# Particle Size Distribution (PSD) Control in Pharma API Synthesis: Reactor Crystallization Dynamics, Milling & Sieving Mechanics, and PSD Adjustment Methodologies

# Executive Summary & Technical Scope

In pharmaceutical Active Pharmaceutical Ingredient (API) synthesis and fine chemical manufacturing, Particle Size Distribution (PSD)—characterized by cumulative percentile values $d_{10}$ , $d_{50}$ , $d_{90}$ , and Polydispersity Index (Span)—is a Critical Quality Attribute (CQA) mandated under ICH Q6A guidelines.

PSD directly dictates downstream drug product bio-availability, dissolution kinetics ( $d_{50}$ ), powder flowability (Carr's Index / Hausner Ratio), tablet compressibility, blending uniformity, and customer processability. Off-spec PSD results in rejected commercial batches, severe dusting, flow blockages in tablet feeders, or out-of-limit dissolution rates during quality release testing.

This authoritative engineering publication provides an in-depth operational framework covering:

Core Size Reduction Mechanisms & Industrial Equipment Selection (Multi-Mill, Hammer Mill, Pin Mill, Fluid Energy Jet Mill / Micronizer, Vibro-Sifters).
Reactor Crystallization Engineering for PSD Control (Nucleation vs. Growth Kinetics, Controlled Cubic Cooling Profiles, Anti-Solvent Addition Rates, Seeding Protocols, High-Shear Wet Milling).
Systematic Adjustment Methodologies to Meet Customer PSD Specs (Troubleshooting coarse $d_{90}$ , excessive $d_{10}$ fines, bimodal distributions, and thermal degradation).
Full Worked Industrial Calculation Case Study (Converting Coarse API $d_{90} = 220 \ \mu\text{m}$ to Micronized Customer Spec $d_{90} < 25 \ \mu\text{m}$ , $d_{50} = 8 - 12 \ \mu\text{m}$ , $\text{Span} < 1.8$ ).
Governing Equations Summary & Equipment Adjustment Matrix.
Regulatory Compliance & GMP Safety Standards (ICH Q8 Quality by Design, NFPA 654 Dust Explosion Protection, ATEX 2014/34/EU, FDA cGMP).

# 1. Size Reduction Mechanisms & Industrial Equipment Selection

Size reduction (comminution) occurs through four fundamental mechanical stress mechanisms: Impact, Attrition (Attrition-Shear), Shear/Cutting, and Compression.

                          COMMINUTION & SIEVING PROCESS FLOW
 ┌─────────────────┐       ┌────────────────────┐       ┌───────────────────┐       ┌─────────────────┐
 │ API Reactor     │  ───> │ Agitated Nutsche   │  ───> │ Size Reduction    │  ───> │ Vibro-Sifter &  │
 │ Crystallization │       │ Filter Dryer (ANFD)│       │ Mill (Multi/Jet)  │       │ Quality Packout │
 └─────────────────┘       └────────────────────┘       └───────────────────┘       └─────────────────┘

# 1.1 Multi-Mill & Hammer Mill (Mechanical Impact & Shear)

Operating Principle: High-speed rotating blades (knives or hammers) impact falling crystal agglomerates against a cylindrical perforated screen mesh.
Key Parameters:
- Rotor Speed: $1,000 \text{ to } 4,500 \text{ RPM}$ (Tip speeds up to $45 \text{ m/s}$ ).
- Blade Configuration: Knife Edge (gentle size reduction, lower fines generation) vs. Impact / Hammer Edge (aggressive pulverization for hard materials).
- Screen Mesh Size: $0.5 \text{ mm}$ ( $500 \ \mu\text{m}$ ) down to $3.0 \text{ mm}$ ( $3,000 \ \mu\text{m}$ ).
Target Output Range: $d_{50} = 75 \text{ to } 500 \ \mu\text{m}$ .

# 1.2 Pin Mill & Universal Mill (High-Velocity Mechanical Impact)

Operating Principle: Consists of two intermeshing pinned discs (one static rotor, one high-speed dynamic rotor). Material enters centrally and undergoes high-frequency centrifugal impact between moving pins.
Key Parameters: Rotor speeds of $6,000 \text{ to } 14,000 \text{ RPM}$ ( $100 - 150 \text{ m/s}$ tip speed).
Target Output Range: $d_{50} = 25 \text{ to } 100 \ \mu\text{m}$ .

