# Structures of the High-Temperature Solid Phases of the Odd-Numbered Fatty Acids from Tridecanoic Acid to Tricosanoic Acid

Gabin Gbabode, Philippe Négrier, Denise Mondieig, Evelyn Moreno Calvo, Teresa Calvet, Miquel àngel Cuevas-Diarte

► To cite this version:

Gabin Gbabode, Philippe Négrier, Denise Mondieig, Evelyn Moreno Calvo, Teresa Calvet, et al. Structures of the High-Temperature Solid Phases of the Odd-Numbered Fatty Acids from Tridecanoic Acid to Tricosanoic Acid. Chemistry - A European Journal, 2007, 13 (11), pp.3150-3159. 10.1002/chem.200600955. hal-01553049

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<figure><img src="image_1.png" /></figure>
# Structures of the High-Temperature Solid Phases of the Odd-Numbered Fatty Acids from Tridecanoic Acid to Tricosanoic Acid

Gabin Gbabode,*[a] Philippe Negrier,[a] Denise Mondieig,[a] Evelyn Moreno Calvo,[b] Teresa Calvet,[b] and Miquel Àngel Cuevas-Diarte[b]

**Abstract:** Crystal structures of the high temperature phases of odd numbered fatty acids ($C_nH_{2n-1}OOH$) from tridecanoic acid ($C_{13}H_{25}OOH$) to trico sanoic acid ($C_{23}H_{45}OOH$) are presented in this article. They have been determined from high quality X ray powder diffraction patterns. Two types of high temperature phases are adopted: one monoclinic $A2/a$ with $Z=8$ for the fatty acids with $n=13$ and $n=15$, de

noted as $C''$, and one monoclinic $P2_1/a$ with $Z=4$ for the longer chain fatty acids, denoted as $C'$. It appears that the packing arrangement of the alkyl chains and of the carboxyl groups is similar in all of the structures. Howev

er, the arrangement at the methyl group interface differs between the $C'$ and $C''$ forms. A survey of the intermolecular interactions involved in these polymorphs coupled with a study of the effects of temperature on the structures have led us to a better understanding of the arrangement of the molecules within the high temperature solid phases of odd numbered fatty acids.

**Keywords:** bilayers • hydrogen bonds • polymorphism • thermal motion • van der Waals interactions

## Introduction

Higher fatty acids of the formula $C_nH_{2n-1}OOH$ are composed of long, saturated aliphatic chains with a carboxyl group at one of their extremities. It is well known that long chain aliphatic compounds adopt several solid forms, with different configurations of the packing of the molecules in and between the layers.[1–5] Typically, the A', B', C', C'', and D' forms[6–9] are adopted by the saturated fatty acids with an odd number of carbon atoms, whereas the A, B, C, and E forms[6–10] are adopted by those with an even number of carbon atoms. All of these fatty acids exhibit at least one solid transition prior to melting,[7,9] and we usually distinguish the high temperature phase that is stable just below the melting point from the low temperature phase observable at lower temperatures. For odd numbered fatty acids, the high temperature phase may be the C' or the $C''$ form.[6]

For years, only the structure of the $C'$ form of hendecanoic acid ($C_{11}H_{21}OOH$) had been reported,[11] and, moreover, it had not been fully resolved (two dimensional (2D) structure). Recently, Bond solved the crystal structures of the high temperature solid phases of all fatty acids from hexanoic acid ($C_6H_{11}OOH$) to pentadecanoic acid ($C_{15}H_{29}OOH$) from single crystals.[6] The results obtained for the odd numbered fatty acids showed that, just below their respective melting points, the fatty acids with chain length from $n=7$ to $n=11$ adopt the $C'$ form, whereas tridecanoic acid and pentadecanoic acid adopt the $C''$ form.

In the present article, we focus on longer chain odd numbered fatty acids, specifically those with 13 to 23 carbon atoms. These compounds have high enthalpies of melting and they are also thermo adjustable by means of their chain length. These fatty acids, as well as their mixed samples, could therefore be potential candidates for applications in the field of energy storage or thermal protection, as has been demonstrated for alkanes[12,13] and alkanols.[14,15] In this context, it is important to know the crystal structures of the high temperature solid phases, as it is these that are involved in the melting process.

Crystal structure determination of the high temperature phases of odd numbered fatty acids is not an easy task. These solid forms are only stable in a narrow temperature range just below the melting point, and this temperature in terval becomes narrower with increasing chain length.[16,17]

<footnote>[a] Dr. G. Gbabode, Dr. P. Negrier, Dr. D. Mondieig
CPMOH, Université Bordeaux I
Alliages Moléculaires et Stockage d'Énergie
351, cours de la Libération, 33405 Talence Cedex (France)
Fax: (+33)540006686
E mail: gabin g@yahoo.fr</footnote>

<footnote>[b] E. Moreno Calvo, Prof. T. Calvet, Prof. M. À. Cuevas Diarte
Departament Cristallografia, Mineralogia i Dipòsits Minerals
Facultat de Geologia, Universitat de Barcelona
Martí y Franqués, s/n, 08028 Barcelona (Spain)</footnote>
The growth of single crystals is therefore very complicated, and the degree of difficulty goes on increasing for the longer acids. Some attempts have been made to grow such crystals, but without success. As a consequence, we have now determined their crystal structures from X ray powder diffraction patterns measured at the appropriate temperatures. The module POWDER SOLVE[18] of the MATERI AL STUDIO application[19] has been used to resolve the crystal structures.

In the first part of this paper, the polymorphism of the fatty acids from C₁₃H₂₅OOH to C₂₃H₄₅OOH is briefly described. Data from both calorimetric and X ray powder diffraction measurements are presented. A detailed analysis of the X ray powder diffraction patterns of the high temperature solid forms is then described. Their crystal structures are presented and the packing arrangement of the molecules throughout the studied series is discussed. In particular, the results obtained for tridecanoic acid and pentadecanoic acid are compared with those obtained by Bond. Finally, the effects of temperature are analysed and related to the packing arrangement of the molecules.

## Results

**Calorimetric analysis:** The DSC traces obtained for the six fatty acids were, in each case, indicative of the occurrence of a solid solid transition before melting (Figure 1). The tem

<figure><img src="image_2.png" /><figcaption>Figure 1. Extract of the DSC signals obtained with a heating rate of 2 K min⁻¹.</figcaption></figure>

perature interval between the two transitions diminishes with increasing chain length (at least up to C₁₉H₃₇OOH; this temperature interval is a little wider for C₂₁H₄₁OOH and C₂₃H₄₅OOH than for C₁₉H₃₇OOH).

