Model-Based Optimization of Solid-Supported Micro-Hotplates for Microfluidic Cryofixation

At a Glance

Metadata	Details
Publication Date	2024-08-24
Journal	Micromachines
Authors	Daniel B. Thiem, G. Szabó, Thomas P. Burg
Institutions	Technical University of Darmstadt
Citations	2
Analysis	Full AI Review Included

Executive Summary

This research focuses on optimizing solid-supported micro-hotplates for microfluidic cryofixation, a steady-state system designed to enable time-correlated live imaging and ultra-rapid freezing for structural biology.

Core Achievement: Developed and validated a lumped-element thermal domain model to optimize the geometry and material selection for maximizing cooling rates (CRs) in microfluidic cryofixation systems.
Performance Limits: The maximum achievable cooling rate is fundamentally limited by the thermal diffusivity of water, requiring CR > 10⁶ K s^-1 for pure water vitrification, corresponding to a maximum theoretical thickness of 7.6 µm.
Material Optimization: Replacing standard copper/silicon heat sinks with high-conductivity materials like diamond significantly improves performance. Diamond enables CRs > 10⁶ K s^-1 for pure water samples up to 5.4 µm thick, compared to only 2.8 µm for copper.
Biological Sample Viability: For typical biological samples, which require CRs > 10⁴ K s^-1 due to natural cryoprotection, the model predicts that vitrification is possible for millimeter-scale specimens with thicknesses up to 10 µm.
Model Validation: The domain model showed strong correlation with Finite Element Modeling (FEM) simulations (R² = 0.9279) and accurately predicted experimental results (Modeled CR: 20,407 K s^-1 vs. Measured CR: 23,782 K s^-1 for a 20 µm channel).
Design Insight: Optimal performance requires tuning the insulation layer thickness (h_ins) to balance a high thermal gradient (∆T_ins) with a sufficiently short thermal time constant (τ_ins).

Technical Specifications

Parameter	Value	Unit	Context
Target Cooling Rate (Pure Water Vitrification)	> 10⁶	K s^-1	Required to prevent ice crystal formation.
Target Cooling Rate (Biological Samples)	10⁴ to 10⁵	K s^-1	Sufficient due to cryoprotective effect of cytosol.
Theoretical Max Water Thickness (CR > 10⁶ K/s)	7.6	µm	Limit based on water thermal diffusivity alone.
Max Water Thickness (Diamond Heat Sink, CR > 10⁶ K/s)	5.4	µm	Optimized steady-state system performance.
Max Water Thickness (Copper Heat Sink, CR > 10⁶ K/s)	2.8	µm	Optimized steady-state system performance.
Measured Cooling Rate (20 µm Water Channel)	23,782	K s^-1	Experimental validation using heater resistance method.
Modeled Cooling Rate (20 µm Water Channel)	20,407	K s^-1	Domain model prediction for the fabricated system.
Steady-State Heater Power (Measured)	12.0	W	Power required to maintain 20 °C against LN₂.
Heater Material	Titanium (Ti)	N/A	Used for active heating and resistance sensing (α_R = 0.003 K^-1).
Heat Sink Temperature (T_LN2)	-196	°C	Cryogenic bath temperature.
Water Channel Thickness (h_water)	20	µm	Fabricated system dimension.
PDMS Layer Thickness (h_pdms)	4	µm	Layer between heater and water channel.
Polyimide (PI) Insulation Layer Thickness (h_ins)	4	µm	Layer between heater and silicon substrate.
Silicon Wafer Thickness (h_si)	500	µm	Substrate thickness.
Thermal Conductivity (Diamond)	2900	W m^-1 K^-1	Optimized heat sink material.
Thermal Conductivity (Copper)	401	W m^-1 K^-1	Fabricated heat sink material.
Thermal Conductivity (Silicon, Low Temp)	950 to 1670	W m^-1 K^-1	Temperature-dependent range (115% to 360% increase).

Key Methodologies

The system relies on a combination of microfabrication techniques for the heater and microfluidic channel, coupled with a specialized thermal modeling approach for optimization.

1. Device Fabrication and Assembly

Heater Fabrication:
- Polyimide (PI-2610) insulation layer spin-coated and cured on a double-sided polished silicon wafer.
- PI surface roughened using CF₄ plasma to enhance adhesion.
- Titanium (Ti) heater deposited via electron-beam evaporation and patterned using photolithography.
- Electrical contacts established with an evaporated Gold (Au) layer.
- Backside metallized (Cr/Cu) and soldered to a bulk copper heat sink using Indium for low thermal resistance.
Microfluidic Channel Fabrication:
- SU-8 photoresist master mold created on a separate silicon wafer to define channel geometry.
- Parylene coating applied to the master mold for easy demolding.
- PDMS spin-coated, partially cured, and bonded to a mechanically strong PDMS structure.
- Channels sealed by plasma bonding a thin (4 µm) spin-coated PDMS layer to minimize thermal resistance between the water and the heater.
Assembly:
- PDMS channel structure placed directly onto the heater.
- System fixed in position and copper heat sink immersed in liquid nitrogen (LN₂) to establish the steady-state counter-cooling.

