Polarization involves the alignment of the individual magnetic nuclei in the presence of a static external magnetic field B_o. This external field exerts a torque that forces the spinning nuclei to precess around it by a frequency given by the Larmor frequency given by Equation 1, where γ is the gyromagnetic ratio which is a characteristic property of the nucleus. For hydrogen, γ/2π = 42.58 MHz/Tesla. This value is different for different elements.

(1)

Considering the case of a proton under the influence of an external magnetic field, it will be in one of two possible energy states depending on the orientation of the precession axis. If the axis is parallel to B_o, the proton is in the lower energy state (preferred state) and in the higher energy state if anti-parallel shown by Figure 3.

Figure 3: Alignment of precession axis. Copyright: Halliburton Energy Services, Duncan, OK (1999).

We define the net magnetization per unit volume of material, M_o, from Curie’s law as Equation 2, where N = number of protons, h = Planck's constant (6.626 x 10^-34 Js), I = spin quantum number of the nucleus, k = Boltzmann's constant (1.381 x 10^-23 m² Kg s^-2 K^-1), and T = temperature (K)

The protons are said to be polarized completely once they are all aligned with the static external field. Polarization grows with a time constant called the longitudinal relaxation time (T₁) as shown in Equation 3, where t = time of exposure to B_o, M_z(t) = magnitude of magnetization at time t, with B_o along z-axis and T₁ = time at which M_z(t) reaches 90% of its final value, i.e., M_o.

T₁ is the time at which M_z(t) reaches 63% of its final value, M_o. A typical T₁ relaxation experiment involves application of a 90^o RF pulse that rotates the magnetization to the transverse direction. With time, the magnetization returns to its original value in the same fashion described by the above equation.

Click on the video below to see the mechanism of T₁ relaxation

Once the polarization is complete, the magnetization direction is tipped from the longitudinal plane to a transverse plane by applying an oscillating field B₁ perpendicular to B_o. The frequency of B­­₁ must equal the Larmor frequency of the material from B_o. This oscillating field causes a possible change in energy state, and in-phase precession. The total phenomenon is called nuclear magnetic resonance as shown in Figure 4.

Figure 4: A schematic representation of the phenomenon of nuclear magnetic resonance. Copyright: Halliburton Energy Services, Duncan, OK (1999).

The oscillating field is generally pulsed in nature and so terms in books such a 180^o pulse or 90^o pulse indicates the angle through which the net magnetization gets tipped over. Application of a 90^o pulse causes precession in the transverse phase. When the field B­₁ is removed, the nuclei begin to de-phase and the net magnetization decreases. Here a receiver coil detects the decaying signal in a process called free induction decay (FID). This exponential decay has an FID time constant (T₂) which is in the order of microseconds.

The time constant of the transverse relaxation is referred as T₂, and the amplitude of the decaying signal is given by Equation 4, with symbols as defined earlier.

Click on the video below to see the mechanism of T₂ relaxation

The de-phasing caused by T₁ relaxation can be reversed by applying a 180^o pulse after a time τ has passed after application of the initial 90^o pulse. Thus the phase of the transverse magnetization vector is now reversed by 180^o so that “slower” vectors are now ahead of the “faster” vectors. These faster vectors eventually over-take the slower vectors and cause rephasing which is detected by a receiver coil as a spin echo. Thus time τ also passes between the application of the 180^o pulse and the maximum peak in the spin echo. The entire sequence is illustrated in Figure 5. A single echo decays very quickly and hence a series of 180⁰ pulses are applied repeatedly in a sequence called the Carr-Purcell-Meiboom-Gill (CPMG) sequence.

Figure 5: A schematic representation of the generation of a spin echo. Copyright: Halliburton Energy Services, Duncan, OK (1999).

From an atomic stand point, T₁ relaxation occurs when a precessing proton transfers energy with its surroundings as the proton relaxes back from higher energy state to its lower energy state. With T₂ relaxation, apart from this energy transfer there is also dephasing and hence T₂ is less than T₁ in general. For solid suspensions, there are three independent relaxation mechanisms involved:-

These mechanisms act in parallel so that the net effects are given by Equation 5 and Equation 6

The relative importance of each of these terms depend on the specific scenario. For the case of most solid suspensions in liquid, the diffusion term can be ignored by having a relatively uniform external magnetic field that eliminates magnetic gradients. Theoretical analysis has shown that the surface relaxation terms can be written as Equation 7 and Equation 8, where ρ = surface relaxivity and s/v = specific surface area.

Thus one can use T₁ or T₂ relaxation experiment to determine the specific surface area. We shall explain the case of the T₂ technique further as Equation 9.

One can determine T₂ by spin-echo measurements for a series of samples of known S/V values and prepare a calibration chart as shown in Figure 6, with the intercept as 1T2,bulk1T2,bulk size 12{ { {1} over {T rSub { size 8{2, ital "bulk"} } } } } {} and the slope as ρ2ρ2 size 12{ρ rSub { size 8{2} } } {}, one can thus find the specific surface area of an unknown sample of the same material.

Figure 6: Example of a calibration plot of 1/T₂ versus specific surface area (S/V) of a sample.

A result sheet of T₂­ relaxation has the plot of magnetization versus time, which will be linear in a semi-log plot as shown in Figure 7. Fitting it to the equation, we can find T­₂ and thus one can prepare a calibration plot of 1/T₂ versus S/V of known samples.

Figure 7: Example of T₂ relaxation with magnetization versus time on a semi-log plot.

The following are a few of the limitations of the T₂ technique:

A study of colloidal silica dispersed in water provides a useful example. Figure 8 shows a representation of an individual silica particle.

Figure 8: A representation of the silica particle with a thin water film surrounding it.

A series of dispersion in DI water at different concentrations was made and surface area calculated. The T₂ relaxation technique was performed on all of them with a typical T₂ plot shown in Figure 9 and T₂ was recorded at 2117 milliseconds for this sample.

Figure 9: T₂ measurement for 2.3 wt% silica in DI water.

A calibration plot was prepared with 1/T₂ – 1/T_2,bulk as ordinate (the y-axis coordinate) and S/V as abscissa (the x-axis coordinate). This is called the surface relaxivity plot and is illustrated in Figure 10.

Figure 10: Calibration plot of (1/T₂ – 1/T_2,Bulk) versus specific surface area for silica in DI water.

Accordingly for the colloidal dispersion of silica in DI water, the best fit resulted in Equation 10, from which one can see that the value of surface relaxivity, 2.3 x 10^-8, is in close accordance with values reported in literature.

The T₂ technique has been used to find the pore-size distribution of water-wet rocks. Information of the pore size distribution helps petroleum engineers model the permeability of rocks from the same area and hence determine the extractable content of fluid within the rocks.

Usage of NMR for surface area determination has begun to take shape with a company, Xigo nanotools, having developed an instrument called the Acorn Area^TM to get surface area of a suspension of aluminum oxide. The results obtained from the instrument match closely with results reported by other techniques in literature. Thus the T₂ NMR technique has been presented as a strong case to obtain specific surface areas of nanoparticle suspensions.