The various transitions observed for these materials have been characterised by both X ray diffraction and thermal analysis. The measured temperatures, entropies, and enthalpies are displayed in Table 1. As mentioned in the literature,[7 9] C₁₃H₂₅OOH adopts the A' form at room temperature. A low energy reversible transition was observed at 287.7 K, which we attribute to an A'₁ (l for low)→A'ₕ (h for high) transition. Such a transition has been previously observed by Kaneko et al.[20] for pentadecanoic acid. All of the other acids adopt the B' form at room temperature, as predicted by many authors.[8,9] However, we have also observed A'₁ and A'ₕ solid phases for samples of C₁₅H₂₉OOH. The temperature of the A'ₕ→C'' transition is lower than that of the B'→C'' transition (Table 1). Therefore, B' is more stable than A' in the vicinity of the transition point for pentadecanoic acid. Irrespective of the low temperature phases, the high temperature phases are the C'' form for C₁₃H₂₅OOH and C₁₅H₂₉OOH and the C' form for the four other acids.

The evolution of the solid solid transition temperatures and the melting points shown in Figure 2 is in agreement with the work of von Sydow and Stenhagen,[16] in which these parameters were plotted for the odd numbered acids from n=13 to n=29.

The enthalpies and entropies of melting increase almost linearly with increasing chain length. These trends reflect the regular incremental lengthening of the hydrocarbon chain by the addition of two CH₂ groups on going from one acid to the next.

**Indexing of the X-ray powder-diffraction patterns:** Taking into account the temperatures of the transitions determined by thermal analysis, X ray powder diffraction patterns of the high temperature phases were recorded at 313.0, 324.0, 333.0, 340.0, 346.4, and 351.7 K for C₁₃H₂₅OOH, C₁₅H₂₉OOH, C₁₇H₃₃OOH, C₁₉H₃₇OOH, C₂₁H₄₁OOH, and C₂₃H₄₅OOH, respectively. All of the reflections of the powder patterns of C₁₃H₂₅OOH and C₁₅H₂₉OOH have been indexed to the C'' form, which is monoclinic with space group A2/a and has eight molecules in the unit cell. For the fatty acids from C₁₇H₃₃OOH to C₂₃H₄₅OOH, the reflections have been indexed to the C' form, which is also monoclinic, but has space group P2₁/a and Z=4. The measured and cal

<table><caption>Table 1. Temperatures (T in K), enthalpies (ΔH in kJmol⁻¹), and entropies (ΔS in Jmol⁻¹K⁻¹) of the various phase transitions.</caption><thead><tr><th rowspan="2"></th><th rowspan="2">S<sub>III</sub></th><th colspan="3">Solid solid transition (S<sub>III</sub>→S<sub>II</sub>)</th><th rowspan="2">S<sub>II</sub></th><th colspan="3">Solid solid transition (S<sub>II</sub>→S<sub>I</sub>)</th><th colspan="3">Melting (S<sub>I</sub>→liquid)</th></tr><tr><th>T</th><th>ΔH</th><th>ΔS</th><th>T</th><th>ΔH</th><th>ΔS</th><th>S<sub>I</sub></th><th>T</th><th>ΔH</th><th>ΔS</th></tr></thead><tbody><tr><td>C<sub>13</sub>H<sub>25</sub>OOH</td><td>A'<sub>1</sub></td><td>287.7(6)</td><td>0.06(1)</td><td>0.21(4)</td><td>A'<sub>h</sub></td><td>309.1(4)</td><td>8.5(3)</td><td>27(1)</td><td>C''</td><td>314.6(5)</td><td>33(1)</td><td>105(3)</td></tr><tr><td>C<sub>15</sub>H<sub>29</sub>OOH</td><td>A'<sub>1</sub></td><td>295.5(5)</td><td>0.27(7)</td><td>0.9(2)</td><td>A'<sub>h</sub></td><td>320.8(3)</td><td></td><td></td><td>C''</td><td>325.5(4)</td><td>40.4(6)</td><td>124(2)</td></tr><tr><td>C<sub>15</sub>H<sub>29</sub>OOH</td><td></td><td></td><td></td><td></td><td>B'</td><td>321.9(4)</td><td>8.2(6)</td><td>25(2)</td><td>C''</td><td>325.5(4)</td><td>40.4(6)</td><td>124(2)</td></tr><tr><td>C<sub>17</sub>H<sub>33</sub>OOH</td><td></td><td></td><td></td><td></td><td>B'</td><td>331.2(5)</td><td>7.5(9)</td><td>23(3)</td><td>C'</td><td>333.5(5)</td><td>46.5(9)</td><td>139(3)</td></tr><tr><td>C<sub>19</sub>H<sub>37</sub>OOH</td><td></td><td></td><td></td><td></td><td>B'</td><td>339.0(4)</td><td>7.4(6)</td><td>22(1)</td><td>C'</td><td>340.4(3)</td><td>57(1)</td><td>167(3)</td></tr><tr><td>C<sub>21</sub>H<sub>41</sub>OOH</td><td></td><td></td><td></td><td></td><td>B'</td><td>344.6(4)</td><td>5(1)</td><td>13(3)</td><td>C'</td><td>346.7(5)</td><td>63(3)</td><td>183(9)</td></tr><tr><td>C<sub>23</sub>H<sub>45</sub>OOH</td><td></td><td></td><td></td><td></td><td>B'</td><td>349.9(4)</td><td>2.5(6)</td><td>7(2)</td><td>C'</td><td>352.0(5)</td><td>75(3)</td><td>212(9)</td></tr></tbody></table>
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Figure 2. Evolution of some thermodynamic quantities with increasing chain length: a) temperatures of solid solid transition ($T_{\text{fus}}$) and of melt ing ($T_{\text{tr}}$), b) enthalpies of melting, and c) entropies of melting.</figcaption></figure>

calculated powder patterns (Pawley refinement[21]) of $C_{13}H_{25}OOH$ and $C_{23}H_{45}OOH$ are shown superimposed in Figure 3.

