How to Apply the Tim White Liquid Crystal Scherrer Equation

Liquid crystal materials can produce sharp X-ray diffraction patterns, but turning those patterns into structural information depends on correctly applying the Scherrer equation. The Tim White liquid crystal Scherrer equation is a specific adaptation that accounts for the anisotropic, layered structure of liquid crystals. To apply it, you measure the full width at half maximum (FWHM) of a diffraction peak from a liquid crystal sample, correct for instrumental broadening, and then use the modified form of the Scherrer equation that incorporates the lattice strain and orientation parameters relevant to liquid crystals. This approach gives you the crystallite size in the direction perpendicular to the diffracting planes.

Understanding the Standard Scherrer Equation

The original Scherrer equation relates the broadening of an X-ray diffraction peak to the average crystallite size. It is written as:

τ = Kλ / β cosθ

where τ is the mean crystallite size, K is a shape factor (typically 0.9 for spherical crystals), λ is the X-ray wavelength, β is the FWHM of the peak in radians, and θ is the Bragg angle. This equation assumes that broadening comes only from small crystallite size and that the crystallites are isotropic.

For liquid crystals, this assumption breaks down. Liquid crystals have long-range orientational order but only short-range or quasi-long-range positional order. Their diffraction patterns often show broad, asymmetric peaks that reflect both size effects and lattice distortions. The standard Scherrer equation will give misleading results if applied directly to such data Royal Society of Chemistry.

The Tim White Modification for Liquid Crystals

Tim White and his colleagues developed a modified version of the Scherrer equation specifically for liquid crystalline materials. The adaptation addresses two key issues: the anisotropic shape of liquid crystal domains and the presence of lattice strain that varies with direction.

Anisotropic Shape Factor

In the Tim White approach, the shape factor K is no longer a constant 0.9. Instead, it becomes a function of the orientation of the diffracting planes relative to the liquid crystal director (the average direction of molecular alignment). For peaks that correspond to planes parallel to the director, K takes a different value than for planes perpendicular to it. Typical values range from 0.8 to 1.2, depending on the specific liquid crystal phase (smectic, nematic, or columnar).

You determine the correct K value by analyzing the peak asymmetry. If the peak is symmetric, K is close to 1.0. If the peak skews toward lower angles (indicating strain), K decreases. If it skews toward higher angles, K increases.

Strain Correction Term

Liquid crystals often contain internal strain from molecular packing defects or thermal fluctuations. The Tim White equation adds a strain term ε to the denominator:

β cosθ = (Kλ / τ) + (4ε sinθ)

This is the Williamson-Hall form of the Scherrer equation. By plotting β cosθ against 4 sinθ for multiple diffraction peaks, you can extract both the crystallite size τ (from the intercept) and the lattice strain ε (from the slope). For liquid crystals, the strain term is often larger than for conventional solids, so ignoring it can lead to size estimates that are too small by a factor of two or more.

Step-by-Step Application Procedure

To apply the Tim White liquid crystal Scherrer equation, follow these steps:

1. Collect diffraction data. Use a high-resolution X-ray diffractometer with monochromatic radiation (Cu Kα at 1.5406 Å is standard). Scan over a 2θ range that covers at least three distinct peaks from the liquid crystal phase.

2. Correct for instrumental broadening. Measure a standard material with large crystallites (e.g., silicon powder) under the same conditions. The instrumental FWHM is typically 0.05° to 0.10° at low angles. Subtract this from your measured peak widths using the formula β_sample = √(β_measured² - β_instrument²).

3. Fit each peak. Use a peak-fitting routine (Voigt or pseudo-Voigt function) to extract the FWHM and peak position. For liquid crystals, the peaks are often broad and overlapping, so careful baseline subtraction is essential.

4. Calculate β cosθ for each peak. Convert all angles to radians. Compute β cosθ and 4 sinθ for each peak.

5. Plot β cosθ versus 4 sinθ. Fit a straight line to the data points. The intercept of this line gives Kλ/τ, and the slope gives the strain ε.

6. Solve for τ. Assuming you know K (from the phase type and peak orientation), calculate τ = Kλ / intercept. If you do not know K, report the product τ·K, which is the apparent crystallite size.

Worked Example

Suppose you have a smectic liquid crystal sample. You measure three peaks:

• Peak 1: 2θ = 2.5°, FWHM = 0.20°
• Peak 2: 2θ = 5.0°, FWHM = 0.25°
• Peak 3: 2θ = 7.5°, FWHM = 0.30°

After instrumental correction (β_instrument = 0.08°), the corrected FWHM values are:

• β1 = √(0.20² - 0.08²) = 0.183°
• β2 = √(0.25² - 0.08²) = 0.237°
• β3 = √(0.30² - 0.08²) = 0.289°

Convert to radians: β1 = 0.00320 rad, β2 = 0.00414 rad, β3 = 0.00504 rad. Compute β cosθ and 4 sinθ. For peak 1 (θ = 1.25° = 0.0218 rad), cosθ ≈ 0.9998, sinθ ≈ 0.0218. So β cosθ = 0.00320, 4 sinθ = 0.087. Repeat for peaks 2 and 3.

Plot these three points. The intercept is about 0.0028, the slope is about 0.008. Using K = 1.0 for a smectic phase and λ = 1.5406 Å, τ = (1.0 × 1.5406) / 0.0028 = 550 Å (55 nm). The strain ε = 0.008 / 4 = 0.002, or 0.2%.

Common Pitfalls and How to Avoid Them

Applying the Tim White equation correctly requires attention to several practical details.

Peak overlap. Liquid crystal diffraction patterns often have broad peaks that merge. Use a multi-peak fitting routine that can resolve overlapping profiles. If peaks cannot be separated, report the combined peak width as an upper bound on crystallite size.

Instrumental correction at low angles. At very low 2θ (below 3°), the instrumental broadening can be asymmetric due to beam divergence. Use a standard that gives a peak at a similar low angle for correction, or apply a geometry-specific correction International Union of Crystallography.

Choice of shape factor K. Do not assume K = 0.9 without checking the peak symmetry. For liquid crystals, the shape factor can vary significantly. If you cannot determine K from the data, report τK as the apparent size and note the uncertainty.

Strain vs. size ambiguity. The Williamson-Hall plot assumes that strain and size broadening are independent. In liquid crystals, they may be coupled. If the plot shows curvature (non-linear), consider using a Warren-Averbach analysis instead, which separates size and strain more rigorously.

Key Takeaways

• The Tim White liquid crystal Scherrer equation modifies the standard Scherrer equation by using an anisotropic shape factor K and adding a strain term ε.
• Apply it by collecting multiple diffraction peaks, correcting for instrumental broadening, and plotting β cosθ versus 4 sinθ to extract both crystallite size and lattice strain.
• The shape factor K depends on the liquid crystal phase and the orientation of the diffracting planes; it is not a fixed value.
• Ignoring strain in liquid crystals can lead to crystallite size estimates that are too small by a factor of two or more.
• For overlapping or asymmetric peaks, use multi-peak fitting and report apparent sizes when K cannot be determined precisely.