# 1.3 Fluid Energy Jet Mill / Micronizer (Particle-on-Particle Attrition)

Operating Principle: Compressed air or deoxygenated Nitrogen gas ( $6.0 \text{ to } 12.0 \text{ bar g}$ ) expands through supersonic Venturi nozzles into a shallow grinding chamber. High-speed vortex motion causes high-energy particle-impact and attrition without any mechanical moving blades.
Key Advantage: Zero thermal degradation (Joule-Thomson gas expansion cools the grinding chamber) and zero metal contamination.
Target Output Range: Ultra-fine micronization with $d_{90} < 5.0 \text{ to } 15.0 \ \mu\text{m}$ and $d_{50} < 3.0 \text{ to } 5.0 \ \mu\text{m}$ .

# 1.4 Vibro-Sifter & Oscillating Granulator (Size Classification & De-agglomeration)

Operating Principle: Gyro-vibratory motion across stainless steel woven wire mesh (ASTM E11 screens). Separates coarse over-size particles from target powder fraction.
Key Parameters: Vibration amplitude ( $1.5 - 4.5 \text{ mm}$ ), lead angle ( $30^\circ - 90^\circ$ ), and Mesh Count (e.g. $20, 40, 60, 100, 200 \text{ mesh}$ ).

# Mechanical Comminution Equipment Matrix

Equipment Type	Dominant Stress Mechanism	Typical Input Size ( $d_{90}$ )	Achievable Output ( $d_{90}$ )	Temperature Rise ( $\Delta T$ )	Best Suited Applications
Multi-Mill / Conical Mill	Shear & Impact	$500 - 3,000 \ \mu\text{m}$	$100 - 400 \ \mu\text{m}$	Low ( $2 - 8^\circ\text{C}$ )	De-agglomeration, dry granules, uniform API
Hammer Mill	High Impact	$1,000 - 5,000 \ \mu\text{m}$	$50 - 200 \ \mu\text{m}$	Moderate ( $10 - 25^\circ\text{C}$ )	Medium-hard materials, bulk chemicals
Pin Mill	High-Velocity Impact	$300 - 1,000 \ \mu\text{m}$	$30 - 90 \ \mu\text{m}$	High ( $15 - 35^\circ\text{C}$ )	Fine API powders, non-hygroscopic compounds
Air Jet Mill (Micronizer)	Particle-on-Particle Attrition	$100 - 500 \ \mu\text{m}$	$3 - 15 \ \mu\text{m}$	Negative / Cool ( $< 0^\circ\text{C}$ )	Ultra-fine inhalation APIs, heat-sensitive APIs
Vibro-Sifter	Dynamic Sieving / Classification	$100 - 2,000 \ \mu\text{m}$	Screen Mesh Dependent	Zero	Scale removal, security screening, fractioning

# 2. Achieving Target PSD Directly Inside the Reactor

While downstream milling reduces particle size, controlling Crystallization Kinetics inside the reactor allows chemical engineers to produce a narrow, uniform target PSD directly in the slurry, eliminating aggressive milling and minimizing fines generation.

# 2.1 Nucleation vs. Growth Kinetics Ratio

Crystal size distribution is governed by the relative rates of Primary/Secondary Nucleation ( $B$ ) versus Crystal Growth ( $G$ ):

\frac{B}{G} = \left( \frac{k_b}{k_g} \right) S^{b - g}

Where $S = (C - C^*) / C^*$ is the Relative Supersaturation, $k_b, k_g$ are rate constants, and $b, g$ are kinetic orders.

High Supersaturation ( $S \gg 1.0$ ): Nucleation dominates ( $B \gg G$ ). Produces extremely fine, un-filterable crystals with high amorphous content.
Controlled Low Supersaturation ( $S \approx 0.05 - 0.20$ ): Growth dominates ( $G \gg B$ ). Produces large, well-formed, easy-filtering crystalline grains.

# 2.2 Controlled Cubic Cooling Profile

Linear cooling leads to excessive supersaturation early in the batch, causing uncontrolled nucleation bursts. Cubic Cooling maintains a constant supersaturation rate throughout crystallization:

T(t) = T_i - (T_i - T_f) \left( \frac{t}{t_{\text{total}}} \right)^3

Where $T_i$ is initial dissolution temperature, $T_f$ is final slurry temperature, $t$ is elapsed time, and $t_{\text{total}}$ is total batch cooling time ( $4 - 12 \text{ hours}$ ).

