Sample Preparation
Samples with different average carbon chain lengths were prepared as different mixtures of several common fatty acids (Table 1). The fatty acids for this study were selected by their melting points, which should be lower than 25 °C to perform the measurements at room temperature. The commercially available fatty acids were used without further purification. Their properties (molecular weights, melting points and carbon chain lengths) are given in Table 1.
Table 1 Fatty acids used for samples preparation
Average carbon chain length CLmix values of the samples can be calculated knowing the molecular weights and mass parts of fatty acids using the following equations:
$${\text{CL}}_{\text{mix}} = \mathop \sum \limits_{i = 1}^{n} a_{i} {\text{CL}}_{i } ,$$
(1)
where a
i
is the part of molecules of i-acid in mixture, n is the number of acids in mixture and CL
i
is the carbon chain length of i-acid. The a
i
is obtained from the equation:
$$a_{i} = \frac{{N_{i} }}{{\mathop \sum \nolimits_{i = 1}^{n} N_{i} }} ,$$
(2)
where the number of molecules of i-acid, N
i
, is calculated from Eq. (3):
$$N_{i} = \frac{{m_{i} N_{A} }}{{M_{{{\text{w}}i}} }} ,$$
(3)
where m
i
is the mass part of i-acid in the mixture (in g), M
wi
is the molecular weight of i-acid (in g/mol) and N
A
is the Avogadro constant.
Twenty samples were selected for the current study. The samples had the CLmix values within the range from 4 to 18. These values are found in Table 2.
Table 2 Experimental values of carbon chain lengths CLmix and spin–spin relaxation rates R
2 of fatty acid mixtures
TD-NMR Measurements
The TD-NMR measurements were done using the portable NMR analyser Spin Track [24] with 1H resonance frequency of 25.7 MHz. The diameter of the sensor hole was 10 mm. The digital receiver console provided quadrature acquisition with the ringing time of around 10–12 μs. CPMG (Carr–Parcell–Meiboom–Gill) pulse sequence [25] was applied for the measurement of spin–spin relaxation rate R
2. Echo time was equal to 6 ms; the number of 180° pulses in the sequence was 600. Relaxation delay was 2 s, and the amount of scans was equal to 8. Durations of 90° and 180° RF pulses were 7 and 14 μs, respectively. All measurements were performed at room temperature (25 °C).
The experimental relaxation rate R
2 is dependent on the composition of different fatty acid molecules in a sample [26]:
$$R_{2} \; = \;\mathop \sum \limits_{i = 1}^{n} \left( {P_{i} R_{2i} } \right),$$
(4)
where P
i
and R
2i
are the proton populations and spin–spin relaxation rates of i-th fatty acid molecule; n is the number of fatty acid types in mixture.
The individual R
2i
values corresponding to different fatty acid molecules in a mixture were not observed due to fast proton exchange [26]. It takes place as the magnetization exchange rate k
ex between fatty acid molecules is much higher than the measured relaxation rate. Thus, one R
2 distinct value was obtained for each sample. It was confirmed by applying Laplace transformation to measured CPMG decays, as can be seen in Fig. 1. Therefore, all measured CPMG decays A(t) were fitted by mono-exponential function defined by Eq. (5)
$$A\left( t \right) = A_{0} {\text{e}}^{{ - R_{2} t}} ,$$
(5)
where \(A_{0}\) is the maximal signal magnitude and R
2 the spin–spin relaxation rate. A Matlab software script, written by the authors, was used for determining R
2 values.
Theory
The Bloembergen–Purcell–Pound (BPP) approach [27] is normally used for the description of spin–spin \(T_{2}\) NMR relaxation in liquids. According to BPP theory [25, 27], spin–spin relaxation time \(T_{2}\) or relaxation rate \(R_{2}\) (\(R_{2} = 1/T_{2}\)) is dependent on molecular mobility expressed as correlation time \(\tau_{\text{c}}\), which is the characteristic parameter of molecular mobility. For liquid samples \(\omega_{0} \tau_{\text{c}}\) ≪1 (\(\omega_{0} =\) the resonance frequency of TD-NMR devices) and R
2 linearly increases with the growth of correlation time as follows:
$$R_{2} = 10M_{2} \tau_{\text{c}} ,$$
(6)
where \(M_{2}\) is the value of second moment, which is determined by the strength of dipole–dipole interactions between neighbouring nuclei.
