Polymer Nanopore Translocation
A polymer chain threading through a nanometer-scale pore is the capstone problem of this course. Every concept we have studied — conduction, diffusion, convection, Joule heating, free energy, and polymer dynamics — operates simultaneously in a volume barely larger than the chain itself. Nanopore technology has already made portable DNA sequencing possible, and it is beginning to do the same for synthetic polymers. This chapter gives you the physical tools to understand both.
Reading DNA One Base at a Time
The Oxford Nanopore MinION is a USB-stick-sized device that sequences a human genome in hours. Inside it are thousands of protein pores — each about 1 nm in diameter — embedded in a lipid bilayer membrane. DNA is driven through each pore by an electric field. As each nucleotide passes through the narrowest point, it partially blocks the ion current flowing through the pore. The size of the blockade identifies the base: A, T, G, or C.
Every physics concept in this textbook is present in that pore. The ion current obeys Ohm's law. The electric field does work on the ions, heating the pore by Joule dissipation. The polymer chain loses entropy as it is squeezed into the narrow constriction. Diffusion and drift compete to set the translocation speed. And the temperature at the pore centre — set by the balance between Joule heating and conduction away through the membrane — shifts the rate by an Arrhenius factor.
The same physics applies to synthetic polymer chains: PEG, polystyrene, nylon — any flexible chain can in principle be read at the monomer level through a nanopore with the right chemistry.
Fig. 16.1 — Nanopore translocation. A voltage drives K+ and Cl− ions through the pore, producing a baseline current $I_0$. When a polymer chain threads through, it partially occupies the pore, reducing the available cross-section for ion flow and producing a transient current blockade $\Delta I$.
16.1 Nanopore Geometry and Ionic Conductance
A nanopore is a hole in a thin membrane with a diameter between 1 nm and 100 nm. Biological pores (α-hemolysin, MspA) are protein complexes with precisely defined inner diameters. Solid-state pores are drilled in silicon nitride (SiN), silicon dioxide (SiO2), or graphene membranes using a focused electron or ion beam. Polymer track-etched pores are made by bombarding films of PET, PC, or PI with heavy ions and then etching the damage tracks with NaOH.
Pore Resistance
Model the pore as a cylinder of length $L$ and radius $r$ filled with electrolyte of conductivity $\sigma$ (S/m). The electrical resistance is the same formula as thermal resistance for a cylinder:
The analogy with thermal resistance is exact: $R_{\text{pore}} = L/(\sigma A)$ mirrors $R_{\text{th}} = L/(kA)$. The conductivity of the electrolyte $\sigma$ plays the role of thermal conductivity $k$. This is why all the circuit network tools from Chapters 2–4 apply directly to ionic circuits.
Once the resistance is known, Ohm's law gives the open-pore current:
Access Resistance
For very short or wide pores, the resistance of the hemisphere of solution converging onto the pore entrance on each side is comparable to the pore resistance itself. This is called access resistance:
For a pore with $L$ = 20 nm and $d$ = 10 nm, the ratio of access to pore resistance is $(L/(\pi r)) = 20/(5\pi) \approx 1.3$, so the two are comparable. For biological protein pores that are effectively atomically thin at their narrowest constriction, access resistance can dominate entirely.
Worked Example 16.1 — Pore Conductance
A SiN nanopore has diameter $d$ = 10 nm and length $L$ = 20 nm. The electrolyte is 1 M KCl ($\sigma$ = 10.0 S/m). Calculate the pore resistance $R_{\text{pore}}$ (from Eq. 16.1 alone) and the open-pore current at an applied voltage of 100 mV.
Pore resistance:
$$R_{\text{pore}} = \frac{L}{\sigma \pi r^2} = \frac{20\times10^{-9}}{10 \times \pi \times (5\times10^{-9})^2}$$ $$= \frac{20\times10^{-9}}{10 \times \pi \times 25\times10^{-18}} = \frac{20\times10^{-9}}{7.854\times10^{-16}} = 2.55\times10^7\;\Omega = 25.5\;\text{M}\Omega$$Open-pore current:
$$I_0 = \frac{V}{R} = \frac{0.100}{2.55\times10^7} = 3.92\;\text{nA}$$This is a typical experimental value for a 10 nm pore in 1 M KCl at 100 mV. The measurement precision required to resolve a 4% blockade (a dsDNA molecule, diameter 2 nm) is about 0.16 nA — achievable with a low-noise patch-clamp amplifier.