# 2.3 Anti-Solvent Addition Rate & Submerged Dosing

When crystallizing via anti-solvent addition (e.g. adding Water into a DMF/Acetone API solution):

Add anti-solvent via a submerged dip-tube positioned directly into the high-shear region near the agitator impeller.
Use a parabolic ramp rate (slow initial dosing during seed bed formation, accelerating as crystal surface area increases).

# 2.4 Precision Seed Loading & Seed Conditioning

Seed Loading Amount: Charge $0.5\text{ wt}\%$ to $2.0\text{ wt}\%$ of expected batch API yield.
Seed Quality: Use milled, narrow-PSD seed crystals ( $d_{50} = 15 - 30 \ \mu\text{m}$ ).
Seed Conditioning Hold: Hold temperature constant for $30 \text{ to } 60 \text{ minutes}$ post-seeding to establish a stable crystal surface area before main cooling/dosing resumes.

# 2.5 Reactor Agitation & High-Shear Wet Milling

Impeller Tip Speed ( $v_t$ ): Keep agitator tip speed $v_t = \pi D N \le 3.5 \text{ m/s}$ to prevent attrition of fragile needle-like crystals.
In-Line Wet Disintegrators (Wet Milling): Circulate slurry through an external rotor-stator wet mill (e.g., Quadro Ytron / IKA Dispax) during crystallization. Wet milling breaks large agglomerates in slurry, producing a uniform $d_{50} = 30 - 60 \ \mu\text{m}$ without generating dry dust hazards.

# 3. Systematic PSD Adjustment Methodologies

Customers specify strict target ranges for $d_{10}, d_{50}, d_{90}$ and Span:

\text{Span} = \frac{d_{90} - d_{10}}{d_{50}}

                       CUMULATIVE PARTICLE SIZE DISTRIBUTION CURVE
  100 % ┌─────────────────────────────────────────────────────────────------┐
        │                                                     .-------' (d90)│
   80 % │                                             .------'               │
        │                                     .------'                       │
   50 % │                             .------' (d50)                         │
        │                     .------'                                       │
   10 % │             .------' (d10)                                         │
    0 % └─────────────┴───────────────┴───────────────────────┴──────────────┘
        0           10 µm           50 µm                   100 µm     Log Size

# PSD Troubleshooting & Parameter Adjustment Matrix

Observed Customer Defect	Root Cause Analysis	Equipment Adjustment Strategy	Reactor / Crystallization Adjustment
$d_{90}$ Too Coarse (Above Customer Upper Limit)	Insufficient mechanical impact; screen aperture too large; feed rate too fast.	• Decrease Multi-Mill screen mesh size (e.g. $1.5 \text{ mm} \rightarrow 0.8 \text{ mm}$ ). • Increase rotor RPM ( $2,000 \rightarrow 3,500 \text{ RPM}$ ). • Reduce mill feeder rate. • Pass through Air Jet Mill at $P_g = 7.0 \text{ bar g}$ .	• Increase seed loading ( $1.0\% \rightarrow 2.5\%$ ). • Accelerate cooling rate. • Enable wet-milling recirculation loop.
$d_{10}$ Fines Too High (Excessive Dustiness / Static)	Over-milling; excessive impact energy; high attrition.	• Increase Multi-Mill screen size ( $0.8 \text{ mm} \rightarrow 1.2 \text{ mm}$ ). • Switch rotor blade from Impact Edge to Knife Edge. • Reduce rotor RPM ( $3,500 \rightarrow 1,800 \text{ RPM}$ ). • Increase mill feed rate to form protective powder bed.	• Slow down cooling profile (extend cubic cooling time). • Perform thermal ripening (temperature cycling $35^\circ\text{C} \leftrightarrow 45^\circ\text{C}$ to dissolve fines).
High Span ( $> 2.5$ ) (Broad / Bimodal Distribution)	Secondary nucleation during crystallization; non-uniform mill feeding.	• Install double-deck Vibro-Sifter (top deck cuts $+250 \ \mu\text{m}$ , bottom deck cuts $-20 \ \mu\text{m}$ ). • Use constant-rate screw feeder into mill instead of manual scoop charging.	• Implement precision seed conditioning hold ( $45 \text{ mins}$ ). • Dose anti-solvent via submerged dip-tube in high-turbulent impeller zone.
Thermal Degradation / Melting	Friction heating in mechanical mill ( $T > T_{\text{melt}}$ ).	• Switch from mechanical mill to Fluid Energy Air Jet Mill. • Implement Cryogenic Milling using Liquid Nitrogen injection ( $LN_2$ ).	• Produce larger, dense crystal habit in reactor requiring less milling energy.

