A variety of physical and chemical processes are typically studied in liquid and gas phases. To date many of them are well understood and described. The density, viscosity, diffusion coefficient, dielectric constant, ion product, etc. for liquid and gas are very different. At elevated temperatures and pressures we can get the fluid with intermediate values of each parameter. Despite the large amount of accumulated experimental data and intuitive fluid properties at elevated temperatures and pressures, the understanding of physical and chemical processes on molecular scale that occur under these conditions, which would have predictive power, is not yet sufficient. Unfortunately the behavior of the fluids cannot be explained simply by the variation of a single parameter, such as the equilibrium constants and the corresponding equilibrium shift due to the principle of Le Chatelier.
The following questions arise: What happens in specific cases on molecular scale?
What is the suitable technique to explore it?
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The number of studies of the properties of the sub- and supercritical fluids at the atomic and molecular scale is rather limited because of technical difficulties. From a methodological point of view the construction of an ESR high pressure cell is a quite sophisticated task (in comparison, for example, with a high-pressure cell for other spectral methods). In the case of ESR the sample under pressure must be extended into the resonator having a small internal dimeter and made of a material that does not absorb microwave radiation. The thick-walled quartz or sapphire capillary tubes can serve as a key element of the high pressure cell. Currently, there is no commercial system which allows to study the samples via ESR technique at high pressures (> 30 atm).
The brittleness of quartz often restricts the researches to work with fluids having moderate critical parameters: CO2, alkanes and alcohols. We realized ESR in situ technique at elevated pressures and temperatures including supercritical conditions in two versions. The first approach used the flame sealed thick-walled capillaries with internal diameter 0.3 – 0.6 mm, outer diameter 1 – 2 mm and a length of about 20 – 30 mm. It allows us to reach the pressure and temperature up to 400 atm and 500 °C, respectively, in the mode of constant average density. The second technique is based on a specially designed sapphire ampoule with Dout/Din ~ 4/1 mm. It is known that sapphire is an order of magnitude stronger than quartz. These two cells allow us to apply the ESR method to study the rotational dynamics of paramagnetic particles and clusters, as well as the processes of spin exchange between the radicals in sub- and supercritical conditions.
4. ESR of VOSO4/H2O solution in the temperature range 20 ... 400 °C. Phase stratification at T ~> Tcrit,H2O.
The experimental ESR spectra of an aqueous solution of VO2+ sulfate were registered in a wide range of T and P: from normal conditions to supercritical ones.
Fig. 3a shows the ESR spectra of 0.1 M VOSO4 aqueous solution registered at different temperatures. It has already been mentioned above that the 8 spectral components are equalized when the temperature increases due to fast rotation. At temperatures above
~ 150 °C the spectrum becomes almost isotropic. Upon a further temperature increase, a broadening of all hyperfine structure lines is observed. The observed line broadening can be caused by several reasons: an increase of the spin-rotational interaction and an increase of the spin exchange rate. In both cases, the line width must grow in proportion to the parameter T/η(T). This dependence is observed up to T ~ 300 °C. At higher temperatures the intensity rapidly decreases and the spectrum just disappears. However, a broad line of low intensity appears at supercritical conditions (T ~ 390 °C > Tcrit,H2O = 374 °C). The observed phenomenon points to an increase of the exchange interaction between the VO2+ ions due to their higher local concentrations. For comparison the ESR spectrum of 3 M VOSO4 aqueous solution at normal conditions is also shown. Thus,
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in the supercritical water we observe phase separation of the 0.1 M VOSO4 solution into a 3 M VOSO4 solution and a solution with a lower, may be close to zero, VO2+
ion concentration.
