Energy, strain

THE ORIGIN OF STRAIN ENERGY A. Definition of Strain Energy 

To discuss and compare theoretically defined strain energies with those defined in terms of the experimental heats of formation for the cyclic, bicyclic and propellane molecules, we shall define group energies in the same manner as is done using the experimental heats of formation except that they will not be referenced to the standard states of the elements. The strain energy of a given group in a molecule is obtained by 

The results presented in this section make use of the 6-31G /6-31G calculations, since only for this basis has such a large set of molecules been treated in a consistent manner. However, some cyclic and bicyclic molecules have been calculated using the larger basis sets and, as demonstrated in the text, the results given here quantitatively mirror those obtained using the larger basis sets. 

Schleyer defines strain energy somewhat differently, and uses experimental values for the compounds in question to evaluate it (Schleyer et al., 1970). However, in practice, his calculated strain energies and ours differ by trivial amounts (perhaps with a rare exception). Using our scheme, the calculated values for inherent strain are as shown in Table 6. 

Compound Inherent strain (kcal mole-1) Compound Inherent strain (kcal mole-1) 

The normal and simple alkanes are calculated to have inherent strains of essentially zero. One can obtain from these numbers the actual strain energies, if desired, by adding the amount by which the enthalpy is raised due to admixture of the higher energy conformations. Such information is rarely of interest, however. 

Note that the strain energy of isopentane is small, but that in 

Small ring compounds such as cubane and norbomane arc strained for obvious reasons. Larger rings tend to be less strained, as perusal of Table 6 will show. 

It is obvious that the strain energy in a molecule is an energy difference between the molecule and some (strainless) reference standard. That reference standard is arbitrary and various different standards have been used in the earlier literature. It is desirable to have a useful, clear, and precisely defined way of calculating strain energies. It should be simple, straightforward, and completely transparent. We introduced such a method for use with our 1971 force field ° and used it in subsequent force fields (MM2, MM3, MM4). A detailed summary of the method is given by Burkert.  

The strainless standard for alkanes (MM4 and earlier) is defined by five parameters. These are the C-C and C-H bonds and three branching parameters. Me, Iso, and Neo, where the methylene group is a default quantity (with an energy value of zero). Methane is also given a strain energy of zero. 

However, real compounds frequently contain molecule specific and often large contributions to their energy, and hence to their strain energy, from their vibrational motions. These contributions are given by the terms POP and TOR that are used for thermodynamic calculations in molecular mechanics and in quantum mechanics. The relationship is given by Eq. (11.5). 

TAB LE 11.7. MM4 Heats of Formation, Strain Energies, and Dispersion Energies for Selected Alkanes  

Work must be done by the external forces acting on a body when the latter deforms and in the case of an elastic solid all this work is stored as potential energy of the distorted solid, or strain energy. The whole of this stored energy may be recovered when the external forces are removed from the elastic solid reversibly. 

Let the rod be perfectly elastic, which means that F is a singlevalued function of e. When the changes in the extension take place reversibly, the work done by the external forces as the extension is changed from ej to is  

If Hooke s law is obeyed and the rod has an unstretched length Lq and area of cross-section a, then  

This is a simple example of a situation where the stress and strain are not homogeneous. The problem is tackled by considering an element in which stress and strain can be considered as homogeneous and then integrating over the complete solid. 

The surface area of each end of the element is rd dr, where d0 is the angle subtended at the axis by the element, and the shear strain suffered by the element is 0r/L. The applied shearing forces needed to produce this strain are of magnitude  

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Figure 7-2. Strain energies [kj/mol] of three-membered ring systems.

The interaction potential (R) describes both bonding and non-bon ding in teraction s. Th e bon dm g interactions arc u snally form u -lated as a strain energy that is zero at some ideal configuration of the atoms and describe how the energy increases as the ideal con-figu ration is deform ed. Don d in g in teraction s ii su ally refer to atom s in the following relationships ... 

Fig. 11.40 Distribution of strain energy is two knotted polymer chains containing 35 (left) and 28 (right) carbon atoms. The strain energy is localised and most of the bonds immediately outside the entrance point to the knot. (Figure redrawn from Saitta A M, P D Sooper, E Wasserman and M L Klein 1999. Influence of a knot on the strenght of a polymer strand. Nature 399 46-48.)...

