Hidden Phase Transition in Single-Polymer Stretching under Elongational Flow

Aiqing Zhu; Fushuai Wang; Eleonore Vissol-Gaudin; Qianxiao Li; Beatrice W. Soh

cond-mat.softOct 29, 2026

Hidden Phase Transition in Single-Polymer Stretching under Elongational Flow

Authors: Aiqing Zhu, Fushuai Wang, Eleonore Vissol-Gaudin, Qianxiao Li, Beatrice W. Soh

Organizations: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076, Singapore · Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, 117585, Singapore · Institute for Functional Intelligent Materials, National University of Singapore, 4 Science Drive 2, 117544, Singapore · Fujitsu Research of Europe, 3–9 Albert Street, Slough SL1 2BE, United Kingdom

Abstract

Using machine learning, we identify a second transition in polymer chain stretching driven by planar elongational flow, beyond the well-known coil-stretch transition. We achieve this by developing a conditional extension of Universal and Identifiable Dissipative Dynamics (UIDD) that jointly learns reduced stochastic dynamics from microscopic trajectories across flow strengths and yields the corresponding stationary nonequilibrium free energy landscapes. As the flow strength increases, the evolving landscape exhibits the expected coil-stretch transition, and then a previously unrecognized dynamical transition. In the latter, the stretched states persist, but the distribution along a folding coordinate becomes bimodal when conditioned on small end-to-end separation. Large-deviation analysis of a simplified analytical chain model reproduces both transitions and shows that they are phase transitions in the long-chain limit. Single-molecule DNA experiments provide qualitative support for the dynamical transition through the predicted reversal of the folding response.

Figures & tables

Figure 1: Overview of the polymer system and transition analysis. ( a ) A single polymer chain is represented by a bead-rod model, subjected to planar elongational flow, with chain extension measured along the elongational axis. ( b ) Plot of chain extension as a function of time across varying flow strengths. ( c ) Microscopic trajectories are reduced by PCA and fitted with conditional UIDD to identify phase transitions; large-deviation theory independently supports both transitions, while single-molecule DNA experiments provide qualitative support for the second transition.

Figure 2: Free energy landscapes and transition diagrams. ( a-d ) Learned marginal free energies $F_{2}(Z_{2})$ and sections $F_{23}(0,Z_{3})$ at the indicated $\mathrm{Wi}$ . Insets show the corresponding landscapes $F_{23}(Z_{2},Z_{3})$ . ( e,f ) Near-minimizing sets of the learned $F_{2}$ ( $\eta=0.6$ ) and $F_{23}(0,\cdot)$ ( $\eta=1.0$ ), respectively, defined by $\{q:F(q;\mathrm{Wi})-\min_{q^{\prime}}F(q^{\prime};\mathrm{Wi})\leq\eta\}$ . Color intensity indicates the number of the five independently trained models satisfying this condition. For each model, the transition point $\mathrm{Wi}_{t}$ is determined by the first crossing of $F(0;\mathrm{Wi})-\min_{q}F(q;\mathrm{Wi})$ above $\eta$ . Orange dashed lines and bands indicate the mean and range of $\mathrm{Wi}_{t}$ across models, respectively, with sensitivities shown in the insets. ( g-j ) Free energy profiles per bond of the analytical model, $F_{2,n}(z_{2})$ and $F_{23,n}(0,z_{3})$ . ( k,l ) Minimizing sets of $F_{2,\infty}(z_{2})$ and $F_{23,\infty}(0,z_{3})$ , respectively. Beyond the second transition, the gray region in ( l ) denotes the minimizing interval, with diamonds marking its endpoints.

Figure 3: ( a ) Definition of the image-based folding coordinate $\widetilde{Z}_{3}$ . ( b ) Schematic of the single-well to double-well transition in the free energy landscape and the corresponding increments near the origin. ( c ) Experimental signatures of folding reorganization. Symbols show the odd component of the conditional finite-lag mean increment of $\widetilde{Z}_{3}$ . Error bars indicate uncertainty in the estimated response. Curves show polynomial fits; triangles mark zero crossings interpolated from the binned data at 40 V. Insets show apparent potentials obtained as negative antiderivatives of the fitted polynomials.

Figure 4: Correlations of the PCA coordinates. Top: $Z_{2}$ versus the signed axial endpoint difference, $r_{0,x}-r_{N,x}$ . Bottom: $Z_{3}$ versus the signed foldedness, $(r_{0,x}+r_{N,x})/2-r_{\mathrm{com},x}$ . Correlation coefficients are computed separately for each $\mathrm{Wi}$ interval.

Figure 5: Correlations of the image-based coordinates. The image-based coordinate $\tilde{Z}_{3}$ is extracted from images generated from simulated polymer conformations, whereas the microscopic folding asymmetry $x_{1}+x_{N}-2x_{\mathrm{com}}$ are computed directly from the simulated configurations.

Figure S1: Transient stretching from the fast, medium, and slow initial conformations (column groups) at different $\mathrm{Wi}$ (rows). Each group compares the mean and standard deviation of $Z_{1}$ and the distribution of $n_{\mathrm{u}}$ defined in Eq. ( S23 ) obtained from microscopic simulations (blue) and the conditional UIDD model (orange), using $500$ realizations per ensemble. Horizontal axes are given in output-step units.

Figure S2: Experimental response checks and sensitivity to analysis choices. (a,b) Raw conditional response

f

(gray circles) and odd component

g

(red squares) at

20

and

40

V. (c) Central slopes

s_{0}

20

and

40

V for the

72

admissible analysis settings.

Phase Transitions

Phase transitions are abrupt changes in collective behavior as temperature, density, noise, or other control parameters vary.

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Langevin Flow Maps: Efficient Molecular Dynamics and Transition Path Sampling

Sam McCallum, Niklas Rindtorff, Alexander Tong, James Foster

University of Bath · AITHYRA

Stochastic Differential Equations Machine Learning Interatomic Potentials Langevin Dynamics Flow Map Learning Molecular Dynamics Simulation

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Molecular dynamics simulations proceed by integrating the Langevin equations over many small femtosecond timesteps. This poses a challenge for estimating ensemble properties and transition dynamics that occur on much longer timescales. We introduce Langevin Flow Maps, which extend machine-learned force-fields to additionally learn the stochastic Langevin integrator. We show that Langevin Flow Maps enable large-timestep molecular dynamics and recover accurate dynamical properties of the system, while running an order of magnitude faster than current machine-learned force fields. Further, by training on a diverse molecular dataset, we demonstrate a path towards transferable Langevin Flow Maps.

Abstract

Phase Transitions

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