When the ESR technique is applied at elevated temperatures and supercritical conditions, it is necessary to pay a special attention to the temperature gradients that can appear in the system. It was found that the temperature gradient of about 30 K/cm results in the appearance of an additional anisotropic ESR signal as VOSO4 aqua solution is approaching the critical point (Fig. 3b). The anisotropic signal is well simulated assuming a high rate of spin exchange between VO2+ having the same g-tensor. The g-factor components of this spectrum are in good agreement with the g-factor of anhydrous VOSO4. This phenomenon can be explained by the local density differences in capillary in the presence of a temperature gradient. There is a gradual reduction of this signal upon further heating; at T > Tcrit,H2O, when the entire volume of the ampoule is filled uniformly by supercritical water, a single line with a width of about 300 G is registered as in the case of gradientless heating. Thus, anhydrous VOSO4 transforms into the average state close to a 3M solution of VOSO4 under normal conditions.
3000 3500 4000
x 3 x 3 x 3
3 M aq. sol. VOSO4
335 °C
x 50 x 50
300 °C
20 °C 100 °C 160 °C 235 °C 270 °C 395 °C
H, G
x 3
20 °C
a)
3000 3500 4000
g=2.000
390 °C
× 50
350 °C 320 °C
230 °C
H, G
25 °C 250 °C
200 °C 180 °C
90 °C
b)
Fig. 3. ESR spectra of aqueous solution of VO2+ sealed in a quartz capillary (Din/Dout×L = 0.3mm/0.9mm×20mm, average density is 0.45 g/mL) at various temperatures. External temperature gradient dT/dx = a) 2 K/cm, b) 30 K/cm. In the figure, the temperature of the middle part of the capillary upon recording of spectrum is given.
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5. Vanadyl porphyrins as internal spin probes in oil. Dynamics of vanadyl containing oil molecules in the range (T = 20 °C, P = 1 atm) - (T = 300 °C, P ~> 50 atm). Evaluation of size distribution width of vanadyl containing species.
It is well known that oil consists of a complex mixture of hydrocarbons of various molecular weights and other chemical compounds. In particular, oil contains vanadium, which is usually a part of the porphyrin complexes that in turn are the fragments of macromolecular compounds: resins and asphaltenes. The size and structure of these compounds, their tendency to aggregate and form the deposits are still under discussion.
ESR can provide useful information, for example, about the sizes of rigid bonded vanadyl containing molecular fragments.
It is known that there are two main signals in the ESR spectra of oil originating species: 1) V4+ spectra of vanadyl porphyrin groups of high molecular compounds and 2) a single resonance line that is usually interpreted as the resonance absorption of unpaired electron of aromatic carbon π-systems.
A typical temperature dependence of the ESR spectra of oil and its heavy component dissolved in benzene is shown in Fig. 4a. Comparing the temperature dependence of the ESR spectra of VOTPP/benzene and asphaltene/benzene one can claim that the spectra of asphaltene/benzene apparently is a superposition of the VO2+
containing species spectra at different τ. This leads to the conclusion that the asphaltene/benzene solution contains large molecules or their aggregates with different sizes, which incorporate the VO2+ ions.
Using the SED equation (see above) one can calculate τ and then the ESR spectra of VO2+ containing species with different sizes D in a solution with a given viscosity η.
So we simulated the experimental spectrum of the oil solution assuming a certain size distribution of rotating paramagnetic particles.
The model is based on the following assumptions: (a) the parameters of the spin Hamiltonian: perpendicular and parallel components of the g and A tensors and line width are determined from the spectra simulation of the frozen asphaltene/benzene solution; (b) the temperature dependence of the solution viscosity is similar to the one for pure benzene; (c) the particle size distribution in a first approximation may be described by a step function (dN(D)/dD = const if D lies within the range [Dmin;Dmax] and dN(D)/dD = 0 otherwise).
It was found that these rather crude assumptions allow us to achieve a good agreement between the experimental and calculated spectra over the entire temperature range for Dmin ~ DVOTPP ~ 1.6 nm and Dmax = 12 nm (Fig. 4b).
