Concept of Structure–Property Relationships in Molecular Solids and Polymers

1.1 Introduction

Low molar mass organic molecules and polymeric materials are often found as solids and their physical properties are a consequence of the way in which the molecules are organized: their morphology. The morphology is a result of specific molecular interactions which control the processes involved in the individual molecules packing together to form a solid phase. Depending on the extent of the molecular organization, a crystalline solid, liquid crystals or amorphous solid may be formed. As we shall see later, the organization that is created at a molecular level sometimes also tells us about the macroscopic form of the material, but in other cases it does not, hence the subtitle of the book: 'from nano to macro organization'.

Synthetic polymers, often referred to as plastics, are familiar in the home as furniture, the frames for double glazed windows, shopping bags, furnishings (carpets, curtains and covering for chairs), cabinets for televisions and paper and paint on the walls. Outside the house plastics are used for rainwater pipes, septic, water and fuel storage tanks, garden furniture, water hoses, traffic cones and sundry other items which we see around us. Removal of all articles containing polymers from a room would leave it bare. Synthetic plastics form the basis for many forms of food packaging, containers for cosmetics, soft drink containers and the trays used in microwave cooking of food. Natural polymers such as wood, cotton and wool all exhibit a high degree of order and many biopolymers play a critical role in the human body.

Whilst a focus of this monograph is structure in polymeric materials, many of the factors that control the organization of these big molecules are best studied with lower molecular weight analogues. It is therefore appropriate to spend some time understanding small molecular systems before the consideration of the complexity of polymers is undertaken.

The physical properties of a material are dictated by its ability to self-assemble into a crystalline form. Polymer chemists have for many years sought to establish structure-property relationships that predict various physical properties from of knowledge the chemical structure of the polymer. Staudinger recognized that polymers or macromolecules are constructed by the covalent linking of simple molecular repeat units. This structure is implied in the phrase poly meaning many and mer designating the nature of the repeat unit. Thus poly(ethylene) is the linkage of many ethylene units:

H2C=C2H -> -(CH2-CH2)n-

Recognition of the nature of this process of polymerizationmade it possible to produce materials with interesting and useful properties, and brought about the discipline of polymer science. The value of 'n' indicates the number of monomers in the polymer chain.

In the last forty years, a very significant effort has been directed towards understanding the relation between the chemical structure of the polymer repeat unit and its physical properties. In the ideal situation, knowing the nature of the repeat unit it should be possible to be able to determine all the physical properties of the bulk solid. Whilst such correlations exist, they also require an understanding of the way in which the chemical structure will influence the chain-chain packing in forming the solid. Similar correlations can be created for the understanding of other forms of order in lower molecular weight materials.

1.2 Construction of a Physical Basis for Structure–Property Relationships

Why do structure-property relations apparently work?

1.2.1 Ionic Solids

In order to understand the basis for structure-property correlations, it is appropriate to consider the structure of simple ionic solids. A solid sodium chloride single crystal (Figure 1.1) can be constructed starting from the atomic species of sodium [Na+] and chlorine [Cl-]. Since each ion carries a single charge, pairing the ions to form an NaCl pair would create a lower energy state. This pair will have a minimum separation and would be charge neutral. The line formed between the two atoms can be considered to lie on the x-axis. If another pair of atoms is brought close to the first pair, then once more a minimum energy situation would be created if the two pairs of atoms are aligned but their orientations are in the opposite sense. This arrangement will have a lower energy than the isolated pair and the distance between the atoms will be slightly reduced compared with the isolated pair. A further reduction in energy will accompany bringing to the cluster a further pair of atoms. This latter pair will align in an opposite sense to the pair to which it attaches itself. Once more there will be a small change of separations to reflect the formation of a lower energy state. It is relatively easy to see that this process can be repeated and a sheet of atoms would be formed. As we will see later this same principle is used in considering attachment and growth of molecular and polymeric crystals.

The sheet of atoms formed by the process described above is not the lowest energy structure that can be formed. If this original sheet is sandwiched between similar states such that each of the Na and Cl atoms becomes surrounded by atoms of the opposite sign then a true minimum will be observed. If this order structure cannot be formed, because the entropy (disorder) is high, then the ensemble of atoms will be in the melt or gaseous state.