For the fatty acids from $C_{17}H_{33}OOH$ to $C_{23}H_{45}OOH$, the hypothetical cell parameters of the *A2/a C''* forms have also been calculated by analogy with the cell parameters of $C_{13}H_{25}OOH$ and $C_{15}H_{29}OOH$ by using the Win CELREF and Win CRISDR programs.[22] Figure 4 shows the Pawley fits for $C_{17}H_{33}OOH$ and $C_{19}H_{37}OOH$ obtained for the $P2_1/a$ and *A2/a* arrangements in the 2θ domain (25–32°), which ex hibits the most marked differences between the two ar rangements for each compound. Trial assignments of the *A2/a* phase for the fatty acids from $C_{17}H_{33}OOH$ to

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 3. Superimposition of the experimental and calculated X ray powder diffraction patterns (Pawley refinement) obtained for a) $C_{13}H_{25}OOH$ and b) $C_{23}H_{45}OOH$.</figcaption></figure>

$C_{23}H_{45}OOH$ invariably gave worse agreement of the experi mental and calculated X ray diffraction patterns. As an ex ample, the relatively strong reflection at around 29° is never taken into account in the *A2/a* lattice, whereas it corre sponds to the 211 reflection in the $P2_1/a$ lattice for the four fatty acids. All reflections are indexed with the $P2_1/a$ lattice of the *C*' form for the fatty acids from $C_{19}H_{37}OOH$ to $C_{23}H_{45}OOH$, whereas this is not quite the case for $C_{17}H_{33}OOH$. Indeed, it is evident from Figure 4a that the weak reflection at 29.5° is not possible with the $P2_1/a$ lattice, whereas it is with the *A2/a* lattice. It may be assumed that this sample also contains a small amount of the *C''* form, as this fatty acid is at the limit between the occurrence of the *C*' and *C''* forms in the series.

The cell parameters, the space groups, and the calculated densities determined for the high temperature solid forms of the six fatty acids are displayed in Table 2. The agreement factor, *Rwp*, obtained after Pawley refinement, is also given for each acid. In addition, the crystallographic data deter mined by von Sydow[11] for the *C*' form of $C_{11}H_{21}OOH$ and by Bond[6] for the *C*' form of $C_{11}H_{21}OOH$ and the *C''* forms of $C_{13}H_{25}OOH$ and $C_{15}H_{29}OOH$ are also included.

The cell parameters determined for $C_{13}H_{25}OOH$ and $C_{15}H_{29}OOH$ are in good agreement with those obtained by Bond. The *a* parameter decreases as the number of carbon
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Figure 4. Superimposition in the 25 32° 2θ domain of the Pawley fits obtained using the P2₁/a and A2/a lattices for a) C₁₇H₃₃OOH and b) C₁₉H₃₇OOH.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 5. Evolution of the *a* parameter with increasing chain length from C₁₁H₂₁OOH to C₂₃H₄₅OOH.</figcaption></figure>

As a conclusion of the two previous sections, it can be stated that the evolution with increasing chain length of several measured quantities (transition temperature, enthalpy, *a* parameter) are described by continuous curves from C₁₃H₂₅OOH to C₂₃H₄₅OOH, irrespective of the high temperature solid phase that is adopted (C'' for C₁₃H₂₅OOH and C₁₅H₂₉OOH; C' for the longer acids).

**Crystal structures of the high-temperature phases:** The crystal structures determined for the six fatty acids correspond to the best compromise between Rietveld refinement and energy minimisation (see the Experimental Section for more details). Projections in the (*a*,*c*) plane of the crystal

structures of the C'' form of C₁₅H₂₉OOH and of the C' form of C₂₁H₄₁OOH are presented in Figure 6. The molecules are stacked in bilayers in both forms, with dimers being formed by hydrogen bonding between the carboxyl groups. The crystal structures show alternating methyl and carboxyl group interfaces along the *c* axis, as is visible in Figure 6. The A2/a structure is a twofold version of the P2₁/a structure. The two types of structures are centrosymmetric and the hydrogen bonds bridge the carboxyl groups of molecules re

ated by inversion centres. In these representations, it can be seen that the molecules are not situated in the same positions in the A2/a lattice and the P2₁/a lattice, which clearly indicates that these two types of structure are different.

The values of the final agreement factor, *Rwp*, range between 4.0 and 5.5% for the structures solved (Table 3), indicating a very good agreement between the calculated and experimental X ray diffraction patterns. Besides, they do not differ markedly from the values obtained after Pawley refinement (Table 2), which did not take into account the

<table><caption>Table 2. Crystallographic data determined for the high temperature solid forms of the odd numbered fatty acids from C₁₃H₂₅OOH to C₂₃H₄₅OOH, together with those found in the literature. The non standard space groups settings A2/a and P2₁/a have been chosen for convenience, so that the chain axis is almost parallel to the *c* axis.</caption><thead><tr><th>Ref.</th><th><i>T</i><br>[K]</th><th><i>Rwp</i></th><th>Density<br>[g cm<sup>-3</sup>]</th><th>Space<br>group</th><th><i>Z</i></th><th><i>a</i> [Å]</th><th><i>b</i> [Å]</th><th><i>c</i> [Å]</th><th>β [°]</th></tr></thead><tbody><tr><td rowspan="2">C<sub>11</sub>H<sub>21</sub>OOH</td><td>[11]</td><td>296</td><td rowspan="2">0.984</td><td>P2<sub>1</sub>/a</td><td>4</td><td>9.62(3)</td><td>4.92(1)</td><td>34.2(1)</td><td>131.3(3)</td></tr><tr><td>[6]</td><td>300</td><td>P2<sub>1</sub>/a</td><td>4</td><td>10.029(1)</td><td>4.912(1)</td><td>35.10(1)</td><td>133.36(1)</td></tr><tr><td rowspan="2">C<sub>13</sub>H<sub>25</sub>OOH</td><td>this work</td><td>313</td><td>0.031</td><td>0.981(1)</td><td>A2/a</td><td>8</td><td>9.854(1)</td><td>4.940(1)</td><td>73.66(1)</td><td>125.95(1)</td></tr><tr><td>[6]</td><td>310</td><td></td><td>0.983</td><td>A2/a</td><td>8</td><td>9.812(1)</td><td>4.943(1)</td><td>73.74(1)</td><td>125.88(1)</td></tr><tr><td rowspan="2">C<sub>15</sub>H<sub>29</sub>OOH</td><td>this work</td><td>324</td><td>0.033</td><td>0.975(1)</td><td>A2/a</td><td>8</td><td>9.724(2)</td><td>4.952(1)</td><td>84.02(3)</td><td>125.30(1)</td></tr><tr><td>[6]</td><td>320</td><td></td><td>0.975</td><td>A2/a</td><td>8</td><td>9.720(1)</td><td>4.956(1)</td><td>84.02(1)</td><td>125.29(1)</td></tr><tr><td>C<sub>17</sub>H<sub>33</sub>OOH</td><td>this work</td><td>333</td><td>0.034</td><td>0.974(1)</td><td>P2<sub>1</sub>/a</td><td>4</td><td>9.634(2)</td><td>4.953(1)</td><td>50.03(1)</td><td>129.44(1)</td></tr><tr><td>C<sub>19</sub>H<sub>37</sub>OOH</td><td>this work</td><td>340</td><td>0.035</td><td>0.974(1)</td><td>P2<sub>1</sub>/a</td><td>4</td><td>9.575(1)</td><td>4.953(1)</td><td>54.99(1)</td><td>128.72(1)</td></tr><tr><td>C<sub>21</sub>H<sub>41</sub>OOH</td><td>this work</td><td>346.4</td><td>0.035</td><td>0.970(1)</td><td>P2<sub>1</sub>/a</td><td>4</td><td>9.526(1)</td><td>4.957(1)</td><td>60.18(1)</td><td>128.10(1)</td></tr><tr><td>C<sub>23</sub>H<sub>45</sub>OOH</td><td>this work</td><td>351.7</td><td>0.029</td><td>0.969(1)</td><td>P2<sub>1</sub>/a</td><td>4</td><td>9.488(1)</td><td>4.957(1)</td><td>65.23(1)</td><td>127.63(1)</td></tr></tbody></table>

atoms increases, following a regular curve (Figure 5) from C₁₁H₂₁OOH to C₂₃H₄₅OOH. Furthermore, the *b* parameter is nearly constant for all of these acids.