It should be noted that there is a certain increase in the contribution to the spectrum of anisotropic component when the temperature increases from 300 to 330 °C. At such temperatures in the conditions of our experiments benzene passes to supercritical state,
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and its dissolving capacity can strongly decrease. In this way in asphaltene/benzene solution a partial association of asphaltene molecules or phase separation of the solution may take place, so a large part of the asphaltenes appears in the phase with the higher viscosity.
a) b)
Fig. 4. a) ESR spectra of asphaltenes dispersed in benzene with the volume ratio Vasphaltene/Vbenzene ∼ 1:3, registered at different temperatures in situ. The sample was sealed inside the capillary Din/Dout×L = 1.2 mm/2.8 mm×40mm. The average density of the benzene inside the capillary was equal to the critical value with the accuracy of 10%. b) Simulated ESR spectra of VO2+ containing species having a uniform size distribution in the range from 1.6 to 12 nm.
Of course, the developed approach can be optimized assuming a more realistic size distribution in order to achieve the maximum coincidence between the experimental and simulated spectra. In this case, it is necessary to take into account the changes of the spin Hamiltonian parameters, because the local environment of VO2+ for different molecules may be slightly different. We believe that the further development of the approach will open opportunity to determine the temperature variation of the size distribution function in situ, which nowadays is a great challenge for understanding the behavior and physical and chemical peculiarities of complex heterogeneous multicomponent system like crude heavy oils.
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Multifunctional rapid scan EPR imaging
Mark Tseytlin
University of West Virginia, HSCS 5523 One Medical Center Drive, Morgantown, WV 26506, USA
E-mail: mark.tseytlin@hsc.wvu.edu
The digital revolution that we have been witnessing the last decades enables the development of new spectroscopy and imaging, driving a transition from experiments that require little or no data processing but may be not very informative to experiments that generate rich, complex, multidimensional data that require adequate mathematical models of the underlying physical and biological phenomena, robust and efficient algorithms, and computational power to gain both in quantity and in quality, acquiring better images in shorter time and obtaining novel information from them. Hardware innovations, such as Arbitrary Waveform Generators (AWG) to shape the desired pulses, fast digitizers to acquire the responses, Field-Programmable Gate Arrays (FPGA) for embedded designs, highly capable CPU and GPU closing the distance between specialized workstations and desktop PCs, and sophisticated software environments such as LabView to provide convenient control over hardware and MatLab for simulation of experimental data, have opened pathways to novel experimental designs providing new levels of sensitivity, resolution and functionality. However, in commercial EPR digital progress has been mostly in replacing analog parts by their digital counterparts, while conceptually the EPR spectrometer has not changed much in 50 years.
Similar to NMR, the progress in usefulness and versatility in EPR spectroscopy, backed up by technical development, is mostly about the new ways to generate, acquire, and process data. It is even more so in EPR imaging.
EPR images are reconstructed from a set of projections, which are absorption spectra measured in the presence of linear gradients of the external magnetic field. An imposed gradient, G, broadens EPR spectrum by the amount equal to ∆B= G ∆L, where ∆L is the characteristic linear size of the imaged object. On one hand, the larger the gradient, the better the spatial resolution is. On the other hand, the downside of increasing the resolution is weakening the signal intensity. As the spectrum gets broader, its intensity drops as 1/G2 in the standard CW EPR experiment. This is because the first derivative spectrum is measured.
Geometrical explanation would be in considering a right triangle with sides H and W, and the slope (derivative) equal to H/W. Reducing H by a factor of two will result in increasing W by x2, provided that the area is the same (number of spins in the spectrum is constant). As a result, the slope is reduced by x4. The problem of sensitivity is especially acute for in vivo imaging, in which EPR at down to 250 MHz is performed to provide sufficient depth of tissue penetration (several cm) and generate images that are not distorted by dielectric loss effects, with the ensuing decrease in sensitivity due to lower Boltzmann factor.