In the case of the NaCl crystal, this lowest energy structure is a cubic close-packed structure and results from each atom having six neighbouring atoms of opposite sign. Changes in the size of the ions and their charges lead to different types of packing being favoured. However, it is relatively easy to see that an average energy can be ascribed to a basic unit of the structure and this will reflect the physical properties of the bulk. Whilst the energy of the first pair can be calculated explicitly, adding additional elements means that the force field has to be averaged and will give rise to the problem of how one calculates the interaction of many bodies all interacting. The energy is the result of electrostatic (Coulombic) interactions between unit charges and in principle can be calculated by averaging all the interactions that will act on an atom chosen as the reference. The above example illustrates not only the lowering of the energy by surrounding an atom by other atoms but it illustrates that atoms in the surface will have a higher energy and we will meet this concept again when we consider polymers organizing at interfaces.

The total number and relative magnitudes of the Coulombic interactions and whether they are attractive or repulsive are taken into account by using a factor known as the Madelung constant, A. The lowest energy for the lattice ΔU(0K) (Coulombic) can for an ionic lattice be expressed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where |z+| and |z-| are, respectively, the modulus of the positive and negative charges, e is the charge on the electron (=1.602 x 10-19 C), ε0 is the permittivity of a vacuum (=8.854 x 10-12 F m-1), r is the intermolecular distance between the ions (in m), L is Avogadro's number (6.022 x 10-23 mol-1) and A is the Madelung constant.

The Madelung constant takes into account the different Coulombic forces, both attractive and repulsive, that act on a particular ion in a lattice. In the NaCl lattice, six Cl- atoms surround each Na+ atom. The coordination number -6 describes the number of atoms which surround the selected reference atom. X-ray analysis indicates that each atom is a distance of 281 pm from its nearest neighbour. To calculate the Madelung constant we consider the four unit cells that surround the selected reference atom. Firstly there are twelve Cl- ions each at a distance a from the central ion, and the Cl- ions repel one another. The distance a is related to r by the equation

a2 = r2 + r2 = 2r2and thus a = r [square root of] 2 (1.2)

Next there are eight Na+ ions each at a distance b from the central Cl- ion, giving rise to attractive forces. Distance b is related to r by the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

Further attractive and repulsive interactions occur, but as the distance involved increases, the Coulombic interactions decrease.

The Madelung constant, A, contains terms for all the attractive and repulsive interactions experienced by a given ion, and so for the NaCl lattice the Madelung constant is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

where the series will continue with additional terms for interactions at greater distances. In general, the larger the distances involved the smaller the contribution to the energy and the magnitude of A is dominated by the first and second neighbour interactions. Note that r is not included in the equation and the value of A calculated is for all sodium chloride types of lattices. Thus the Madelung constant is a single parameter that describes with other constants the energy of the lattice. In this system, the dominant forces are electrostatic and hence the picture of the atoms as spheres is a reasonable approximation to reality. The physical properties of sodium chloride can be calculated on the basis of a knowledge of the interaction between the atoms. This simple principle can be extended to molecular species and to polymers. Obviously as the molecular structure becomes more complex the problem of the calculation increases dramatically; however, the additivity principle often applies and reflects the appropriateness of mean field approximations in many cases. The A parameter is associated with a specific atom pair and changing the atoms will give another characteristic value. Examination of a number of pairs of such systems allows specific interactions to be identified which can be used additively to predict the properties of an unknown system. In the case of an atomic solid the dominant forces are electrostatic. In most organic materials, short-range van der Waals repulsive and attractive interactions are dominant and longer range electrostatic and dipolar interactions play a very important role in defining the final structure.

1.2.2 The Crystal Surface

A further important feature of physical predictions can be obtained from this simple model. If we consider the surface of the solid, it is relatively easy to see that the atoms in this sheet will have a slightly different energy from those in the bulk of the material. This excess energy was recognized by Gibbs and discussed in terms of surface tension for a liquid. Bringing a further layer of atoms to this surface — crystal growth — can lower the energy of the atoms in the surface or if the atoms are different this process is usually considered as absorption. NaCl is a very simple model and the question we will next address is whether this concept can be applied to covalently bonded systems.