HEAT OF FORMATION AND STRAIN ENERGY CALCULATIONS (UNIT=KCAUMOLE)... 

File Segment 5-1 Heat of Fomiation and Strain Energy Output. 

The consistent force field (CFF) was developed to yield consistent accuracy of results for conformations, vibrational spectra, strain energy, and vibrational enthalpy of proteins. There are several variations on this, such as the Ure-Bradley version (UBCFF), a valence version (CVFF), and Lynghy CFF. The quantum mechanically parameterized force field (QMFF) was parameterized from ah initio results. CFF93 is a rescaling of QMFF to reproduce experimental results. These force fields use five to six valence terms, one of which is an electrostatic term, and four to six cross terms. 

PC Model has some features that are not found in many other molecular mechanics programs. This is one of the few programs that outputs the energy given by the force field and the heat of formation and a strain energy. Atom types for describing transition structures in the MMX force field are included. There is a metal coordination option for setting up calculations with metal atoms. There are also molecular similarity and conformation search functions. 

If we assume that there are certain ideal val ues for bond angles bond distances and so on itfol lows that deviations from these ideal values will destabilize a particular structure and increase its po tential energy This increase in potential energy is re ferred to as the strain energy of the structure Other terms for this increase include steric energy and steric strain Arithmetically the total strain energy ( ) of an alkane or cycloalkane can be considered as... 

Once requiring minicomputers and worksta tions many molecular mechanics programs are avail able for personal computers The information that strain energy calculations can provide is so helpful... 

Make molecular models of c/s and trans 1 2 dimethylcy clopropane and compare their strain energies... 

Mesitylene (13 5 trimethylbenzene) is the most stable of the tnmethylbenzene isomers Why2 Which isomer do you think is the least stable" Make a molecular model of each isomer and compare their calculated strain energies with your predictions Do space filling models support your explanation" ... 

Make molecular models of the two chair conformations of cis 1 tert butyl 4 phenyl cyclohexane What is the strain energy calculated for each conformation by molecular mechanics Which has a greater preference for the equatorial onentation phenyl or tert butyD... 

The distorted structure can be replaced by a more reasonable structure using an empir ical molecular mechanics calculation This calculation which is invoked m Spartan Build by clicking on Minimize, automatically finds the structure with the smallest strain energy (m this case a structure with realistic bond distances and a boat conformation for the SIX membered ring)... 

Molecular mechanics strain energies have another use They can also be used to com pare the energies of models that share the same molecular formula that is models that are either stereoisomers or different conformations of a single molecule (allowed com parisons are shown here)... 

SpartanBuild reports strain energies m kilocalories per mole (1 kcal/mol = 4 184 kJ/mol) m the lower left hand corner of the SpartanBuild window... 

The stmcture of DPXN was determined in 1953 from x-ray diffraction studies (22). There is considerable strain energy in the buckled aromatic rings and distorted bond angles. The strain has been experimentally quantified at 130 kj/mol (31 kcal/mol) by careful determination of the formation enthalpy through heat of combustion measurements (23). The release of this strain energy is doubtiess the principal reason for success in the particularly convenient preparation of monomer in the parylene process. 

Fracture mechanics (qv) tests are typically used for stmctural adhesives. Thus, tests such as the double cantilever beam test (Fig. 2c), in which two thick adherends joined by an adhesive are broken by cleavage, provide information relating to stmctural flaws. Results can be reported in a number of ways. The most typical uses a quantity known as the strain energy release rate, given in energy per unit area. 

ASTM D5045-91, "Plane Strain Fracture Toughness and Strain Energy Release Rate of Plastic Materials," A.nnualBook ofyiSTM Standards, ASTM Puhhcations, Philadelphia, 1993. 

Criteria of Elastic Failure. Of the criteria of elastic failure which have been formulated, the two most important for ductile materials are the maximum shear stress criterion and the shear strain energy criterion. According to the former criterion, from equation 7... 

If it is assumed that yield and subsequent plastic flow of the material occurs in accordance with the maximum shear stress criterion, then /2 may be substituted for in the above and subsequent equations. For the shear strain energy criterion it may be assumed, as a first approximation, that the corresponding value is G j fz. Errors in this assumption have been discussed (11). 

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