Expansion Kinetics of Flexible Polymers upon Release from a Disk-Shaped Confinement

A general theory is developed to explain the expansion kinetics of a polymer released from a confining cavity in a d-dimensional space. At beginning, the decompressed chain undergoes an explosive expansion while keeping the structure resembling a sphere. As the process continues, the chain transitions to a coil conformation, and the expansion significantly slows down. The kinetics are derived by applying Onsager’s variational principle. Computer simulations are then conducted in a quasi-two-dimensional space to verify the theory. The average expansion of the chain size exhibits a distinctive sigmoidal variation on a logarithmic scale, characterized by two times and associated exponents that represent the fast and the slow dynamics, respectively. Through an analysis of the kinetic state diagrams, two important universal behaviors are discovered in the two expansion stages. The intersection of the expansion speed curves allows us to define the crossover point between the stages and study its properties. The scaling relations of the characteristic times and exponents are thoroughly investigated under different confining conditions, with the results strongly supporting the theory. Additional calculations conducted in a three-dimensional (3D) space demonstrate the robustness of the proposed theory in describing the kinetics of polymer expansion in both 2D and 3D spaces.

The factor z appearing at the end of Eq. ( S2), due to the differentiation, can be disregarded when studying scaling, in order to avoid distraction.It can be considered as part of the prefactor in Eq. (S3), which is not explicitly expressed there.Given the initial chain size R 0 , the differential equation can be solved: Here, the factor 1 z+1 , which is related to the integration on the left-hand side of the equation, has been omitted following the same principle used in scaling derivation.We can rewrite the equation in a more precise form: by reintroducing the prefactor, referred to as c 1 .The equation is then simplified to where α 1 = 1 z+2 = dν b −1 2(dν b −1)+d and τ 1 = c −1 . For a well-confined chain, the initial chain size R 0 scales with the cavity diameter D. Therefore, we can relate R 0 to the confining volume fraction ϕ 0 using the , which can be deduced to be τ . Since η0σ 2 kBT represents the Rouse time, which is responsible for the time unit, we know that A 1 is indeed a dimensionless prefactor.
In the second stage of expansion, the chain adopts a coil conformation and the Flory free energy is utilized.The kinetics can be solves from the equation, dR dt + kBT R d+1 ∼ 0, derived from Eq. (S1).It takes the basic form of a Bernoulli differential equation as dR dt + aR = bR p , and the solution is given by where c is a factor.Therefore, the evolution of the chain size can be solved and takes the form as a recovery function powered by a specific exponent α 2 = 1 d+2 : In the solution, R F represents the final chain size and the well-known scaling relation R F ∼ σN ν with ν = 3 d+2 is recovered.The characteristic time τ 2 scales as where A 2 is the prefactor.
Eq. (S7) also includes a factor a c , for which I propose setting the value to 1.This choice is based on the observations from simulations in order to effectively describe expansion behavior for the case of long chain with a high confining volume fraction.Further explanation will be provided at the end of Section S4, near Figure S11.