2. Thermal Modeling and Optimization

Domain Model Development:
- A two-dimensional heat conduction model was established, assuming infinite domain in the z-direction.
- The geometry was simplified into a combined linear domain (for the wide channel center) and a semi-cylindrical domain (for the edges).
- The system was represented as a lumped-element thermal equivalent circuit (Cauer-type network) consisting of thermal resistors (R_th) and capacitors (C_th) for each layer (water, PDMS, insulation, heat sink).
- The model was implemented and automated using ngspice (circuit simulation) and Julia (data processing).
Temperature-Dependent Modeling:
- The thermal conductivity of the silicon layer was modeled using voltage-dependent resistors to account for the drastic increase in silicon conductivity at cryogenic temperatures (down to -180 °C).
Cooling Rate Estimation:
- An analytical estimator was derived based on the time constants of the water layer (τ_w) and the insulation layer (τ_ins) to quickly predict the cooling rate (CR) based on the temperature difference across the insulation layer (∆T_ins).
Model Validation:
- FEM Comparison: The domain model was validated against full Finite Element Modeling (FEM) simulations (COMSOL Multiphysics) across a wide range of geometric parameters (Table 2), showing high accuracy (R² = 0.9279).

3. Experimental Validation

Heater Resistance Method: The temperature change at the heater was measured by monitoring the change in the heater’s electrical resistance during cooldown, leveraging the known temperature coefficient of resistance of Titanium. This data was used to derive the heat flow and calculate the cooling rate in the water channel.
Fluorescent Dye Method: Rhodamine B, a temperature-dependent fluorescent dye, was used to optically indicate the initial temperature change in the water channel during cryofixation.

Commercial Applications

The optimized micro-hotplate technology provides ultra-rapid, time-correlated thermal control, making it valuable for applications requiring precise temperature dynamics and high cooling rates.

Structural Biology & Microscopy:
- Cryogenic Electron Microscopy (cryo-EM): Preparation of biological samples (cells, tissues) for cryo-EM without structural artifacts caused by ice crystal formation.
- Time-Resolved Cryofixation: Capturing fast biological processes (sub-second dynamics) by coupling cryofixation with live optical imaging (e.g., Time Resolved Cryo-Correlative Light and Electron Microscopy).
Microfluidics and Lab-on-a-Chip:
- Rapid PCR: Achieving fast heating and cooling cycles to significantly improve the cycle time for Polymerase Chain Reaction (PCR) analysis.
- Micro-Actuation and Mixing: Utilizing rapid temperature changes for thermal micromixers, microbubble actuation, and actuation using thermally expandable polymers.
Separation Science:
- Temperature Gradient Focusing (TGF): Creating steep, controlled temperature gradients at the edge of the heater for separating ionic species in solution.
Digital Microfluidics:
- Droplet Actuation: Generating steep temperature gradients to increase thermocapillary effects, enabling precise control and actuation of pico- and nano-droplets.

View Original Abstract

Cryofixation by ultra-rapid freezing is widely regarded as the gold standard for preserving cell structure without artefacts for electron microscopy. However, conventional cryofixation technologies are not compatible with live imaging, making it difficult to capture dynamic cellular processes at a precise time. To overcome this limitation, we recently introduced a new technology, called microfluidic cryofixation. The principle is based on micro-hotplates counter-cooled with liquid nitrogen. While the power is on, the sample inside a foil-embedded microchannel on top of the micro-hotplate is kept warm. When the heater is turned off, the thermal energy is drained rapidly and the sample freezes. While this principle has been demonstrated experimentally with small samples (<0.5 mm2), there is an important trade-off between the attainable cooling rate, sample size, and heater power. Here, we elucidate these connections by theoretical modeling and by measurements. Our findings show that cooling rates of 106 K s−1, which are required for the vitrification of pure water, can theoretically be attained in samples up to ∼1 mm wide and 5 μm thick by using diamond substrates. If a heat sink made of silicon or copper is used, the maximum thickness for the same cooling rate is reduced to ∼3 μm. Importantly, cooling rates of 104 K s−1 to 105 K s−1 can theoretically be attained for samples of arbitrary area. Such rates are sufficient for many real biological samples due to the natural cryoprotective effect of the cytosol. Thus, we expect that the vitrification of millimeter-scale specimens with thicknesses in the 10 μm range should be possible using micro-hotplate-based microfluidic cryofixation technology.

Tech Support

Original Source

DOI: https://doi.org/10.3390/mi15091069

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