1.2.3 Molecular Solid

The next step in the development of an understanding of the physical properties of polymers is to consider how a molecule such as dodecane forms single crystals. Crystals of dodecane are usually grown from a solution or from the melt by slow cooling. The dodecane molecule, CH3-(CH2)10-CH3 (Figure 1.2), has an all-trans conformation as a consequence of nonbonding repulsive interactions between hydrogen atoms on neighbouring carbon atoms. A higher energy gauche state exists in which the interaction between neighbouring atoms is greater than in the trans conformation. The distribution between the gauche and trans conformations is predictable in terms of statistical mechanics.

Following the process used to create the NaCl crystal, a low-energy state can be achieved if two of these all-trans dodecane molecules are brought close together and aligned. Following the logic presented above, a lowering of the energy will occur and a further reduction will be observed when a third and fourth molecule are brought up to the first two. This process would produce a layer of molecules extending in the y–z plane. A further reduction in energy would be achieved by the addition of another sheet of molecules on top of the first and so forth. The forces that govern the interaction between the molecules are now van der Waals interactions rather than the stronger electrostatic Coulombic interactions.

A question that could be asked is whether an even lower energy would arise if the molecules were to pack in a staggered array rather than being perfectly aligned. The proposed difference in order gives rise to nematic and smectic phases in liquid crystalline phases (Figure 1.3). The staggered form is analogous to the nematic phase, with the molecules aligned in one direction but disordered in at least one other direction. In the smectic phase, the molecules are aligned in a plane but may be misaligned between the layers and are closer to the lowest energy crystalline ordered structure than the nematic phase. The nematic will have a number of methyl-ethylene bond interactions and these will be less favourable than the ethylene-ethylene bond interactions. The topic of liquid crystals is discussed more fully in Chapter 3.

Dodecane can exist in a number of higher energy forms in which one or more gauche structures are incorporated into the chain backbone. The process of conformational change will involve the hydrogen atoms on neighbouring carbon atoms being brought into an eclipsed conformation. This eclipsed conformation is a higher energy state and inhibits the free exchange between the trans and the gauche conformation in which the energy has once more been minimized.

At any temperature above absolute zero there will be a finite population of the higher energy gauche state dictated by the Boltzmann distribution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

where ΔE is the energy difference between the gauche and trans states, g1 and g2 are, respectively, the degeneracy of the trans and gauche states at the temperature T and R is the gas constant. Since there are two gauche states, which are energetically degenerate, then the statistical factor is 1/2 and ΔE is the energy difference between the trans and gauche states. This temperature dependence of the conformation of many molecular species plays a critical role in determining their behaviour when cooled to form a solid. At the temperature of the melt phase, there will be expected to be a significant population of gauche conformations. The trans conformation is the lowest energy state and is able to nucleate crystal growth.

The molecules which are in the surface will also have the lowest energy state and whilst there will be defects in the crystal there is no reason to believe that the surface structure should not be different from that of the bulk. The energy of interaction between the dodecane chains can be seen to be the average of the interaction of one methylene chain with another. As with the case of NaCl, a single interaction parameter should describe the physical properties of the solid. In creating this average energy the interactions of the end chains needs to be included. Studies of the melting points for the paraffin homologous series indicate that the lower members lie on different curves depending whether they have odd or even numbers of carbon atoms. The equilibrium crystal structure is also different and can in part be explained by the way in which the methyl groups at the end of the molecules interact. As the chains become longer this odd–even effect disappears and is nonexistent for n greater than about 20 (Figure 1.4).

1.2.4 Low Molar Mass Hydrocarbons

The dodecane molecule in the liquid state will be expected to have on average one gauche state per molecule at room temperature but in the solid, however, it will have a structure that is predominantly made up of the all-trans form. The enthalpy of interaction compensates for the required loss of entropy in the crystallisation process. In the case of the n-alkanes, the bond lengths for the C–C and C–H bonds are, respectively, 1.5 and 1.10 Å and the C–C–C bond angle is 112° from studies of the solid. The H–C–H bond angle has been found to be 109°. The conformational changes can be described by a potential energy diagram (Figure 1.5).

Abe et al. have shown that the potential energy profile can be reproduced by selecting a barrier to the interchange between the trans and gauche forms of 12 540 J mol-1 and the energy difference between the two conformations has a value of 2090 J mol-1. The energy and barrier to rotation are a result of nonbonding repulsive and attractive interactions between the hydrogen–hydrogen and hydrogen-carbon atoms on neighbouring carbon atoms.