S2. SIMULATION DETAILS AND SNAPSHOTS
In this study, a chain is modeled by using a bead-spring chain approach, and the 2D expansion behavior of the chain is investigated within a slit confinement.To comply with the geometrical constraint in a two-dimensional space, the slit height H has to be sufficiently small to prevent the occurrence of passing of one portion of the chain over the other portion in the confinement.The threshold height for the passing-over is Illustration showing a critical situation where a monomer, the middle yellow bead near the top wall (represented as a blue line), is allowed to pass through the upper cleavage space created by the connected monomers near the bottom wall of a slit confinement.The threshold height of the slit can be calculated to be selected to be small enough to prevent the passing-over, while also allowing for the system to relax at a fast rate.A disk-shaped cavity is created by enclosing it with a large cylindrical side wall perpendicular to the slit confinement.The diameter of the cylindrical wall is D. For each studied (N , ϕ 0 ) case, 1000 independent configurations of polymer confined in the disk cavity are prepared prior to the releasing process.A configuration is obtained by loading a chain through a small tube perpendicular to the confining disk, by pumping the monomers downward in the tube, as illustrated in Figure S2(a).After loading, the chain is equilibrated in the disk-shaped cavity (see Figure S2(b)) for a minimum of 10 8 timesteps.Random forces given by the Langevin thermostat are generated by using different seeding numbers in order to produce independent configurations.Snapshots of chain expansion during a releasing process from ϕ 0 = 0.6 are presented in Figure S3 .The topview images show that the chain initially expands with a disk-like structure.It gradually evolves and adopts a coil conformation over time.
The bonding between adjacent monomers on a chain in the simulations is modeled using a harmonic potential with a large spring constant, k=6000 [k B T /σ2 ].The bond length, denoted by b, has been studied within the confining cavity during the equilibrating phase.The mean value is obtained: ⟨b⟩ = 1.00 with a standard deviation of 0.01.The maximum b detected in all of the runs is 1.049956.For the selected slit height H = 1.8, the passing-over of a monomer through the upper cleavage space of two connected monomers at the bottom (refer to the schematic in Figure S1) would require a stretching of the bond length between the two monomers to be 1.2.This represents an extremely rare event and is highly unlikely to happen because it involves stretching the bond length by about twenty times the standard deviation.I note that there have been no detection of such passing-over events in any of the runs.I remark that despite the utilization of the large spring constant, the chain retains its flexibility due to the modeling of the monomer beads using the Weeks-Chandler-Anderson potential, which does not involve any frictional force between the beads.As a result, a bead is able to move freely in any direction tangent to the connected bead while maintaining the bond length of about 1.0.Therefore, the chain remains flexible.

