Poisson–Boltzmann Equation for Microfluidic Transport Phenomena with Statistical Thermodynamics Approach
Bibliographic record
Abstract
With the advent of microfluidics and lab-on-chip systems, DNA and protein separation technologies are being developed for biology, diagnostics, and health purposes. Fully realizing these applications requires developing numerical models for sample transport. In this paper, a thorough investigation of electrokinetics and microfluidics transport phenomena reviews the background of the Poisson–Boltzmann equation. A detailed derivation of the equation is presented, which is not available in the microfluidic literature at one place, with the view of providing a more consolidated and comprehensive understanding of it. This equation is then applied to find the electric potential and charge density distributions in the electric double layer (EDL). The present study provides a detailed derivation of the Boltzmann distribution, and principles of probability are used to identify the most-probable ion distribution. This distribution is subject to constraints of constant number of particles and total energy of the system; Lagrangian multipliers are used to solve the resulting constrained optimization problem. Classical thermodynamics is shown to be consistent with the distribution of ions: the Boltzmann distribution. Then, based on Coulomb’s law, the derivation of Poisson’s equation, and its special form of Laplace’s equation, the electric potential distribution in the EDL and in the bulk flow is derived and presented. By applying classical thermodynamics and integrating the Boltzmann distribution and Poisson equation together, the Poisson–Boltzmann equation is achieved.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".