Time-Domain NMR (TD-NMR) Relaxometry Lab
Model nuclear spin relaxation, CPMG spin-echo decay trains, multi-component $T_2$ pore size distributions, surface relaxivity, and analytical laboratory DCF economics.
Instrument & Sample Controls
Phase & Relaxation Attribution
| Phase Compartment | Fraction & $T_2$ | Physical Origin |
|---|
CPMG Spin-Echo Decay Train ($M_{xy}(t)$)
Transverse Relaxation$T_2$ Relaxation & Pore Size Distribution ($P(T_2)$)
Inverse Laplace SpectrumBenchtop TD-NMR Characterization Facility DCF Model
20-Year discounted cash flow projection for contract TD-NMR laboratories serving battery gigafactories, food processors, and petrophysics clients.
| Year | Volume | Revenue | OpEx | EBITDA | Free Cash Flow | Cumulative Cash Flow |
|---|
Engineering & Physical Chemistry Guide
1. Nuclear Magnetization & Larmor Precession
Hydrogen nuclei ($^1\text{H}$) possess an intrinsic nuclear spin $I = 1/2$ and magnetic dipole moment $\mu = \gamma \hbar \mathbf{I}$. Inside a static magnetic field $B_0$, spins precess at the Larmor frequency $\omega_0 = \gamma B_0$ ($\gamma/2\pi \approx 42.58\text{ MHz/T}$). A permanent NdFeB or SmCo magnet array ($0.50\text{ T}$) creates a macroscopic net longitudinal equilibrium magnetization $M_0$ along the $z$-axis without liquid helium cryogens.
2. CPMG Spin Echo Refocusing Mechanics
Applying an on-resonance $90^\circ_x$ radiofrequency pulse tips $M_0$ into the transverse $xy$-plane. Spins immediately begin dephasing due to molecular interactions ($T_2$) and static field inhomogeneities ($\Delta B_0$). A train of $180^\circ_y$ pulses applied at intervals $2\tau$ repeatedly mirrors spin phases, reversing the static dephasing and producing an echo train whose decaying envelope isolates true transverse relaxation $T_2$.
3. Surface Relaxivity & Brownstein-Tarr Theory
In porous rock cores and colloidal battery slurries, molecules diffuse rapidly between the bulk pore fluid and the solid particle boundary. According to the Brownstein-Tarr relation in the fast-diffusion limit: $\frac{1}{T_2} = \frac{1}{T_{2,\text{bulk}}} + \rho_2 \left(\frac{S}{V}\right)$. Because smaller pores or highly dispersed nanoparticles have higher surface-to-volume ratios, their relaxation times are dramatically shortened, allowing direct non-destructive pore size mapping.
4. Slurry Rheology & Battery Electrode Quality Control
In lithium-ion manufacturing, cathode and anode slurries must achieve uniform particle dispersion. The ratio of restricted solvent ($T_2 \approx 5\text{--}15\text{ ms}$, bound to PVDF binder and graphite surfaces) to bulk free NMP ($T_2 > 200\text{ ms}$) provides an instantaneous, non-destructive index of binder coverage and slurry agglomeration, preventing costly pinhole defects and coating delamination on roll-to-roll coater lines.
Frequently Asked Questions
Time-Domain NMR Knowledge Quiz
1. Why do fluid molecules relax much faster (display shorter $T_2$) inside smaller pores?
2. How does the CPMG pulse sequence eliminate the dephasing effect of static magnetic field inhomogeneity ($\Delta B_0$)?
3. What is the $^1\text{H}$ proton resonance frequency in a benchtop permanent magnet of $B_0 = 0.50\text{ Tesla}$?