16.2 Barriers to Translocation
Driving a polymer chain through a nanopore requires overcoming two types of resistance: enthalpic barriers from polymer-wall interactions, and entropic barriers from the loss of conformational freedom inside the pore.
For a neutral polymer in an uncharged pore, the enthalpic term is often small. The dominant barrier is entropic: the chain loses a large number of accessible conformations when it is forced into the narrow cylindrical space of the pore.
The Blob Model and Entropic Confinement
de Gennes introduced the blob model to describe polymer confinement in a pore of diameter $d$. The key idea: if the pore is narrower than the polymer's free-solution radius $R_g$, the chain splits into a linear sequence of blobs, each of size $\xi \approx d$.
Fig. 16.2 — The blob model. A polymer coil in free solution (radius $R_g$) entering a pore of diameter $d \ll R_g$ reorganises into a linear string of blobs, each occupying one pore diameter. The number of blobs is $N_{\text{blob}} = N/g$ where $g$ is the number of monomers per blob.
Each blob contains $g$ monomers and has diameter $\xi \approx d$. From the scaling of a self-avoiding walk in good solvent ($\nu = 0.588$):
The number of blobs in the confined chain is $N_{\text{blob}} = N/g$. Each blob costs roughly $k_BT$ of free energy to confine to size $d$, so the total entropic free energy barrier is:
The Free Energy Landscape
The full free energy profile as a function of the number of monomers already inside the pore, $s$, includes both the entropic confinement cost and the work done by the applied electric field on the charged backbone:
$$F(s) = \frac{N-s}{g}\,k_BT - s \cdot q E b$$where $qEb$ is the electrostatic energy gained per monomer in field $E$ and $q$ is the effective charge per Kuhn segment. When the electric driving force exceeds the entropic cost per monomer ($qEb \gt k_BT/g$), translocation becomes thermodynamically downhill once the chain tip enters the pore. The critical field is:
Below $E_c$ the chain retracts; above $E_c$ it threads through. In practice, applied voltages of 50–200 mV across a 5–50 nm membrane give fields of 1–40 MV/m, which comfortably exceed $E_c$ for DNA in 1 M KCl.
16.3 Joule Heating in Nanopore Membranes
The ionic current flowing through the pore dissipates electrical power as heat. This is the same Joule heating you encountered in the context of electrical cables and resistive heaters — the principle is identical, just at a nanometre scale.
For the pore in Worked Example 16.1 at 100 mV: $P = (3.92\times10^{-9})^2 \times 2.55\times10^7 = 3.92\times10^{-10}$ W = 392 pW. This is tiny in absolute terms, but it is deposited in a volume of only $\pi r^2 L \approx 1.6\times10^{-24}$ m3, giving a volumetric heat generation rate of about $2.4\times10^{14}$ W/m3 — far exceeding even a nuclear reactor core (≈ 108 W/m3).
The steady-state temperature rise at the pore centre is moderated by conduction into the surrounding solution and membrane material. Using the result for a cylindrical heat source in an infinite medium:
For the 10 nm pore at 100 mV: $\Delta T \approx 392\times10^{-12}/(4\pi \times 0.6 \times 5\times10^{-9}) = 392\times10^{-12}/(3.77\times10^{-8}) \approx 10\;\text{mK}$. At 100 mV the heating is negligible. However, at 1 V the power scales as $V^2$, giving 100$\times$ more heating (≈ 1 K) — measurable and significant.
Translocation is an activated process: the chain must overcome the barrier $\Delta F_{\text{ent}}$ to enter the pore. The rate follows an Arrhenius-type relation, $k \propto \exp(-\Delta F / k_B T_{\text{local}})$. A local temperature rise of just 5 K (achievable at higher voltages) shifts the translocation rate by $\exp(\Delta F \times \delta T / k_B T^2)$. For $\Delta F \sim 10\,k_BT$ and $T$ = 300 K, a 5 K rise changes the rate by roughly 17% — a measurable artefact if not accounted for in calibration.
