structure which packs less well in the crystal and needs much more space torotate into a conformationally disordered structure As a result of this different

structure, the cis polymer melts at a lower temperature, and in one step.

2.5 Phases and Their Transitions 175

Fig 2.113

2.5.4 Mesophase Glasses

All mesophases have some large-amplitude type of motion On cooling themesophases have two paths to the solid state as seen in Fig 2.103 Either themesophase orders to the crystal state, or the large-amplitude motion changes to thecorresponding vibrations without change of order In the second case there is noentropy of transition to the solid state (see Sect 2.5.6), only the heat capacity changes

to the vibration-only case, discussed in Sect 2.3 The mesophase glass differs fromthe amorphous glass by possessing the order of the mobile mesophase Eachmesophase, thus, has a corresponding glass, as was discussed in Sect 2.5.1 andshown in Fig 2.103 Liquid crystal glasses were seen already in the 1930’s [28].Liquid crystals are quite similar to the liquids in their mobility, their glasstransition is similar to the glass transition of an amorphous liquid Figure 2.114shows a special example The monomer of the chosen polymer, acryloyloxybenzoicacid, does not have a liquid crystalline phase On polymerization, an amorphousliquid results that changes with time to a liquid crystal on dimerization of theoxybenzoic acid side-group via hydrogen bonds The thermal analysis of Fig 2.114shows in its upper trace a sample quenched to the amorphous glass before ordering

A normal glass transition occurs on heating, followed by ordering to the liquid crystal

at To On the second cooling, the LC glass is formed, which, on reheating (bottomtrace) shows a glass transition of similar magnitude, but at higher temperaturebecause of the dimerization Glass transitions also have been observed for plasticcrystals and condis crystals Depending on the degree of cooperation necessarybetween neighboring molecules, narrow or broad glass transition regions result Forsome additional information, see Sect 5.5

Fig 2.114

2.5.5 Thermodynamics and Motion

Thermodynamics and motion can be used as a base for an operational definition ofthe solid state A solid is a phase below its glass- or melting-transition temperaturewhere the molecular motion is almost completely restricted to small-amplitudevibrations Both transitions are easily determined by thermal analysis (the operation).Recently it has become possible by simulation on supercomputers to establish the linkfrom the microscopic thermal motion of macromolecules to the macroscopic thermalanalysis By solving the equation of motion, one can produce a detailed movie ofmolecular motion (see Sect 1.3.4, Fig 1.47) At high temperature, conformationaldisorder is seen, i.e., the crystal can change to a condis state Note that even theconformational motion occurs in a picosecond time scale (see Sect 5.3.4)

The meaning of the word transition is not specific In Fig 2.115 a dictionary

definition is listed, which states: Transition means just a passing from one condition

to another The basic transitions of interest to thermal analysis are described in theclassification scheme of Fig 2.103 In the bottom half of Fig 2.115 the properties

of the condensed phases are reviewed and their transitions identified On the left side,the solid states are characterized by their order The mobility and degrees of packingare listed at the bottom, and the physical states above the diagram The mobilemesophases appear at higher temperatures via glass transitions (at Tg) or disorderingtransitions (at Td) The liquid phase (melt) is reached with an additional transition,the isotropization (at Ti), or by bypassing the mesophases from the glass via a glasstransition (at Tg) or a one-step melting from the crystal (at Tm) The liquid is still acondensed phase, meaning that the constituent motifs, the atoms, ions, or molecules,are more or less touching The gas phase is dilute, in contrast to the condensed

2.5 Phases and Their Transitions 177

Fig 2.115

phases Large empty spaces exist between the motifs Its connection to thecondensed phases is shown in Fig 2.115 by boiling from the liquid phase (at Tb) orthe direct transition from the solids by sublimation (at Ts)

The motions observed for a specific material are dependent on the molecularstructure Spherical atoms cannot exist as mesophases, except when included as asolvent (as in lyotropic liquid crystals) Only small, rigid molecules can exist asplastic crystals, while condis crystallinity is reserved for flexible molecules Sinceall degrees of freedom are at low temperature vibrational with a limiting heat capacity

of 3R per mole of atoms, and all large-amplitude motion decreases to R/2 per degree

of freedom when losing the potential energy contribution, one would expect that theheat capacity decreases at every transition to higher-temperature phases that possessmore large-amplitude motion The experimental data are not in agreement with thisprediction (see Sect 2.3) The main reason for the often observed increase in heatcapacity at glass and disordering transitions is the need of additional potential energy

to create the increased volume for the large-amplitude motion

The macroscopic, thermodynamic description of states is achieved through theirmajor functions and variables of state listed in Fig 2.116 One can deduce from themicroscopic picture of the states of matter that the enthalpy of the gas must be largest,and of the solid, smallest The larger the packing density, the stronger are theinteractions between the motifs decreasing the enthalpy On transition to a new,higher-temperature phase, this interaction must be overcome by an endothermic heat

of transition

The magnitude of the entropy can also be estimated from the microscopicdescription One expects the entropy of the gas to be largest because of the largedisorder in this dilute state The crystal, on the other hand, because of its order,should have the smallest entropy; in fact, the third law of thermodynamics sets the