Looking from a different perspective, technically CW EPR employs very slow sweeps of the magnetic field and low amplitude modulation, so that only a very small portion of the spins contribute to the signal. The larger the overall spectral width, the fewer spins contribute. From the engineering standpoint, CW EPR signal detection is very badly optimized. It is because the EPR signal is detected at a single modulation frequency fm, which normally does not exceed 100
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kHz, and therefore the signal BW is just twice the fm, all harmonics beyond the first, or, in special cases, second, are rejected by the detection channel of the spectrometer:
𝑅(𝐵, 𝑡) = 𝑅1(𝐵)cos(𝜔𝑚𝑡) + 𝑅⏟ 2(𝐵)cos(2𝜔𝑚𝑡) + 𝑅3(𝐵)cos(3𝜔𝑚𝑡) + ⋯
𝑤𝑎𝑠𝑡𝑒𝑑 𝐸𝑃𝑅 𝑠𝑖𝑔𝑛𝑎𝑙
However, quite often EPR resonator would have a bandwidth, BW, of tens of MHz, which is much larger than the signal BW, and, e.g., in pulse EPR all available detection bandwidth is used to boost sensitivity. On the other hand, the major advantage of CW EPR is the simplicity of experiment interpretation, as field scan produces the familiar first derivative spectrum.
To overcome the CW EPR limitation, a method called rapid scan (RS) EPR is being developed [1]. As compared to CW ESR, where modulation amplitude is selected to be a fraction of the narrowest feature in the spectrum, in RS EPR the amplitude exceeds the spectral width. Typical RS setup uses a self-resonating coil to efficiently provide a sinusoidal field sweep with amplitude of units to tens of Gauss (capable of covering the entire spectrum) and frequency of the order of tens of kHz. Rapid scan is still continuous wave method on the side of excitation, but it is closer to pulse EPR from the detection standpoint. In fact, a free induction decay type of signal is observed because of the rapid passage effect. Wide scans generate multiple harmonics of the modulation frequency and the signal is directly detected.
A comparison of CW and RS EPR is sketched below:
In RS EPR, the absorption spectra are measured, and as a result, the signal drops as 1/G.
In the above triangle example, H is reduced by x2. All the spins contribute to the signal twice over the modulation period, and the spin system is not as readily saturated, which provides more than tenfold gain in Signal/Noise ratio while being a factor of 100 faster than CW EPR, with the upper gain limit approaching that of pulse EPR. On the flip side, The RS EPR spectra are more complex and not intuitively interpreted, as a fast scan through each line is similar to a short
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chirp pulse producing a superposition of free induction decay type responses from which the spectrum need to be reconstructed:
EPR imaging is essentially EPR spectroscopy with spatial resolution, and EPR Imager is essentially an EPR spectrometer + x, y, z gradients + software. The spatial dimension is very useful as tumors are highly heterogonous with respect to pO2, pH and other physiological parameters, while spectroscopy may give you average values that could be meaningless.
Resolved in space EPR images produce pO2 maps that can be used, e.g., for ‘dose painting’ in cancer radiation treatment, as hypoxic cells are less sensitive to radiation and thus a higher local dose is needed. Multifunctional in vivo spectroscopy and imaging require the use of spin probes with complex multiline EPR spectra. The positions and widths of the lines report local pH, pO2, and other important physiological parameters. CW EPR does not provide sufficient sensitivity for in vivo imaging. The use of pulse EPR would require prohibitory high power pulses to cover the wide multiline spectra. RS EPR is then the method of choice.
If the environment of an electron spin probe is spatially heterogeneous, the EPR spectrum is the sum of contributions from all locations. Spectral-spatial imaging divides the sample volume into an array of small spatial segments (voxels) and calculates the EPR spectrum for each of these segments. Magnetic field gradients are used to encode spatial information into EPR spectra, which are called projections. The spectral-spatial image is then reconstructed from these projections. The most commonly-used EPR image reconstruction algorithm is Filtered Back Projection (FBP) [2], initially developed for X-ray tomography, which involves rotation of the sources and detectors around the imaged object. The FBP algorithm, which was adapted for EPR spectral-spatial imaging, imposes additional experimental constraints:
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it imposes certain restrictions on the gradients, limits spectral width, and does not allow incorporation of relevant a priori information (biology, sample geometry, spin probe lineshape, pO2 limits, etc.), in return requiring little computational resources. Alternatives to FBP include maximum entropy (MEM) [3] and regularized optimization (RO) [4] iterative algorithms, which are less restrictive, have demonstrated substantial improvement in the quality of the reconstructed images, but are computationally inefficient (compared to FBP) and require large amounts of computer memory. Computational efficiency is a limiting factor for 3D spatial and 4D spectral-spatial imaging.