Worked Example 16.2 — Joule Heating Power
Using the pore from Example 16.1 ($R$ = 25.5 MΩ, $I_0$ = 3.92 nA at 100 mV), calculate the Joule heating power and express it in picowatts (pW).
Method 1 (use P = IV):
$$P = I_0 \times V = 3.92\times10^{-9} \times 0.100 = 3.92\times10^{-10}\;\text{W} = 392\;\text{pW}$$Method 2 (use P = V²/R):
$$P = \frac{(0.100)^2}{2.55\times10^7} = \frac{0.0100}{2.55\times10^7} = 3.92\times10^{-10}\;\text{W} = 392\;\text{pW}$$Both methods agree. To put 392 pW in context: a standard LED consumes about 1 W = 1012 pW. The nanopore dissipates ten billion times less power, yet in its tiny volume the energy density exceeds most engineering heat sources.
16.4 Polymer Dynamics Under Confinement
Once the chain has entered the pore, two competing processes govern its motion: the electrophoretic drift (electric force on the charged backbone) and thermal diffusion (Brownian motion). The ratio of drift to diffusion is the Péclet number for the chain:
$$\text{Pe} = \frac{v_{\text{drift}} \cdot L}{D_{\text{chain}}} = \frac{\mu_{\text{ep}} E \cdot L}{D_{\text{chain}}}$$where $\mu_{\text{ep}}$ is the electrophoretic mobility of the chain. For dsDNA at 100 mV across a 20 nm pore, Pe ≈ 10–100: drift dominates over diffusion for most of the translocation event.
Ionic Current Blockade
When the polymer chain occupies the pore, it physically displaces ion-carrying solution from the pore volume. The fractional reduction in cross-sectional area available to ions gives the fractional current blockade:
The blockade depth thus gives a direct measurement of the chain's cross-sectional size. Different monomers along a synthetic copolymer chain may have different radii, causing a time-varying blockade signal that encodes the monomer sequence.
Translocation Time Scaling
The time for an $N$-monomer chain to cross the pore scales with chain length. In the drift-dominated regime (high voltage, Pe ≫ 1), the chain moves at roughly constant velocity, so the translocation time scales linearly:
$$\tau_{\text{trans}} \sim \frac{N \cdot b}{v_{\text{drift}}} \quad (\text{Pe} \gg 1)$$In the diffusion-dominated regime (low voltage, Pe ≪ 1), Rouse dynamics give the translocation time:
$$\tau_{\text{trans}} \sim \frac{N^{1+2\nu}}{D_{\text{chain}}} \quad (\text{Pe} \ll 1)$$For dsDNA ($\nu \approx 0.6$ in free solution), the diffusion-limited scaling gives $\tau \sim N^{2.2}$: doubling the chain length more than quadruples the translocation time. In practice, most nanopore experiments operate in a crossover regime, and calibration with known-length standards is essential.
Worked Example 16.3 — Blob Count and Free Energy Barrier
A neutral polymer chain with $N$ = 200 monomers (Kuhn length $b$ = 0.5 nm) must enter a pore of diameter $d$ = 2.0 nm. Using the blob model with $\nu$ = 0.5 (theta solvent), calculate: (a) the number of monomers per blob $g$, (b) the number of blobs $N_{\text{blob}}$, and (c) the entropic free energy barrier $\Delta F$ in units of $k_BT$.
(a) Monomers per blob (Eq. 16.4 with ν = 0.5):
$$g = \left(\frac{d}{b}\right)^{1/\nu} = \left(\frac{2.0}{0.5}\right)^{1/0.5} = 4^2 = 16$$(b) Number of blobs:
$$N_{\text{blob}} = N/g = 200/16 = 12.5$$(c) Free energy barrier (Eq. 16.5):
$$\Delta F = N_{\text{blob}} \cdot k_BT = 12.5\,k_BT$$A barrier of 12.5 $k_BT$ corresponds to a Boltzmann factor of $e^{-12.5} \approx 3.7\times10^{-6}$. Without an electric driving force, a neutral polymer chain of this length would enter this pore only 4 times in every million attempts. The electric field or a chemical gradient is essential.
16.5 Synthetic Polymer Applications
Nanopore sensing is not limited to DNA. Any polymer chain that can thread through a pore produces a blockade signal, and any variation in the chain along its length — different monomers, branches, charges, or attached labels — produces a variation in the blockade.