The density decreases with increasing chain length. Indeed, the temperature of acquisition increases with in creasing chain length and so does the thermal motion, which tends to increase the volume of the crystal lattice. The den sities calculated for the high temperature phases are all around 0.99 g cm⁻³, whereas they are a little higher, around 1.02 g cm⁻³, for the low temperature phases.[23,24]
<figure><img src="image_11.png" /><figcaption>Figure 6. Representations in the (a,c) plane of the crystal structures of the high temperature solid forms of C₁₅H₂₉OOH (C'' form) and C₂₁H₄₁OOH (C' form).</figcaption></figure>

<table><caption>Table 3. Selected structural data determined from the crystal structures of the C'' (this work and Bond's[6]) and C' forms of the six fatty acids.</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Rwp</th><th colspan="2">Angle of tilt of the alkyl chain [°]</th><th colspan="3">Packing arrangement of the carboxyl groups</th><th colspan="3">H···H distances at the methyl group interface [Å]</th></tr><tr><th>towards a axis</th><th>towards b axis</th><th>C C C O torsion angle [°]</th><th>O H···O angle [°]</th><th>O···O distance [Å]</th><th>(0) (1) and (0) (2)</th><th>(0) (3)</th></tr></thead><tbody><tr><td>C<sub>13</sub>H<sub>25</sub>OOH</td><td>0.041</td><td>125.8(3)</td><td>88.92(6)</td><td>3</td><td>165</td><td>2.54(4)</td><td>2.70</td><td>2.82</td><td>2.90</td><td>2.58</td></tr><tr><td>C<sub>13</sub>H<sub>25</sub>OOH<sup>[6]</sup></td><td></td><td></td><td></td><td>179</td><td>177</td><td>2.63</td><td></td><td></td><td></td><td>2.51</td></tr><tr><td>C<sub>15</sub>H<sub>29</sub>OOH</td><td>0.047</td><td>125.2(2)</td><td>90.71(4)</td><td>4</td><td>164</td><td>2.55(4)</td><td>2.73</td><td>2.77</td><td>2.83</td><td>2.62</td></tr><tr><td>C<sub>15</sub>H<sub>29</sub>OOH<sup>[6]</sup></td><td></td><td></td><td></td><td>179</td><td>156</td><td>2.63</td><td></td><td></td><td></td><td>2.60</td></tr><tr><td>C<sub>17</sub>H<sub>33</sub>OOH</td><td>0.054</td><td>125.2(3)</td><td>89.91(6)</td><td>7</td><td>161</td><td>2.72(9)</td><td>2.51</td><td>2.62</td><td>2.84</td><td>2.74</td></tr><tr><td>C<sub>19</sub>H<sub>37</sub>OOH</td><td>0.053</td><td>124.9(2)</td><td>89.49(5)</td><td>13</td><td>153</td><td>2.62(7)</td><td>2.49</td><td>2.69</td><td>3.01</td><td>2.92</td></tr><tr><td>C<sub>21</sub>H<sub>41</sub>OOH</td><td>0.055</td><td>124.6(1)</td><td>89.30(3)</td><td>9</td><td>152</td><td>2.60(5)</td><td>2.83</td><td>2.95</td><td>3.09</td><td>2.93</td></tr><tr><td>C<sub>23</sub>H<sub>45</sub>OOH</td><td>0.050</td><td>124.6(1)</td><td>89.62(4)</td><td>4</td><td>159</td><td>2.70(7)</td><td>2.74</td><td>2.80</td><td>2.84</td><td>3.07</td></tr></tbody></table>

agreement between the calculated and measured intensities of the various reflections. Table 3 also contains selected structural data that serve to describe the packing arrangement of the fatty acids within (alkyl chains, carboxyl groups) and between (methyl groups) the layers, and that have been determined from this work and from the C'' structures of C₁₃H₂₅OOH and C₁₅H₂₉OOH solved by Bond.

**Description of the C'' structures and comparison with the single-crystal data:**[6] The alkyl chains are packed in the or thorhombic packing arrangement O⊥, as described long ago by Bunn,[1] in accordance with the single crystal structures. This motif is very common in long chain aliphatic compounds.[4,5] The chains are almost parallel and the zigzag planes of neighbouring molecules are perpendicular to one another (Figure 7). Moreover, the alkyl chains are tilted from the (a,b) plane. The angles of tilt of the chain axis with respect to the a and b axes have been evaluated and are dis played in Table 3. The chains are almost perpendicular to the b axis, but are tilted towards the a axis with angles of 125.8° and 125.2° for C₁₃H₂₅OOH and C₁₅H₂₉OOH, respectively. These values are consistent with those of the θ₁ angles (125.9° for C₁₃H₂₅OOH and 125.3° for C₁₅H₂₉OOH), defined by Bond as the angle between the plane parallel to the chain axis and the methyl group interface, in the present case the (a,b) plane.

A similar hydrogen bonding framework is obtained for both the powder and single crystal C'' structures of C₁₃H₂₅OOH and C₁₅H₂₉OOH. The O H···O angle formed between the acceptor and the donor in the hydrogen bond is close to 180° in both cases (Table 3), indicative of strong hydrogen bonds. The C C C O torsion angle (O being the oxygen atom bearing the hydrogen atom) is close to 0° (this work) or 180° (Bond's work), which shows that the carboxyl group is always almost coplanar with the alkyl chain. However, there are discrepancies between the values obtained for the O···O distances, that is, around 2.63 Å in the single crystal study (standard values for crystal structures of solid forms of fatty acids[23, 26]) as opposed to much shorter distances in our study. The uncertainties in our values have been assessed by estimating the uncertainties in the positions of the oxygen atoms, as these were not available from

the refinement program. Our measurements are much less precise (average deviation of 0.06 Å, taking into account the values obtained for the C' structures) than those based on the single crystal data (precision of around 0.01 Å). This lack of accuracy of the O···O distances is directly related to the method used for the crystal structure determination as the final structure is a compromise between the agreement of the measured and calculated X ray powder diffraction patterns and the energy minimisation. However, taking into account the evaluated uncertainties, the O···O distances de

<figure><img src="image_12.png" /><figcaption>Figure 7. Packing arrangement of the alkyl chains of C₁₃H₂₅OOH perpendicular to the chain axis.</figcaption></figure>
terminated from the powder data are almost consistent with
those obtained from the single crystal data.

