136
6 Melting of Molecular Crystals
systems consisting of molecules having a different isotopic composition of hydrogens, CH 4−n D n (n = 0 − 4) [35]. Although much can be learned from such models,
we do not treat them here because they deeply involve details specific to substances.
6.3 Molecular Deformation
The melting process described in the previous sections from the perfectly ordered
crystal to isotropic liquid is the one during which molecules acquire structural
entropy. If we consider the properties of molecules with the shape, another way of
acquiring structural entropy can be imagined. Such acquiring can happen in reality.
Molecular conformation plays an important role here.
A representative series of molecules, the properties of which are deeply related
to molecular flexibility, is n-alkanes, H–(CH 2 ) n –H (n = 2, 3, . . .). Compounds with
sufficiently large n are known as poly(ethylene), a representative of linear polymers.
Properties of linear polymers are understood based on their flexibility coming from
conformational degrees of freedom [40]. The C–C bonds in n-alkanes are the socalled single bonds and allow the internal rotation. To be concise, we first focus on
a molecule of ethane (CH 3 –CH 3 ). The molecular energy has a threefold symmetry
for the mutual rotation of methyl groups due to the tetrahedral bonding of the carbon atoms. Dihedral angles between planes defined by H a C a C b and C a C b H b (a and
b distinguish carbon atoms) in the molecular ground state are either of π or ±
π
3
(while neglecting the width coming from quantum effects). In the case of n-alkanes
with n ≥ 4, similarly, the dihedral angles are essentially either of π or ±
π
3
for two
planes defined by C a C b C c and C b C c C d . The conformations of C a C b C c C d with a dihedral angle of π and ±
π
3
are called trans and gauche, respectively.
9 The trans form
is more stable in energy than the gauche form in n-alkanes (and n-alkyl groups).
Figure 6.6 shows cumulative entropies of transition starting from an ordered crystal to isotropic liquid of n-alkanes (n = 2 − 29 with n being the number of carbon
atoms in a molecule). As is evident, the dependence on n is linear with an averaged slope (in the sense of least-squares) of 10.1 J K
−1 mol
−1 though they split into
two depending on the parity of n (even or odd) if inspecting in detail [49, 50]. The
dependence on the parity is known as the odd-even effect, which in thermodynamic
properties generally comes from the difference in aggregation structure due to the
different orientation of terminal –CH 2 –CH 3 bonds between molecules with odd and
even n, as explained in Sect. 9.4.1. The magnitude of the slope is comparable with
and slightly larger than R ln 3 (≈ 9.1 J K
−1 mol
−1 ), which is the expected magnitude
for the threefold disorder (over a trans and two gauche conformations) around a C–C
single bond. Namely, the n-dependence of the cumulative entropy indicates that the
threefold disorder realizes certainly in the isotropic liquid.
Acquiring entropy by the internal structural degrees of freedom in the course
of melting also occurs in compounds having alkyl groups. Figure 6.7 illustrates the
assignment of acquired entropies to individual degrees of freedom [51]. The assign9 Italic style is usually used for trans and gauche.
6 Melting of Molecular Crystals
systems consisting of molecules having a different isotopic composition of hydrogens, CH 4−n D n (n = 0 − 4) [35]. Although much can be learned from such models,
we do not treat them here because they deeply involve details specific to substances.
6.3 Molecular Deformation
The melting process described in the previous sections from the perfectly ordered
crystal to isotropic liquid is the one during which molecules acquire structural
entropy. If we consider the properties of molecules with the shape, another way of
acquiring structural entropy can be imagined. Such acquiring can happen in reality.
Molecular conformation plays an important role here.
A representative series of molecules, the properties of which are deeply related
to molecular flexibility, is n-alkanes, H–(CH 2 ) n –H (n = 2, 3, . . .). Compounds with
sufficiently large n are known as poly(ethylene), a representative of linear polymers.
Properties of linear polymers are understood based on their flexibility coming from
conformational degrees of freedom [40]. The C–C bonds in n-alkanes are the socalled single bonds and allow the internal rotation. To be concise, we first focus on
a molecule of ethane (CH 3 –CH 3 ). The molecular energy has a threefold symmetry
for the mutual rotation of methyl groups due to the tetrahedral bonding of the carbon atoms. Dihedral angles between planes defined by H a C a C b and C a C b H b (a and
b distinguish carbon atoms) in the molecular ground state are either of π or ±
π
3
(while neglecting the width coming from quantum effects). In the case of n-alkanes
with n ≥ 4, similarly, the dihedral angles are essentially either of π or ±
π
3
for two
planes defined by C a C b C c and C b C c C d . The conformations of C a C b C c C d with a dihedral angle of π and ±
π
3
are called trans and gauche, respectively.
9 The trans form
is more stable in energy than the gauche form in n-alkanes (and n-alkyl groups).
Figure 6.6 shows cumulative entropies of transition starting from an ordered crystal to isotropic liquid of n-alkanes (n = 2 − 29 with n being the number of carbon
atoms in a molecule). As is evident, the dependence on n is linear with an averaged slope (in the sense of least-squares) of 10.1 J K
−1 mol
−1 though they split into
two depending on the parity of n (even or odd) if inspecting in detail [49, 50]. The
dependence on the parity is known as the odd-even effect, which in thermodynamic
properties generally comes from the difference in aggregation structure due to the
different orientation of terminal –CH 2 –CH 3 bonds between molecules with odd and
even n, as explained in Sect. 9.4.1. The magnitude of the slope is comparable with
and slightly larger than R ln 3 (≈ 9.1 J K
−1 mol
−1 ), which is the expected magnitude
for the threefold disorder (over a trans and two gauche conformations) around a C–C
single bond. Namely, the n-dependence of the cumulative entropy indicates that the
threefold disorder realizes certainly in the isotropic liquid.
Acquiring entropy by the internal structural degrees of freedom in the course
of melting also occurs in compounds having alkyl groups. Figure 6.7 illustrates the
assignment of acquired entropies to individual degrees of freedom [51]. The assign9 Italic style is usually used for trans and gauche.