The correlation time in BPP equations can be described by Stokes–Einstein–Debye equation [28, 29]:
$$\tau_{\text{c}} \; = \;\frac{{C_{\text{r}} \eta V}}{{k_{\text{B}} T}},$$
(7)
where \(V\) is the molecule’s effective volume [30], \(\eta\) is viscosity, T is temperature, \(k_{\text{B}}\) is Boltzmann constant and \(C_{\text{r}}\) is fitting parameter determined by experiment [28,29,30,31]. The Stokes–Einstein–Debye equation is often used in the form modified for homogeneous fluids with molecules described as spheres with hydrodynamic or Stokes radius, not molecular.
Rotational movements are the most perceptible to NMR relaxation contribution, whereas the frequency range of translational motions makes a little contribution to the NMR relaxation proton spectra, and vibrations due to their high frequency are not observed by NMR relaxation.
Correlation times for different molecular motions are the time parameters in specific correlation functions. Complex modelling is required to describe them separately. On the other hand, rotational correlation times of molecules can be calculated by known corresponding diffusion coefficients, which are described by the Stokes–Einstein equation [29]:
$$D_{\text{r}} = \frac{{k_{\text{B}} T}}{{8\pi \eta R^{3} }},$$
(8)
where \(D_{\text{r}}\) is rotational diffusion coefficient, \(k_{\text{B}}\) is Boltzmann constant, T is temperature, \(\eta\) is viscosity and R is hydrodynamic (Stokes) radius of molecule.
When describing the rotational movement of molecules, it is convenient to consider that molecules have a spherical shape with hydrodynamic radius R. In the consideration that the molecule’s effective volume is equal to
$$V = \frac{{4\pi R^{3} }}{3},$$
(9)
the combination of the Eqs. (7–9) gives the following result [28]:
$$\tau_{\text{c}} \sim \frac{1}{{D_{\text{r}} }}.$$
(10)
Thus, it is obviously seen that the rotational correlation time is dependent on the corresponding diffusion coefficient (Eq. 10). Equation (10) is the widely used ratio for calculation of the NMR relaxation correlation time [22], as rotational molecular movements are the major contributors to NMR relaxation.
Generally molecules of different shapes can be characterised by hydrodynamic radii, thus Eq. (10) is probably applicable both to unbranched, rod-like molecules, which are mostly represented in liquid biofuels and to different types of paraffins and aromatic hydrocarbons, which are presented in petroleum fuels.
To connect the rotational diffusion coefficient with the molecular weight, the well-known ratio for self-diffusion D
self behaviour description for untangled polymer chains [32, 33] can be used. In case of elongated, unbranched molecules, there is direct dependence between these two parameters:
$$D_{\text{self}} \sim M_{\text{w}}^{ - \alpha } \;\;{\text{for}}\;M_{\text{w}} < M_{\text{wc}} ,$$
(11)
where M
wc is the entanglement coupling molecular weight [32], α is a coefficient, equal to 1.
Combining the expressions (6), (10) and (11), it is clearly seen the linear relationship between τ
c and, therefore, of R
2 with molecular weight M
w:
$$R_{2} \sim M_{\text{w}}$$
(12)
It is easy to suppose that in the case of unbranched hydrocarbons with similar chemical organisation the molecular weight is proportional to molecular size, and the latter one can be evaluated by CL. Relying on the dependence of R
2 on molecular weight M
w (Eq. 12), it is assumed the linear-like dependence of R
2 on CL:
$$R_{2} \sim {\text{CL}} .$$
(13)