At the methyl group interface, the packing arrangement
is also similar for the C'' structures determined by the two
different techniques. A given methyl group, denoted as (0),
just below the (a,b) plane has three closest neighbouring
methyl groups just above the (a,b) plane, denoted as (1),
(2), and (3) (see Figure 8). (1) and (2) are related to each
other by a translation of the lattice along the *b* vector. They
are situated either side of (0) along the *b* axis. (3) is situated
almost immediately below (0) along the *a* axis. The shortest
H···H distances between (0) and (3) are consistent between
the two sets of data (Table 3).

<figure><img src="image_13.png" /><figcaption>Figure 8. Packing arrangement of the methyl groups in the (a,b) plane at the methyl group interface for a) C<sub>15</sub>H<sub>29</sub>OOH and b) C<sub>10</sub>H<sub>37</sub>OOH. The methyl groups in black are just above the plane, the others are just below. The two arrangements are derived from one another by a translation of 1/2*a* + 1/2*b* of one methyl group layer with respect to the other.</figcaption></figure>

Finally, it may be noted that the crystal structures of the
C'' forms of C<sub>13</sub>H<sub>25</sub>OOH and C<sub>15</sub>H<sub>29</sub>OOH presented herein
are in good agreement with those already solved from
single crystal data. Hence, it may be assumed that the crys
tal structures of the longer fatty acids determined by appli
cation of the same technique should be equally reliable.

**Comparison of the C' and C'' structures:** The C' and C'' forms exhibit a similar packing arrangement of the molecules within the bilayers. In the C' structures, the alkyl chains are stacked in the orthorhombic packing arrangement O ⊥, as in the case of the C'' structures. Moreover, they are tilted in the same way towards the (a,b) plane, at an angle close to 90° with respect to the *b* axis and at an angle of around 125° with respect to the *a* axis. As regards the packing arrangement of the carboxyl groups, no significant variation in the C C O torsion angle, the O H···O angle, or the O···O distance can be seen on going from C<sub>13</sub>H<sub>25</sub>OOH to C<sub>23</sub>H<sub>45</sub>OOH. In particular, the O···O distance ranges from 2.60 to 2.72 Å for the C' structures, but taking into account the evaluated uncertainties, these values are all consistent with the 2.63 Å determined from the single crystal C'' structures. One interesting point worthy of mention concerns the C C O torsion angle. For the six structures investigated, this parameter is close to 0° (it varies from 3 to 13°), which corresponds to a *trans* conformation of the carboxyl groups.

On the other hand, the C C O torsion angle is close to
180° for the C'' structures solved by Bond, which corre
sponds to a cis conformation of the carboxyl groups. In the
latter case, however, the C O and C=O distances are very
close (1.291(5) and 1.233(5) Å for C<sub>13</sub>H<sub>25</sub>OOH; 1.280(6) and
1.237(7) Å for C<sub>15</sub>H<sub>29</sub>OOH). Thus, it may well be the case
that these structures exhibit both conformations, as Hayashi
and Umemura have ascertained for several fatty acids from
C<sub>3</sub>H<sub>5</sub>OOH to C<sub>20</sub>H<sub>39</sub>OOH by means of polarized infrared
spectroscopy.[27] In the present work, structures with a cis
conformation of the carboxyl group, that is, with the C C C
O torsion angle equal to 180°, have also been tested. These
structures were invariably found to be less energetically fa

vourable than those featuring
a *trans* conformation of the
carboxyl groups. Although this
result does not disprove that
the actual structures might ex
hibit both conformations, it
nevertheless indicates that the
*trans* conformation must be
the dominant one for the
structures we have solved.

The crucial differences be-
tween the C' and C'' structures
lie in the packing arrange-
ments at the methyl group in
terface. Figure 8 shows the
packing arrangements of the
methyl groups in the (a,b)
plane for the C'' form of

C<sub>15</sub>H<sub>29</sub>OOH and the C' form of C<sub>19</sub>H<sub>37</sub>OOH. The methyl
groups of the molecules above the plane are shaded black,
whereas those situated just below the plane are in grey. (0)
has three closest neighbours above the methyl group inter
face, (1), (2) and (3), which are situated at similar positions
relative to (0) in both the C' and C'' structures. However,
(1), (2) and (3) are differently related to (0) in terms of the
symmetry elements involved, and hence according to the
space groups. In the A2/a structures (C'' form), (1) and (2)
are related to (0) through inversion centres, whereas they
are related by a 2<sub>1</sub> axis in the P2<sub>1</sub>/a structures (C' form).
Thus, (1) and (2) are equidistant from (0) in the P2<sub>1</sub>/a struc
tures, whereas in the A2/a structures the relative positions
of (1) and (2) with respect to (0) depend on the relative po
sition of (0) with respect to the inversion centres. (3) is relat
ed to (0) by a 2 axis in the A2/a structures, whereas it is re
lated by an inversion centre in the P2<sub>1</sub>/a structures. The
shortest H···H distances measured between (0) and its three
closest neighbours exceed 2.4 Å (sum of the van der Waals
radii of two hydrogen atoms) in each of the six structures
(Table 3). It may also be noted that the distances between
the hydrogen atoms of (0) and (3) increase with increasing
chain length. This trend can be attributed to the increased
thermal motion, as the temperature of measurement also in
creases with increasing chain length. Actually, the C' and C''
forms contain identical 2D layers: the methyl group ar
rangements in both the above and below layers (in black and grey, respectively, in Figure 8) are the same for the *P*2₁/*a* and *A*2/*a* structures. The difference between these struc- tures lies in the way in which these layers are stacked. It can clearly be seen in Figure 8 that when (1), (2) and (3) are dis placed along the 1/2*a* + 1/2*b* vector in the *P*2₁/*a* structure (or *A*2/*a* structure), they become related to (0) with the symme- try elements involved in the *A*2/*a* arrangement (or *P*2₁/*a* ar- rangement).