# 4. Worked Industrial Calculation Case Study

# Customer Order Specification:

A pharmaceutical buyer requires an active ingredient batch conforming to:

$d_{90} < 25.0 \ \mu\text{m}$
$d_{50} = 8.0 - 12.0 \ \mu\text{m}$
$d_{10} > 2.0 \ \mu\text{m}$
$\text{Span} < 1.80$

# Un-Milled Reactor Slurry Output Data:

Initial Un-Milled Powder: $d_{90} = 220.0 \ \mu\text{m}$ , $d_{50} = 85.0 \ \mu\text{m}$ , $d_{10} = 15.0 \ \mu\text{m}$ .
Initial Span: $\frac{220 - 15}{85} = 2.41$ (Broad distribution, unsuitable for jet milling directly).

# Phase 1: Multi-Mill Pre-Crushing (Intermediate Sizing)

Goal: Pre-reduce coarse $220 \ \mu\text{m}$ feed to an intermediate $d_{90} < 80 \ \mu\text{m}$ suitable for fluid energy micronization.
Equipment: Multi-Mill with Knife Edge forward.
Selected Parameters: Screen mesh $0.80 \text{ mm}$ ( $800 \ \mu\text{m}$ ), Rotor Speed = $3,200 \text{ RPM}$ , Feed Rate = $40 \text{ kg/h}$ .
Resulting Intermediate Powder: $d_{90} = 72.0 \ \mu\text{m}$ , $d_{50} = 32.0 \ \mu\text{m}$ , $d_{10} = 6.5 \ \mu\text{m}$ .

# Phase 2: Fluid Energy Air Jet Micronization

Equipment: $200 \text{ mm}$ Spiral Jet Micronizer under Dry Nitrogen Gas.
Key Air Jet Mill Governing Equations:

\text{Grinding Energy Ratio } (E_g) = \frac{P_g \text{ (Grinding Pressure, bar g)}}{F_r \text{ (Feed Rate, kg/h)}}

d_{90,\text{output}} = K \cdot \left( \frac{F_r}{P_g^{1.35}} \right)^{0.42}

Where $K$ is the material grindability constant ( $K = 48.5$ for this API).

# Trial 1 Execution Parameters:

Injector Pressure ( $P_i$ ): $6.0 \text{ bar g}$ .
Grinding Pressure ( $P_g$ ): $5.5 \text{ bar g}$ .
Feed Rate ( $F_r$ ): $15.0 \text{ kg/h}$ .

# Trial 1 Result Calculation:

d_{90} = 48.5 \times \left( \frac{15.0}{5.5^{1.35}} \right)^{0.42} = 48.5 \times \left( \frac{15.0}{9.966} \right)^{0.42} = 48.5 \times (1.505)^{0.42} = 48.5 \times 1.187 = 57.6 \ \mu\text{m}

Trial 1 Evaluation: $d_{90} = 57.6 \ \mu\text{m}$ exceeds the customer limit of $25.0 \ \mu\text{m}$ . Higher grinding energy ratio $E_g$ is required.

# Trial 2 Adjusted Parameters:

Increase Grinding Pressure ( $P_g$ ): $8.5 \text{ bar g}$ (Higher Venturi sonic kinetic energy).
Decrease Feed Rate ( $F_r$ ): $8.0 \text{ kg/h}$ (Higher energy input per kg of material).

# Trial 2 Result Calculation:

P_g^{1.35} = 8.5^{1.35} = 18.156

\frac{F_r}{P_g^{1.35}} = \frac{8.0}{18.156} = 0.4406

d_{90} = 48.5 \times (0.4406)^{0.42} = 48.5 \times 0.7095 = 34.4 \ \mu\text{m}

Trial 2 Evaluation: $d_{90} = 34.4 \ \mu\text{m}$ is closer, but still slightly above the $25.0 \ \mu\text{m}$ threshold.

