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The nuclear magnetic resonance of a particle depends on the ratio of the magnetic moment (μ) and the angular momentum (I) of the protons, called the gyromagnetic ratio (gamma) [4]:
Pulsed NMR of a Curing Epoxy
Ryder Schmidt and Matt Sperle
University of Minnesota,
Department of Physics and Astronomy
1H Pulsed Nuclear Magnetic Resonance was used to ‘watch’ epoxy cure. By using the spin-lattice T1 and spin-spin T2 relaxation times, the curing process was plotted as a function of time. T1 was found to increase from ~50 ms to ~67 ms over a time period of approximately 180 minutes, while T2 was found to increase from ~8 ms to ~27 ms over the same time period. T1 results are consistent with theory while T2 results contradict theory.
Introduction:
Nuclear Magnetic Resonance is used in a variety of fields, such as molecular physics and medical imaging, and has become the leading technique in determining the structure of molecules [1]. Atomic nuclei are placed in a magnetic field and become magnetized to align their spins.[2] The nuclei are then disturbed by an oscillating magnetic pulse and the relaxation times to attain equilibrium: T1 (spin-lattice relaxation time) and T2 (spin-spin relaxation time) are measured. Some tasks that can be performed through observing the relaxation times include creating images as in MRI, detecting hydrogen bonding, and allowing the phase state of materials to be differentiated [3]. The substance studied in this experiment, cyclohexane C6H12 is a non-magnetic material found in many epoxies. Due to the transition from the liquid state to the solid state taking approximately 50 minutes, measurements of the relaxation times can be taken throughout the entire transition. Since NMR is used to differentiate substances, it is feasible that NMR could also be used to differentiate phases of the same substance, as we explore in our experiment. Hydrogen-1 NMR, which we used in our experiment, is common due to the relative simplicity of a proton’s spin state (only values of ± 1/2), the abundance of hydrogen-1, and the large signals that are produced with relative ease as compared to other nuclei [2].
Theory:
(1)
This ratio is unique for different particles, and thus knowing it is one way to characterize the substance under test. All protons have intrinsic spin (S). The z-component of the magnetic spin and thus the magnetic energy are split into two discrete levels given by [4]:
(2)
Equation 2 implies that a proton can change its spin state by absorbing or releasing energy. When the particle is placed in the external magnetic field it will experience a torque (Tau), given by [5]:
(3)
This torque can be seen below in figure 1.
Figure 1: a) shows the particle at the origin with the magnetic moment experiencing a torque that is perpendicular to both μ and B0. (b) Shows the same set-up, but looking from the (+z)-direction into the (-z)-direction. This figure was taken from Stoltenberg [4].
From the geometry of the problem, as seen in figure 1, equation 3 can be rewritten as:
(4)
From equations 2 and 4, we then see that the rate at which the spin precess the external magnetic field is proportional to the energy splitting. This rate of precession is known as the Larmor frequency:
(5)
If a material is placed in an external magnetic field, the spins will align and create a net magnetization (M) in the direction of the external field. The net magnetization in the z-direction (Mz) of the substance as a whole is in thermal equilibrium (Mz = M0) after it is given ample time for the net magnetization to align with the external field.
After establishing equilibrium, exposing the protons to a circularly polarized magnetic pulse B1 perpendicular to B0 causes an energy change, and thus Mz changes. By restricting this B1 pulse to be at the Larmor frequency, known as “on resonance,” the torque on the protons does not vary during the duration of the pulse [6]. The energy change caused by the pulse results in a component of the net magnetization to move into the xy-plane. After the B1 pulse, the presence of B0 causes the spins to begin realigning in order to reach equilibrium once again. When the B1 pulse tips the spins 90° so the net magnetization is completely in the xy-plane (Mz=0), the characteristic time for the spins to realign so that (Mz=M0) once again, is known as the “spin-lattice relaxation time” T1. The rate of change of Mz is proportional to the difference between Mz and M0 [6]:
(6)
The T1 pulse sequence consists of a 180 degree pulse, followed by a 90 degree pulse after some time delay. This pulse sequence is shown below in figure 2.