PEG Analysis
Poly(ethylene glycol) (PEG) produces a remarkably clear nanopore signal. Because PEG chains do not carry an intrinsic charge, they are driven through by electro-osmosis (bulk solvent flow created by the electric field acting on the double layer at the pore walls). Each chain length $N$ produces a characteristic mean blockade depth and dwell time, allowing quantitative mass spectrometry of PEG mixtures without any chromatography.
Track-Etched Polymer Membranes
PET and PC films bombarded with heavy ions from a nuclear accelerator develop latent damage tracks. Etching in NaOH opens each track into a conical or cylindrical nanopore, with the geometry controlled by etching time and temperature. The conductivity and selectivity of the resulting pores depend on the polymer chemistry: carboxyl groups on PET give pH-dependent charge. These track-etched membranes are used both as model systems for nanopore research and as ion-selective membranes in sensors and batteries.
Copolymer Fingerprinting
A block copolymer A-B-A produces a three-level current blockade as it translocates: a deep blockade while the wider-A block is in the pore, shallower while the narrower-B block passes, then deep again. This "fingerprint" identifies the copolymer architecture without destroying the chain. For alternating copolymers, the blockade oscillates with a period set by the monomer length and translocation velocity. Encoding information in the monomer sequence — synthetic information polymers readable by nanopore — is an active research frontier.
Residual monomer in a 3D-printed resin part is a quality concern — too much unreacted acrylate is a skin sensitiser. Nanopore sensing of dissolved monomer oligomers is being explored as an in-line quality control tool. A nanopore embedded in the print bath could, in principle, detect the monomer concentration continuously without sampling or chromatography.
Interactive: Nanopore Conductance Calculator
Explore how pore dimensions and electrolyte conductivity set the open-pore resistance, ionic current, and Joule heating power.
Concept Check
Answer each question in your own words, then click Get AI Feedback for Socratic guidance.
The pore resistance formula $R = L/(\sigma \pi r^2)$ is structurally identical to the thermal resistance of a slab $R_{\text{th}} = L/(kA)$. Identify what plays the role of thermal conductivity $k$ in the ionic case, and explain in physical terms why higher electrolyte concentration lowers the resistance.
Explain in physical terms why a long polymer chain faces an entropic barrier when entering a narrow pore. Why does this barrier scale with the number of blobs $N_{\text{blob}}$ rather than with the total number of monomers $N$?
A researcher doubles the applied voltage from 100 mV to 200 mV. By what factor does the ionic current change? By what factor does the Joule heating power change? If the temperature rise at the pore centre scales as $\Delta T \propto P$, by what factor does $\Delta T$ increase? Comment on whether this is a concern for sensitive translocation timing measurements.
A block copolymer A-B (where A is thicker than B) translocates through a nanopore. Sketch qualitatively the ionic current versus time signal you would expect, labelling which segment causes which current level. How could you determine the length of each block from the signal, assuming you know the translocation velocity?
A scientist wants to slow down DNA translocation by a factor of 10 to improve sequencing accuracy. They propose three strategies: (a) reducing the applied voltage, (b) increasing the solution viscosity with glycerol, (c) lowering the temperature from 37 °C to 4 °C. For each strategy, use the physics in this chapter to predict how it would affect the translocation speed AND the ionic current signal quality. Which strategy would you recommend, and why?
Problem Set
A SiN nanopore has diameter $d$ = 10 nm and length $L$ = 20 nm. The electrolyte is 1 M KCl with conductivity $\sigma$ = 10.0 S/m. Calculate the pore resistance $R_{\text{pore}}$ using Eq. 16.1. Express your answer in MΩ.
$R = L/(\sigma \pi r^2)$. Radius $r = d/2 = 5$ nm $= 5\times10^{-9}$ m. Length $L = 20\times10^{-9}$ m.
$R = (20\times10^{-9}) / (10 \times \pi \times (5\times10^{-9})^2) = (20\times10^{-9}) / (7.854\times10^{-16}) = 2.546\times10^7\;\Omega$ = 25.5 MΩ.