Image reconstruction boils down to solution of a system of coupled linear equations {
𝛼1,3𝑥3+ 𝛼1,14𝑥14+ 𝛼1,15𝑥15+ 𝛼1,26𝑥26+ ⋯ + 𝛼1,60𝑥60= 𝑝1
…
𝛼𝑘,31𝑥31+ 𝛼𝑘,42𝑥42+ 𝛼𝑘,43𝑥43+ 𝛼𝑘,54𝑥54+ ⋯ + 𝛼𝑘,77𝑥77= 𝑝𝑘
of the type 𝐴𝑥 = 𝑝, where A is model matrix with dimensions N x M, not necessarily square, x is the unknown vector (image) in RN and p is the known vector (experiment) in RM. While unique solutions can be readily obtained for well-posed problems such as textbook {𝑥 + 𝑦 = 2
𝑥 − 𝑦 = 0 | 𝑁 = 𝑀
𝑥 = 𝑦 = 1, or over-determined noisy {
𝑥 + 𝑦 = 2 + 𝑛𝑜𝑖𝑠𝑒 𝑥 − 𝑦 = 0 + 𝑛𝑜𝑖𝑠𝑒 3𝑥 − 2𝑦 = 1 + 𝑛𝑜𝑖𝑠𝑒
|
𝑀 > 𝑁 𝑥 = 1 + 𝛿𝑥 𝑦 = 1 + 𝛿𝑦
by
using least-squares solution, image reconstruction is often an ill-posed problem of the type {𝑥 + 𝑦 = 2
− − − | 𝑁 < 𝑀
𝑦 = 1 − 𝑥 or { 𝑥 + 𝑦 = 2 + 𝑛𝑜𝑖𝑠𝑒
𝑥 + 1.001𝑦 = 1.001 + 𝑛𝑜𝑖𝑠𝑒 | 𝑁 = 𝑀
𝑦 = 1 + 1400 ∗ 𝑛𝑜𝑖𝑠𝑒. In other words, many images are formally consistent with the data. The trick is to find a reconstruction method that would produce an image that is stable with respect to small variations of p (i.e., noise), converges into the true image with more data (M → N), and, if possible, can use additional (a priori) information, which would effectively increase the number of equations in 𝐴𝑥 = 𝑝 (M↑), such as the condition of non-negativity. In practice a solution for matrix equation 𝐴𝑥 = 𝑝 with non-square matrix can be obtained using Moore-Penrose pseudo-inversion with a standard Matlab routine pinv, or Tikhonov regularization with identity operator as the regularization operator. The images produced using Tikhonov regularization and Moore-Penrose pseudo-inversion are very similar.
Another challenge is inversion of large matrices using an ‘average’ desktop PC. A 3D problem with 34 voxels per side for a total of 343= 39304 elements produces a 40k x 40k matrix with the size of 12.8 GB, which takes about 21 minutes to generate and invert on available desktop PC. Adding the 4-th dimension for 4D spectral-spatial imaging inflates the problem to intractable N=344, 14 TB.
Currently used FBP image reconstruction method does not work with wide and complex spectra, especially if it is spectral-spatial imaging that provide the experimenter with functional information. A new algorithm is being developed to overcome this limitation [5]. The algorithm solves the ill-posed problem of image reconstruction using Tikhonov regularization in the frequency domain. A 4D large scale problem can be solved in parallel as a series of independent 3D problems and produces the image in one iteration. A mathematical algorithm has also been developed to transform the observed RS EPR signals into spectra and projections for imaging.
The method has demonstrated one to two orders of magnitude signal enhancement vs. CW EPR,