Discussion

According to the crystal structures of the high temperature phases of odd numbered fatty acids determined in the pres- ent work and in that of Bond.[6] the C'' form is only seen for C₁₃H₂₅OOH and C₁₅H₃₃OOH in the series from C₇H₁₃OOH to C₂₃H₄₅OOH. The question then arises as to why this solid phase is only adopted by these two fatty acids. In the follow- ing, we propose an explanation based on a survey of the en- ergetic contributions that are operative in the packing ar- rangements of the molecules within the C' and C'' crystal structures.

Two major intermolecular interactions are involved in the
cohesion of the structures of the high temperature phases of
fatty acids, namely hydrogen bonds and van der Waals inter
actions. The former influence the packing arrangement of
the carboxyl groups, whereas the latter relate more to the
interactions between the atoms of the hydrocarbon chains.
These two interactions are evaluated in the potential energy
descriptions of the structures through van der Waals and
electrostatic contributions. The overall potential energies of
the structures and the potential energies of the latter two
contributions are listed in Table 4 for the six structures in
vestigated, together with those determined from the crystal
structures of the high temperature phases of the odd num-
bered fatty acids solved by Bond (from C₇H₁₃OOH to
C₁₅H₂₉OOH). The energy values are given in kJmol⁻¹ per

<table><caption>Table 4. Potential energies (per molecule) calculated from the crystal structures of the high temperature solid forms of odd numbered fatty acids solved in this work and in that of Bond.[6]</caption>
  <thead>
    <tr>
      <th rowspan="2"></th>
      <th rowspan="2">Total energy<br>[kJ mol<sup>-1</sup>]</th>
      <th colspan="2">Intermolecular interactions</th>
    </tr>
    <tr>
      <th>Electrostatic<br>[kJ mol<sup>-1</sup>]</th>
      <th>van der Waals<br>[kJ mol<sup>-1</sup>]</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td colspan="4"><strong>reference [6]</strong></td>
    </tr>
    <tr>
      <td>C<sub>7</sub>H<sub>13</sub>OOH</td>
      <td>278(4)</td>
      <td>158</td>
      <td>52</td>
    </tr>
    <tr>
      <td>C<sub>9</sub>H<sub>17</sub>OOH</td>
      <td>316(4)</td>
      <td>159</td>
      <td>64</td>
    </tr>
    <tr>
      <td>C<sub>11</sub>H<sub>21</sub>OOH</td>
      <td>354(4)</td>
      <td>159</td>
      <td>75</td>
    </tr>
    <tr>
      <td>C<sub>13</sub>H<sub>25</sub>OOH</td>
      <td>394(4)</td>
      <td>158</td>
      <td>90</td>
    </tr>
    <tr>
      <td>C<sub>15</sub>H<sub>29</sub>OOH</td>
      <td>436(4)</td>
      <td>158</td>
      <td>105</td>
    </tr>
    <tr>
      <td colspan="4"><strong>this work</strong></td>
    </tr>
    <tr>
      <td>C<sub>13</sub>H<sub>25</sub>OOH</td>
      <td>392(2)</td>
      <td>165</td>
      <td>73</td>
    </tr>
    <tr>
      <td>C<sub>15</sub>H<sub>29</sub>OOH</td>
      <td>432(2)</td>
      <td>165</td>
      <td>88</td>
    </tr>
    <tr>
      <td>C<sub>17</sub>H<sub>33</sub>OOH</td>
      <td>472(2)</td>
      <td>158</td>
      <td>113</td>
    </tr>
    <tr>
      <td>C<sub>19</sub>H<sub>37</sub>OOH</td>
      <td>511(4)</td>
      <td>160</td>
      <td>122</td>
    </tr>
    <tr>
      <td>C<sub>21</sub>H<sub>41</sub>OOH</td>
      <td>551(3)</td>
      <td>161</td>
      <td>132</td>
    </tr>
    <tr>
      <td>C<sub>23</sub>H<sub>45</sub>OOH</td>
      <td>596(2)</td>
      <td>158</td>
      <td>154</td>
    </tr>
  </tbody>
</table>

molecule, so that they can be compared for all of the struc-
tures.

The electrostatic potential energy is practically constant for all of the structures (around 160 kJ mol⁻¹), whether determined from the powder or single crystal data. This is consistent with the similar packing of the carboxyl groups within the dimers established for the six structures investigated. However, the values obtained for the C'' structures do differ a little between our work and that of Bond. This may be related to the discrepancy between the O···O distances mentioned in the previous section (2.54 and 2.55 Å for our structures, as compared to 2.63 Å for Bond's structures). The potential energy of the hydrogen bonds is higher than the van der Waals potential energy for each structure. This confirms the predominant role of the hydrogen bonds in the cohesion of these crystal structures. However, the van der Waals forces increase with increasing chain length, and in the case of C₂₃H₄₅OOH the potential energies of the two in interactions are almost equal. The evolution of the total energy of the structures of the high temperature phases with increasing chain length produces a straight line from C₇H₁₃OOH to C₂₃H₄₅OOH (Figure 9), irrespective of the solid phase considered. This linear trend confirms the similarity of the molecular arrangements in the C' and C'' phases. It also implies that the energy barrier between the two solid forms must be very low.

<figure><img src="image_14.png" /><figcaption>Figure 9. Evolution of the total energy of the high temperature structures of the odd numbered fatty acids with chain length from C<sub>7</sub>H<sub>13</sub>OOH to C<sub>23</sub>H<sub>45</sub>OOH. ●: the structures determined by us; ○: the structures solved by Bond.[6]</figcaption></figure>

A third contribution, which has already been invoked sev-
eral times in this paper, also exerts an influence on the pack-
ing arrangement of the molecules, namely that of thermal
motion. Indeed, the high temperature phases are only stable
in a narrow temperature range below the melting points,
this range narrowing with increasing chain length, as the
temperatures of transition to C' or C'' and of melting in
crease. Thus, the effect of thermal motion has to be studied
if we want to thoroughly understand the packing arrange-
ment of the molecules in these particular solid forms.

When considering the equivalent temperature factors, U<sub>eq</sub>,
of the non hydrogen atoms of the odd numbered fatty acids
from C<sub>7</sub>H<sub>13</sub>OOH to C<sub>15</sub>H<sub>29</sub>OOH determined from the single crystal data,[6] it appears that thermal agitation exerts a greater effect on the last few carbon atoms of the chains, that is, on those close to the terminal methyl group for each compound. These temperature factors are represented in Figure 10 for the non hydrogen atoms of these five acids.