Using the pore from P1 ($R_{\text{pore}}$ = 25.5 MΩ), an applied voltage of $V$ = 100 mV is applied across the membrane. Calculate the open-pore ionic current $I_0$ in nanoamperes (nA).
$I_0 = V/R = 0.100\;\text{V} / (25.5\times10^6\;\Omega) = 3.92\times10^{-9}$ A = 3.92 nA.
Calculate the Joule heating power dissipated in the pore from P1 and P2 (V = 100 mV, I₀ = 3.92 nA). Express your answer in picowatts (pW).
$P = I_0 \times V = 3.92\times10^{-9} \times 0.100 = 3.92\times10^{-10}$ W.
$3.92\times10^{-10}$ W $\times 10^{12}$ pW/W = 392 pW. Alternatively, $P = V^2/R = (0.100)^2/(25.5\times10^6) = 3.92\times10^{-10}$ W.
A neutral polymer chain with $N$ = 200 monomers (Kuhn length $b$ = 0.5 nm) must thread through a nanopore of diameter $d$ = 2.0 nm. Using the blob model with $\nu$ = 0.5 (theta solvent conditions): calculate the entropic free energy barrier $\Delta F$ in units of $k_BT$.
With $\nu = 0.5$: $g = (d/b)^{1/\nu} = (2.0/0.5)^{1/0.5} = 4^2 = 16$ monomers per blob.
$N_{\text{blob}} = N/g = 200/16 = 12.5$ blobs.
$\Delta F = N_{\text{blob}} \cdot k_BT = \mathbf{12.5}\,k_BT$. Each blob costs 1 $k_BT$ of entropic penalty for confinement.
A dsDNA molecule (diameter $d_{\text{DNA}}$ = 2.0 nm) translocates through the 10 nm nanopore from P1. Using Eq. 16.9, calculate the expected fractional current blockade $\Delta I / I_0$ and express it as a percentage.
$\Delta I / I_0 = (r_{\text{chain}}/r_{\text{pore}})^2 = (d_{\text{DNA}}/d_{\text{pore}})^2 = (2.0/10)^2 = 0.04$.
$0.04 \times 100\% = \mathbf{4.0\%}$. This means detecting a $4\%$ decrease in the 3.92 nA baseline — a change of about 0.16 nA, which requires a low-noise amplifier with sub-pA noise floor.
Chapter Summary
The nanopore is the perfect capstone problem because it unifies every mode of transport we have studied. The ionic current obeys Ohm's law with a pore resistance $R = L/(\sigma \pi r^2)$ that mirrors the thermal resistance formula of Chapter 2. The ionic current dissipates power $P = V^2/R$ by Joule heating (Chapter 3), elevating the local temperature in a tiny volume and shifting the translocation rate via an Arrhenius factor.
A polymer chain entering the pore loses conformational entropy — each blob of $g = (d/b)^{1/\nu}$ monomers costs one $k_BT$ of free energy, giving a total barrier $\Delta F = N_{\text{blob}} \cdot k_BT$. For realistic chain lengths and pore diameters, this barrier is tens to hundreds of $k_BT$: an applied electric field (for charged polymers) or pressure (for neutral ones) is essential to drive translocation at measurable rates.
The current blockade during translocation, $\Delta I / I_0 = (r_{\text{chain}}/r_{\text{pore}})^2$, reports on the chain's cross-sectional size — and hence on the identity of the monomer or base currently inside the pore. This is the physical basis of nanopore sequencing. The same principle applies to synthetic polymers: any variation in monomer size, charge, or attached group is readable from the time-varying blockade signal.
The translocation time scales as $\tau \sim N$ at high driving force and $\tau \sim N^{1+2\nu}$ at low force (diffusion limited). The crossover is set by the Péclet number Pe = $\mu_{\text{ep}} E L / D_{\text{chain}}$. Slowing translocation for sequencing accuracy (Pe $\rightarrow$ 1) must be balanced against the need for a clean, high-amplitude blockade signal.
Course complete. You have now developed the full toolkit of heat and mass transfer for polymer science: conduction, convection, radiation, diffusion, dimensional analysis, transient transfer, mass transfer analogies, membrane transport, and coupled phenomena. These tools appear in every area of polymer engineering — from the nozzle of a 3D printer to a DNA sequencer smaller than your thumb. The physics is the same; only the scale changes.