Fig 2.116

entropy of an ideal, equilibrium crystal equal to zero at the absolute zero oftemperature (Sect 2.2.4) The free enthalpy, finally, can be derived from enthalpyand entropy as shown Its significance lies in the fact that it tells which state is moststable at a given temperature as shown in Figs 2.84–88

The temperature dependencies of enthalpy, free enthalpy, entropy, and heatcapacity of glasses and crystals are schematically indicated in the three drawings atthe bottom of Fig 2.116 (see also Fig 2.23) Enthalpy, entropy, and heat capacityincrease with temperature, while the free enthalpy decreases The reasons for thetrends of these functions with temperature can be derived from a detailed understand-ing of heat capacity (see Sects 2.3) The link between heat capacity and enthalpy isobvious from its definition (Fig 2.10) Entropy is similarly related to heat capacity

in Fig 2.19 (see also Fig 2.22) The free enthalpy is then fixed by G = H TS

2.5.6 Glass Transitions

The glass transition is a much more subtle transition than melting or evaporation InFig 2.117 the change in heat capacity on going through the glass transitiontemperature is drawn after a typical polymer There is a jump in the heat capacity,but there is no indication of a heat of transition If there is no heat of transition, therecan also be no entropy of transition

The increase in heat capacity Cpalways occurs over a temperature range of 5 to

20 K, and the jump is often 11 J K1mol1of mobile units in the liquid This means,for a monatomic liquid the decrease in heat capacity at the glass transition is

11 J K1mol1 Macromolecules, such as polyethylene, (CH2)x, change in heatcapacity by approximately 11 J K1(mol of chain atom)1(see Appendix 1) Todescribe the glass transition, the temperature of half-vitrification, T, should be

2.5 Phases and Their Transitions 179

A study of large lists of glass transition temperatures reveals that the ratio of themelting temperature to the glass transition temperature is often (but not always)between 1.5 and 2.0 The glasses of small molecules, such as water and ethylalcohol, fulfill this rule also Their ratios are 1.91 and 1.64, respectively The glasstransition, thus, is reached at a much lower temperature than the melting transition,

so that for materials applications, crystalline substances should be preferred Again,exceptions are known Poly(oxy-2,6-dimethyl-1,4-phenylene) (PPO™, GeneralElectric) has a Tgof 482 K and a Tmof 580 K (ratio 1.20), sufficiently low that onpartial crystallization, Tgincreases due to strain caused by the small crystals, and Tmdecreases because of the small size of the crystals, so that Tgmay exceed Tm(see therigid amorphous fraction of PPO in Sect 6.2.2) Similar closeness of Tgand Tmisobserved in long-side-chain polymethacrylates, where the glass transition is fixed bythe backbone of the polymer and the melting transition by its side-chain length.From the measurement of heat capacity one can derive the free enthalpy as drawnschematically in Fig 2.118 Since the heat capacity of the liquid is always larger thanthe heat capacity of the glass (see 2.117), the free-enthalpy curve of the liquid musthave the larger curvature since (02G/0T2

) = C/T (see Fig 2.19) and when

Fig 2.118

extrapolated to below Tg, it lies lower than the glass free enthalpy At the glasstemperature, both free enthalpies are equal, i.e., the two curves touch In addition, theslopes of the curves are identical because of their identical heat capacities (seeSect 2.3)

The question that one must answer is then: How is it possible for such a glasstransition actually to occur, since G of the liquid is less and becomes equal to G ofthe glass only at Tg? In order to stay in equilibrium, it should be forbidden to crossfrom Gliquidto Gglassat Tg This paradox can be resolved only by knowing that below

Tgthe liquid state does not exist Microscopically, Tgis the temperature where theliquid-like large-amplitude motion stops The nature of the stopping precludessupercooling on cooling, and similarly, superheating on heating Whenever theexperimental time is too short for the molecules to readjust to the changes intemperature, one reaches the glass transition This time-dependence is an indicationthat the transition does not occur under equilibrium conditions Different coolingrates produce different glasses, and each glass will have a different free enthalpy

In contrast to the well-defined equilibrium crystals, there exist a multitude ofglasses of the same chemical structure, differing only in the cooling or annealinghistory Also, an analogous extrapolation of the liquid and solid free enthalpiesbeyond the transition temperature, as possible for crystals is not permitted as implied

by the parentheses in Fig 2.118

A detailed discussion of the dependence of the glass transition not only oncooling, but also on heating is given in Sect 6.3 For the present, it suffices tosuggest that a recorded glass transition corresponds only to a glass cooled at a givenrate, i.e., with a fixed thermal history For full specification of a glass transition, thecooling rate or other experimental parameters of the thermal history, and possiblyalso mechanical and electrical history must be specified