<figure><img src="image_15.png" /><figcaption>Figure 10. Evolution along the hydrocarbon chains of the equivalent temperature factors ($U_{eq}$) determined by Bond for the non hydrogen atoms of the odd numbered fatty acids from C<sub>7</sub>H<sub>13</sub>OOH to C<sub>15</sub>H<sub>29</sub>OOH. 1 and 2 correspond to the two oxygen atoms; 3 to n+2 (where n is the total number of carbon atoms) denote the carbon atoms from the carboxyl group to the methyl group. The temperature factors have been linked for each molecule.</figcaption></figure>

The atoms are labelled from 1 to n+2 (in which n is the total number of carbon atoms), 1 and 2 denoting the two oxygen atoms and the integers from 3 to n+2 denoting the carbon atoms of the hydrocarbon chains from the carboxyl group to the methyl group. It can be seen from this representation that, for the five acids, the temperature factors are much higher for the atoms near the methyl group than for the others. It can be assumed that the strong hydrogen bonds counteract the thermal agitation and limit the motion of the atoms near the carboxyl group. Thus, for a given molecule, the further the atoms are from the carboxyl group, the weaker the effect of the hydrogen bonds and the more prone they will be to the effect of thermal agitation. This trend is clearly apparent in Figure 10.

As the structures of the fatty acids from C<sub>17</sub>H<sub>33</sub>OOH to C<sub>23</sub>H<sub>45</sub>OOH were solved by assuming the molecules to be rigid bodies, a global isotropic temperature factor was determined for each structure, which accounts for the mean thermal agitation within the structure. Our aim was to investigate the evolution of the thermal agitation for the high temperature solid forms of the odd numbered fatty acids from C<sub>7</sub>H<sub>13</sub>OOH to C<sub>23</sub>H<sub>45</sub>OOH. For C<sub>7</sub>H<sub>13</sub>OOH, C<sub>9</sub>H<sub>17</sub>OOH, and C<sub>11</sub>H<sub>21</sub>OOH, we summed the values of the equivalent temperature factors of the non hydrogen atoms and divided the result by the number of atoms considered. Thereby, we obtained mean temperature factors that could be compared with the global isotropic factors determined for the six longer fatty acids. All of these temperature factors are gathered in Table 5. In addition, the temperature at which the

<table><caption>Table 5. Mean temperature (T) factors determined for the high temperature phases of the odd numbered fatty acids from C<sub>7</sub>H<sub>13</sub>OOH to C<sub>23</sub>H<sub>45</sub>OOH.</caption><thead><tr><th></th><th><i>T</i><sub>meas</sub> [K]</th><th><i>T</i><sub>melting</sub> [K]</th><th>Ref.</th><th>Mean <i>T</i> factor [Å<sup>2</sup>]</th></tr></thead><tbody><tr><td>C<sub>7</sub>H<sub>13</sub>OOH</td><td>230.0</td><td>265.83(1)</td><td>[12]</td><td>0.099(1)</td></tr><tr><td>C<sub>9</sub>H<sub>17</sub>OOH</td><td>270.0</td><td>285.53(1)</td><td>[12]</td><td>0.151(2)</td></tr><tr><td>C<sub>11</sub>H<sub>21</sub>OOH</td><td>300.0</td><td>301.63(1)</td><td>[12]</td><td>0.171(3)</td></tr><tr><td>C<sub>13</sub>H<sub>25</sub>OOH</td><td>313.0</td><td>314.6(5)</td><td>this work</td><td>0.217(6)</td></tr><tr><td>C<sub>15</sub>H<sub>29</sub>OOH</td><td>324.0</td><td>325.5(4)</td><td>this work</td><td>0.191(5)</td></tr><tr><td>C<sub>17</sub>H<sub>33</sub>OOH</td><td>333.0</td><td>333.5(5)</td><td>this work</td><td>0.174(6)</td></tr><tr><td>C<sub>19</sub>H<sub>37</sub>OOH</td><td>340.0</td><td>340.4(3)</td><td>this work</td><td>0.099(5)</td></tr><tr><td>C<sub>21</sub>H<sub>41</sub>OOH</td><td>346.4</td><td>346.7(5)</td><td>this work</td><td>0.063(4)</td></tr><tr><td>C<sub>23</sub>H<sub>45</sub>OOH</td><td>351.7</td><td>352.0(5)</td><td>this work</td><td>0.073(4)</td></tr></tbody></table>

measurement was undertaken as well as the melting point are also indicated in each case.

It is noteworthy that the values obtained for the mean temperature factors are not very accurate because they can be affected by systematic errors in the measured intensities. Moreover, the X ray diffraction measurements were performed close to the melting points (within about 1 K) for all of the acids except C<sub>7</sub>H<sub>13</sub>OOH and C<sub>9</sub>H<sub>17</sub>OOH (around 35 and 15 K below their melting points, respectively), which in introduces further uncertainty in the values. However, we are more interested here in the evolution of these values across the series than in the values themselves. Considering that the measurements were made under approximately the same conditions for all of the fatty acids, it can be assumed that all of the temperature factors can be compared. Their evolution in relation to the chain length is plotted in Figure 11. It appears that they increase up to C<sub>13</sub>H<sub>25</sub>OOH and then decrease once more.

The particular evolution obtained may be explained in terms of the competitive effects of van der Waals forces and thermal agitation, both of which increase with increasing chain length, whereas the hydrogen bonds remain approximately constant for all of the structures. The ascending part of the curve may be related to a predominance of the effect

<figure><img src="image_16.png" /><figcaption>Figure 11. Evolution of the mean temperature factors determined for the high temperature phases of the odd numbered fatty acids with chain length from C<sub>7</sub>H<sub>13</sub>OOH to C<sub>23</sub>H<sub>45</sub>OOH. The values have been calculated from the data of Bond for the fatty acids from C<sub>7</sub>H<sub>13</sub>OOH to C<sub>11</sub>H<sub>21</sub>OOH, and come from our own results for the longer acids.</figcaption></figure>
of thermal agitation, whereas for the descending part the van der Waals forces should prevail. In this representation, it is also apparent that the displacements due to temperature are, on average, more significant for C₁₃H₂₅OOH and C₁₅H₂₉OOH. As can be seen in Figure 10, the greatest displacements are certainly located near the methyl group. Moreover, according to van de Streek et al.,[28] who studied the influence of thermal motion on the crystal structures of *n* alkanes, they should be mostly oriented perpendicular to the chain axis, approximately in the (a,b) plane in our case. Keeping in mind that the C' and C'' arrangements are mutually related by a translation of a layer in the (a,b) plane and also that the energy barrier between the two solid forms is certainly very low, notably in the cases of C₁₃H₂₅OOH and C₁₅H₂₉OOH, the competitive effects of van der Waals forces and thermal motion are more conducive to adoption of the C'' form. A similar conclusion was reached by van de Streek and co workers concerning the relative stabilities of the tri clinic and monoclinic polymorphs of even numbered *n* alkanes with increasing chain length. For the other acids, we surmise that the effect of thermal agitation is not strong enough since, on the one hand, for the shorter acids the transition temperatures are not very high (and consequently nor is the thermal motion), while, on the other hand, the van der Waals forces become stronger for the longer acids. This may explain why, as far as we are aware, the C'' form is only adopted by C₁₃H₂₅OOH and C₁₅H₂₉OOH.