Figure 2: T1 pulse sequence depiction. There are two graphs here: top one showing the 180 degree and 90 degree pulses (B1), and bottom one showing the net z-magnetization Mz, both on the same time scale. This figure was taken from Stoltenberg [5].
The “spin-spin relaxation time” (T2) is the characteristic time for the xy-plane net magnetization (Mxy) to dissipate due to dephasing [6]. Dephasing arises from protons interacting and repelling one another, partially (or completely) nullifying their xy-magnetization contributions. The rate of spin dephasing is proportional to Mxy [6]:
(7)
The T2 pulse sequence consists of a 90 degree pulse, followed by several 180 degree pulses, this is known as the Car-Purcell-Meiboom-Gill pulse sequence. This T2 pulse sequence is depicted below in figure 3.
Figure 3: Multi pulse sequence used to calculate T2. Here the black arrows at the origin of the coordinate systems represent the net magnetization in the rotating reference frame. Notice that this figure only shows a single 180 degree pulse following the initial 90 degree pulse. This figure was taken from Stoltenberg [5].
Regarding the difference in relaxation times between solid and liquid states: it is predicted that liquid state T1 times will be shorter than solid state T1 times, and liquid state T2 times will be longer than solid state T2 times [7-8]. In a solid, the atoms are tightly bound to a lattice, while they move more freely in a liquid. If the atoms are tightly bound, it will take longer for the external magnetic field to reorient the spins to equilibrium, thus the spin-lattice time should be larger for solid state than liquid state. If the atoms are less bound, they reorient more sporadically. The random nature of reorientation will cause the dephasing of the spins to take longer, hence the T2 relaxation times should be larger for liquid state than for solid state [8].
Apparatus:
Our experiment uses the TeachSpin Pulsed NMR PS2-B system. One of the main components of this instrument is a permanent magnet with a field of (0.500.01) T. The internals of this magnet are shown in figure 4. There are four solenoid coils attached to the sides of the sample probe. These solenoids are the gradient coils responsible for making the magnetic field homogenous. There is also a solenoid perpendicular to the field, this solenoid (or sample coil) has two purposes: it produces the radio frequency (RF) pulse, and measures the net x-y magnetization by measuring the current induced from the precessing spins.
Figure 4: Simple schematic of the permanent magnet and sample coil.
The current induced on the sample coil is then converted to a voltage and displayed on a digital oscilloscope. A PS2 controller regulates the current supply, magnet temperature, and magnetic field gradients of the permanent magnet. The magnet temperature must remain constant throughout the experiment to keep the field constant, with the help of gradient coils to achieve a uniform field. A diagram of the key components of the apparatus for our experiment and the various module connections is shown in figure 5.
Figure 5: Block diagram consisting of the main components of the TeachSpin apparatus.
The pulse programmer determines the pulse length and period as well as triggers the oscilloscope at the appropriate time. The RF pulse is sent from the synthesized oscillator and amplified before being sent to the sample coil, producing a homogeneous circularly polarized magnetic field pulse(B1). The sample sits inside the sample coil and the magnetization then precesses around the external magnetic field (B0). The receiver then reads the induced emf and sends it to two detectors whose signals are both displayed on the oscilloscope. The RF detector outputs a signal proportional to the amplitude of precession which is the signal used to determine T1 and T2 The mixer detector is used to find the resonance frequency of the sample by multiplying the precession signal with the oscillator signal, producing an output frequency proportional to the frequency difference of the two input signals. An output with no “beats” indicates resonance.
Results:
By solving the differential equation for T1 (equation 5) using the appropriate initial conditions for the T1 pulse sequence, an equation for T1 can be derived:
(8)
By using the multi pulse sequence for T1, the following graph was produced, fitting to equation 7.