2.5 Phases and Their Transitions 181

In the case where two phases are in contact and at equilibrium, the stability ofboth phases must be equal This requires that the two free enthalpies per mole, G’,must be the same at the given temperature and pressure In addition, any infinitesimalchange dG’ of each phase must also be equal so that the equilibrium is stable assuggested in Fig 2.79 These conditions of equilibrium between phases I and II arewritten at the top of Fig 2.119 By equating expressions similar to those derived forthe free enthalpy in Fig 2.19, additional information for the phase transitions can beobtained Since the changes dG are expressed per mole, there is no change in thenumber of moles, n, and each side of the equation has only two terms This equationallows only one independent choice between the two variables T and p—i.e., if twophases are in contact and in equilibrium, there is only one degree of freedom for a

one-component system This statement is called the phase rule and can be written

more general as: P + F = C + 2, where P is the number of phases, F is the degrees offreedom, C is the number of components, and 2 represents the two variables p and T.For a one-component system, no degrees of freedom or independently adjustablevariables are possible for three phases in contact The third phase would give rise toanother equation between, let us say, phases II and III, of the same type as written forphases I and II Two equations with two variables permit computation of bothvariables There is thus no degree of freedom In a p–T diagram this situation isrepresented by a point (a triple point) (see also Sects 2.3.5 and 2.3.7 and Sect 1.4.3).The definition of components for linear macromolecules is somewhat compli-cated One must distinguish between processes that occur with parts of molecules,i.e., strongly coupled components which must keep their positions within themolecule (as in copolymers, see Sect 3.4), and processes with whole molecules, i.e.,where there exist only weakly bound components that can easily segregate Moredetails about multi-component systems are given in Chap 7

The most far-reaching transitions are melting and evaporation They involve theloss of order in the case of melting, and the change from a condensed to a dilute state

in the case of evaporation All other transitions are of lesser magnitude In Fig 2.119

it is indicated how the enthalpy changes in going from state I to state II To stay inequilibrium during a transition, the change in Gibbs energy, G, must be zero.The derivatives of G with respect to temperature, on the other hand, do not have

to be zero This suggests a way to characterize the transitions thermodynamically.Ehrenfest suggested that a transition for which (0 G/0T) (=  S, see Fig 2.19) is notequal to zero be called a first-order transition [25] A second-order transition wouldanalogously have the first derivative (0 G/0T) equal to zero, but the secondderivative (02 G/0T2

) would not equal zero The second derivative of G is equal

Fig 2.119

to Cp/T, as shown in Fig 2.119 This condition is superficially fulfilled in a glasstransition at a fixed time scale, but the time-dependence of Tg indicates that thetransition should not be considered an equilibrium transition

To discuss melting and evaporation, one can make use of the schematic drawing

of the free enthalpy in Fig 2.119, which applies to a one-component, pure substance

At low temperature the crystalline state is most stable The crystals are stable up tothe melting temperature, Tm At this temperature, the crystal must change to theliquid, in order to remain stable The change of state is connected with a change inthe slope of G—i.e., the transition must be of first order The system changes stateswith a discontinuity in entropy as well as enthalpy Increasing the temperaturefurther, the limit of stability of the liquid is reached at the boiling temperature Tb.Here again, on going from the liquid to the gas there is a change in the slope of G, sothat it is also a first-order transition with an entropy and enthalpy of evaporation

To describe the thermodynamic equilibrium behavior at the transition, one canwrite the boxed equations in Fig 2.119 Inspection of these equations shows thatthermometry, discussed in Sect 4.1, can give a characterization of a material if there

is information on either the heats or entropies of transition Some help forinterpretation comes from the fact that related materials have similar changes indisorder—the entropies of transition are similar for structurally related materials.Richards’s rule, for example, which applies to solids that melt by producingspherical, mobile motifs, such as noble gases and metals, says that the entropy ofmelting, ... Calorimetry J Thermal Anal and Calorimetry 54: 43 7? ?46 5

Temperature-20 See for example: Hirschfelder JO, Curtis CF, Bird RB (19 54) Molecular Theory of Gasesand Liquids Wiley, New York (§11.4b)

21... discussion of the assignment of the glass transition, see Syler RJ, ed (19 94)

On the Assignment of Glass Transition Temperatures Using Thermomechanical Analysis. ASTM Symposium, Atlanta, March 4? ??5,... 4? ??5, 1993, ASTM STP 1 249 , Am Soc Testing ofMaterials, Philadelphia For the rules of Cpsee: Wunderlich B (1960) J Phys Chem 64:

1052 For larger lists of macromolecular glasses