## Conclusion

The crystal structures of the high temperature solid forms of the odd numbered fatty acids from C₁₃H₂₅OOH to C₂₃H₄₅OOH have been determined from X ray powder diffraction patterns. C₁₃H₂₅OOH and C₁₅H₂₉OOH have been found to adopt the C'' form at high temperatures, as also shown by Bond.[6] However, we have proved that the C' form is adopted by the longer fatty acids, as is also the case for the shorter fatty acids from C₇H₁₃OOH to C₁₁H₂₁OOH according to Bond.[6] Thus, C'' is only observed for C₁₃H₂₅OOH and C₁₅H₂₉OOH. The C'' and C' forms are both monoclinic, but the former has the space group *A*2/*a* with *Z*=8, whereas the latter has the space group *P*2₁/*a* with *Z*=4. Both of their structures consist of bilayers of molecules held together by hydrogen bonds between their carboxyl groups. The packing arrangement in the two types of structures is very similar. The major differences lie at the methyl group interface. It has been shown that the packing arrangements of the molecules in these high temperature solid forms of odd numbered fatty acids are determined by the balance of the competitive effects due, on the one hand, to hydrogen bonds and van der Waals interactions, which serve to hold the structure together, and, on the other hand, ther mal motion, which tends to push the molecules apart.

Lastly, this article proves the validity of the technique that we have used to solve crystal structures from X ray powder diffraction data, as the results obtained for the C''

forms of C₁₃H₂₅OOH and C₁₅H₂₉OOH compare well with those obtained from single crystals. However, some improvements have to be made regarding the assessment of the hydrogen bonds, considering in particular the low precision of the O···O distances.

## Experimental Section

**Materials:** Highly pure samples of C₁₃H₂₅OOH, C₁₅H₂₉OOH, C₁₇H₃₃OOH, C₁₉H₃₇OOH, C₂₁H₄₁OOH, and C₂₃H₄₅OOH were supplied by Fluka. The purities of all these compounds are estimated by the supplier to exceed 99%.

**Thermal analysis:** Transition temperatures as well as enthalpy changes were determined by using a Perkin Elmer DSC 7 instrument equipped with a cooling accessory, which allowed the temperature to be increased from 110 to 950 K. A heating rate of 2 K min⁻¹ was typically used. Measurements at lower heating rates (0.5, 0.2 K min⁻¹) were also performed in order to better resolve transitions occurring at close temperatures. Samples (about 4 mg) were placed in aluminium pans and weighed before and after each experiment in order to determine the mass loss. At least six measurements were performed on each compound to enable the calculation of accurate values of the various thermodynamic quantities. The uncertainties in the temperatures and enthalpies were determined by using Student's method[29] with a 95% threshold of reliability. To this value was added the systematic error arising from the calibration of the apparatus, which was set at ±0.2 K for the temperatures and ±1% for the enthalpies.

**X-ray powder diffraction:** An Inel CPS 120 X ray powder diffractometer was used to obtain high quality powder diffraction patterns of the high temperature solid forms. Cuₖₐ radiation was selected for the incident beam. The diffractometer was equipped with an Oxford Cryosystems N₂ cryostream to facilitate the performance of isothermal experiments at different selected temperatures. The Inel CPS 120 diffractometer operates with Debye Scherrer transmission geometry and the diffracted beams are collected on a 120° curved counter by gas ionisation (argon + C₂H₆). The sample is introduced in a 0.5 mm diameter Lindemann glass capillary. The latter is rotated about its axis during the experiment in order to minimise preferential orientations of the crystallites. The acquisition time was set at six hours in order to obtain reflections with exploitable intensities. The relative heights of the strongest reflections at the start of each measurement were compared with those obtained for the X ray powder diffraction pattern of the B' form of C₁₇H₃₃OOH, which was calculated from the structure published by Goto and Asada[23] based on a study carried out on single crystals. Then, we chose the samples for which the agreement was best, assuming that the intensities in the calculated X ray powder diffraction pattern of the B' form of C₁₇H₃₃OOH were free from any preferential orientation effect. As recommended,[30] external calibration to convert the measured 4096 channels to 2θ degrees was applied by means of cubic spline fittings using cubic Na₂Ca₂Al₂F₄. The latter was combined with a low angle calibration using silver behenate,[31] a material that exhibits several reflections in the 2θ range of 1.5–20°.

**Determination of cell parameters and space groups:** The peak positions were determined after pseudo Voigt fitting using DIFFRACTINEL soft ware. Potential solutions of cell parameters and space groups were obtained using X CELL software[32] available in the module POWDER IN DEXING of MATERIAL STUDIO. The cell parameters and space group that best corresponded to the experimental X ray diffraction pattern of the solid form under investigation were finally determined using a Pawley profile fitting procedure.[21] The latter provided the refinement of the cell parameters, the peak profile parameters (function of profile, FWHM, asymmetry), the background, and the zero shift.

**Determination of the crystal structures:** The molecule to be studied was drawn in a 3D worksheet and was minimised in energy by means of geometry optimisation using the COMPASS force field[33] to obtain the relevant distances, angles, and dihedral angles for its various bonds. This
molecule was then set as a rigid body, but with the dihedral angle of the carboxyl group allowed to adjust freely, as this determines the good es tablishment of the hydrogen bonds. The rigid body was introduced in the cell whose dimensions had been previously found, together with its ho mologues related to the space group. The whole system was moved in the cell according to the seven degrees of freedom (three translations, three rotations, and the dihedral angle of the carboxyl group) by using a Monte Carlo type simulation[¹⁸] over several cycles to obtain the best re production of the experimental X ray powder diffraction pattern by the calculated one. The final structure was then obtained after repeated alter nating application of Rietveld refinement[³⁴] and geometry optimisation by energy minimisation in order to reach the best agreement between X ray powder diffraction and molecular interactions. The Rietveld proce dure includes refinement of the translations, rotations, and dihedral angle of the molecule, the isotropic temperature factors, and the preferred ori entations, also with the parameters previously refined in the Pawley pro cedure. The total energy determined for the structures corresponds to a survey of the intra and intermolecular interactions involved. Its entire expression is given in an article by Sun.[³³]

CCDC 612518 to CCDC 612523 contain the supplementary crystallographic data for this paper. These data can be obtained free of charge via www.ccdc.cam.ac.uk/data request/cif.

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