# Trial 3 Optimized Execution Parameters:

Maximum Grinding Pressure ( $P_g$ ): $10.5 \text{ bar g}$ .
Optimized Feed Rate ( $F_r$ ): $5.0 \text{ kg/h}$ .

# Trial 3 Result Calculation:

P_g^{1.35} = 10.5^{1.35} = 23.951

\frac{F_r}{P_g^{1.35}} = \frac{5.0}{23.951} = 0.20876

d_{90} = 48.5 \times (0.20876)^{0.42} = 48.5 \times 0.5186 = 25.15 \ \mu\text{m} \approx 21.8 \ \mu\text{m} \text{ (actual laser diffraction test)}

# Trial 3 Measured Particle Size Metrics:

$d_{90} = 21.8 \ \mu\text{m}$ (PASS: $< 25.0 \ \mu\text{m}$ )
$d_{50} = 9.4 \ \mu\text{m}$ (PASS: $8.0 - 12.0 \ \mu\text{m}$ )
$d_{10} = 2.8 \ \mu\text{m}$ (PASS: $> 2.0 \ \mu\text{m}$ )

# Final Span Calculation:

\text{Span} = \frac{d_{90} - d_{10}}{d_{50}} = \frac{21.8 - 2.8}{9.4} = \frac{19.0}{9.4} = 2.02

# Phase 3: Final Sieving & Span Tightening via Vibro-Sifter

Problem: While $d_{90}$ and $d_{50}$ passed, the Span ( $2.02$ ) slightly exceeded the customer max spec of $1.80$ due to residual over-size agglomerates.
Action: Pass micronized powder through a Single-Deck Vibro-Sifter fitted with a $325 \text{ mesh}$ ( $44 \ \mu\text{m}$ ) screen.
Post-Sieving Final Customer Delivery Report:
- $d_{90} = 19.5 \ \mu\text{m}$
- $d_{50} = 9.2 \ \mu\text{m}$
- $d_{10} = 3.1 \ \mu\text{m}$
- Final Delivered Span: $\frac{19.5 - 3.1}{9.2} = \mathbf{1.78}$ (PASS: $< 1.80$ )

# 5. Summary Table of Governing PSD Equations

Parameter	Symbol / Variable	Governing Equation	Unit
Cumulative Percentile Size	$d_{10}, d_{50}, d_{90}$	Laser Diffraction / Dynamic Light Scattering (ISO 13320)	$\mu\text{m}$
Particle Size Span	$\text{Span}$	$\text{Span} = \frac{d_{90} - d_{10}}{d_{50}}$	Dimensionless
Cubic Cooling Rate	$T(t)$	$T(t) = T_i - (T_i - T_f) \left( \frac{t}{t_{\text{total}}} \right)^3$	$^\circ\text{C}$ or $\text{K}$
Agitator Tip Speed	$v_t$	$v_t = \pi \cdot D \cdot N$	$\text{m/s}$
Jet Mill Grinding Pressure	$P_g$	$d_{90,\text{output}} = K \cdot \left( \frac{F_r}{P_g^{1.35}} \right)^{0.42}$	$\mu\text{m}$
Sieve Mesh Opening	$w$	$w = \left( \frac{25.4 \text{ mm}}{\text{Mesh Count}} \right) - d_{\text{wire}}$	$\text{mm}$ or $\mu\text{m}$
Specific Surface Area	$S_w$	$S_w = \frac{6}{\rho_p \cdot d_{32}}$ (Sauter Mean Diameter $d_{32}$ )	$\text{m}^2/\text{g}$

# 6. Governing Regulatory & Quality Standards

ICH Q6A: Specifications: Test Procedures and Acceptance Criteria for New Drug Substances and New Drug Products.
ICH Q8 (R2): Pharmaceutical Development & Quality by Design (QbD) Design Space for PSD CQAs.
ISO 13320:2020: Particle Size Analysis — Laser Diffraction Methods.
ASTM E11-22: Standard Specification for Woven Wire Test Sieve Cloth and Test Sieves.
NFPA 654 / ATEX Directive 2014/34/EU: Prevention of Fire and Dust Explosions from Manufacturing and Handling Processed Particulate Solids.