Figure 6: Voltage amplitude of the 90 degree pulse (which is proportional to Mz) for different delay times. The delay times range from 0.001 to 0.1, at which point the magnetization has almost completely returned to equilibrium. This figure was produced approximately 7.5 minutes after mixing. Error bars are not shown as they are too small to be seen. [original figure]
By linearizing equation 7, figure 7 was produced fitting to:
(9)
Figure 7: Linearization of the data from figure 6 using equation 8. Error bars are not shown here as they are too small to be seen. [original figure]
By using LSQ-fitting, a value for T1 was calculated at different times in the curing process. By plotting T1 as a function of curing time, figure 8 was produced.
Figure 8: T1 plotted as a function of cure time. As predicted, T1 increased as the epoxy dried. Notice that T1 continues to increase well beyond the manufacturer quoted cure time of 50 minutes. Error bars are not shown as they are too small to be seen here. [original figure]
By solving equation 6 with the appropriate initial conditions for the CPMG pulse sequence, the following equation was produced:
(10)
By plotting the amplitude of the spin echoes produced by the CPMG pulse sequence, figure 9 was produced, fitting to equation 9.
Figure 9: A plot of the spin echo amplitudes produced by the CPMG pulse sequence, taken approximately 53 minutes after mixing. [original figure]
By linearizing equation 9, the following equation was produced, an figure 9 was then linearized and plotted in figure 10:
(11)
Figure 10: Linearized plot of the data from figure 9. Using equation 10 and LSQ-fitting, the slope of the trendline is used to calculate T2. Notice that the x-axis is the time from the 90 degree pulse to the spin echo. [original figure]
By then plotting T2 as a function of drying time, figure 11 was produced.
Figure 11: T2 plotted as a function of cure time. T2 increased as the epoxy cures, contradicting the predicted T2 behavior. Notice that T2 seems to reach an asymptote after about 80 minutes past mixing, suggesting it is done transitioning from a liquid to a solid. [original figure]
Conclusion:
Using pulsed NMR, the spin-lattice times of cyclohexane present in a curing epoxy were found to be (49.6± 0.3) ms for the liquid state, and (67.5± 0.6) ms for the solid state; while the spin-spin relaxation for the liquid state was found to be (8.4± 0.1) ms, and for the solid state (27.2± 2.4) ms.
The behavior of T1 observed in this experiment is consistent with theory: increasing with cure time. However, there is no apparent asymptote being approached. This may be due to the epoxy needing more time to cure, however this is likely not the case because T1 values were calculated up to 170 minutes past mixing, which is three times the manufacturer quoted cure time. In addition, the data collection process for T1 took several minutes to construct a single plot similar to that of figure 7, meaning the epoxy was curing during each T1 data taking process and potentially skewing the results for T1.
T2 increased as the epoxy cured, contradicting theory. The cause of this contradiction is not yet known, however further research could be done in order to determine the cause. It is hypothesized that heating due to chemical reactions within the epoxy resin is the main source of error. Although T2 contradicts theory, the plot of T2 vs cure time (figure 11) still shows T2 approaching an asymptote, meaning that the epoxy has finished the transition from liquid to solid. With further research, it is plausible based off of the results obtained in this experiment that the curing process of epoxy could be modeled using Pulsed NMR.
Sources:
1. R. M. Silverstein, G. C. Bassler and T. C. Morrill, Spectrometric Identification of Organic Compounds, 5th Ed., Wiley, 1991.
2. Safety Data Sheet Parbond 5011 Hardener. (n.d.). Retrieved May 07, 2017, from https://www.mcmaster.com/#1650500-ghs-5011-072015/=17iz7be
3. “NMR Applications.” Michigan State University 900 MHz NMR Facility. Michigan State University Chemistry Department. Web. 5 Mar. 2017.
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