S3. EVOLUTION OF CHAIN SIZE PLOTTED ON LINEAR SCALE
The chain size is characterized by calculating the radius of gyration, defined by illustrates the time evolution of R g for N = 512 in five single runs, released from the initial confinement condition ϕ 0 = 0.6.These curves are plotted using colored thin lines.Fluctuations of the curves can be smoothed out by calculating the average R(t) = R 2 g 1/2 over 1000 single runs.The average curve has been plotted by using the thick black line in the figure for comparison.Initially, the chain exhibits rapid expansion, leading to a sharp increase in the evolution curve that may not be clearly discernible on a linear scale.On the other hand, the recovery of the chain to its natural size during the later stage requires a relatively longer time, which is strongly dependent on the chain length N .Therefore, studying the expansion behavior on a logarithmic scale, as shown in Figure 1(a) in the paper, is more suitable.
The average curves for the case of (N, ϕ 0 ) = (512, 0.6) calculated from 10, 100, and 1000 independent single runs are presented in Figure S6, plotted using both a linear and a logarithmic time scale.One can observe that the fluctuations decrease as the number of runs involved in the averaging increases.Notably, significant fluctuations persist still in the average curve over 100 runs.It indicates that tens of measurements of individual expansion curves are usually insufficient for the study of the kinetics with a high level of precision.
In this study, the damping time parameter for the Langevin thermostat is set as τ L = 0.05.Recall that the physical quantities are described in terms of the simulation units where m represents mass, k B T represents energy, and σ represents length.To map the simulation to a real system, like a single-strand DNA chain, one can choose the following values: m = 320g/mol = 5.33 × 10 −25 kg, k B T = 300k B = 4.14 × 10 −21 J, and σ = 3.4 Å= 3.4 × 10 −10 m.Using these values, the simulation time unit is t u = σ m kBT = 3.86ps.The dynamic viscosity ξ of solvent can be then estimated from the Stokes' law η 0 = 3πσξ, where the friction coefficient η 0 of a monomer is equal to m τL = 2.76 × 10 −12 kg • s −1 .The obtained value is ξ = 0.862cP, which aligns with the dynamic viscosity of water at room temperature.Therefore, this model effectively simulates expansion of a ssDNA chain in an aqueous environment.It is expected to exhibit comparable variations and fluctuations in behavior.The scaling behavior of the R curves are studied by calculating the two quantities: (1) R ≡ R R0 and (2) R ≡ R RF .The first quantity measures the expansion ratio and evolves from the initial value of 1.The second one describes the progress for the completion of the expansion and tends toward the final value, one.The two kinds of curves can be horizontally shifted by multiplying time by a factor ω 9−g 1,2 .Here ω 1 and ω 2 are two parameters that determine the amount of shift applied to the respective curves, and the value of g = log 2 N represents the exponent associated with the change of chain length in the study.By appropriately selecting values for ω 1 and ω 2 , the curves in the small and in the large time regions can be separately collapsed onto the one for N = 512.
The values were determined by calculating the mean distribution widths, ⟨W 1 ⟩ and ⟨W 2 ⟩, for the two sets of collapsed curves in the respective time regions, t ω1 < 10τ 1 and t ω2 > 0.5τ 2 .The minimum mean widths were then searched.Figure S7 illustrates how the widths of the collapse in the two regions change with the choice of ω 1 and ω 2 for the case ϕ 0 = 0.6.In the middle panel of Figure S7(a), we can see that ω 1 = 2.03 gives a small width of collapse within the time region t ω1 < 10τ 1 (in light-gray color), compared to ω 1 = 1.53 in the top panel, which yields a larger width owing to an insufficient shift of the R curves along the time axis, and ω 1 = 2.53 in the bottom panel, which gives too much shift.For the second expansion stage, a small mean width ⟨W 2 ⟩ of collapse can be achieved by choosing ω 2 = 6.36, as shown in the middle panel of Figure S7(b) within the light-gray region.Decreasing or increasing the value of ω 2 results in under-shift or over-shift of the curves, respectively, leading to a larger distribution width of those R curves, as seen in the top and the bottom panels.
The calculated mean widths ⟨W 1 ⟩ vs. ω 1 and ⟨W 2 ⟩ vs. ω 2 are plotted in Figure S8 for various ϕ 0 .The ω 1 and ω 2 values for the occurrence of the minimum mean width, accurate to the second decimal place, have been reported in Figure 2 of the paper, and in the next two figures, Figure S9 and S10.
Figure S9 is a replot of Figure 2 in the paper, with an enlarged time region for small R and an enlarged time region for large R. The zoom-in views provide a clearer visualization of the collapse of the curves and the corresponding fit for the two releasing cases with ϕ 0 = 0.6 and ϕ 0 = 0.3.
The collapses of the two sets of curves for the cases of loose confinement, ϕ 0 = 0.15 and 0.075, are shown in Figure S10.As we can see, the procedures still work well, even if ϕ 0 is small.The fitting parameters α 1 , τ 1 , α 2 , and τ 2 are reported in the respective panels in the figure.
In the simulations, it is observed that the set of the chain size evolution curves for a give length N , but released from different confining conditions ϕ 0 , collapse together as t increases, specifically when t becomes comparable to the second characteristic time τ 2 (refer to Figure S11(a)).Moreover, the R(t) curves progressively curve downwards in the small time region as ϕ 0 increases and approach a limiting curve.This limiting curve can be interpreted as deducing from the extreme scenario of ϕ 0 → ∞.In this hypothetical situation, the chain should have a size of zero We can see that the function represents an appropriate limiting curve for the simulation curves, effectively describing the variations of chain size in the large time region, i.e. in the second expansion stage.It is important to note that the setting should be applicable for the case of long chains with a high confining volume fraction.However, in the simulations, not all values of N and ϕ 0 correspond to the case.The constant use of a c = 1 allows us to test the robustness of the analysis.Despite this approximation, the results remain consistent and satisfactory, demonstrating the validity of using a c = 1 in the study.
. The chain length N can be found in the legend of Panel (a2).The optimal values of ω 1 and ω 2 for achieving the best collapse, as well as the fit parameters α 1 , τ 1 , α 2 , and τ 2 , are reported in the respective panels.The fitting curves for the first and second expansion stages are represented by gray and magenta dashed lines, respectively.The same plot as in panel (a), but with the chain size replaced by the normalized chain size R = R/R F .The magenta dash-dotted line represents the predicted curve for the case of N = 512, by using Eq.(S7) (or Eq. (3) in the paper) with a c = 1.

S5. KINETICS OF EXPANSION
The kinetics of expansion can be studied by calculating the speed of expansion V = dR/dt as a function of the transition coordinate, the chain size R.According to my theory, the speed should behave as in the first stage.Figure S12(a) plots d R/dt vs. R for ϕ 0 = 0.6.We can see that the curves show scaling variations in the small R region and are parallel to each other for different N with a scaling exponent equal to −8.68(7).Similarly, in the second stage of expansion, the theory predicts R1− 1 α 2 − R , for different values of N have been plotted as magenta dashed lines for comparison, with α 2 set to the 2D value of 0.25.These curves exhibit an asymptotic behavior of R−3 as R becomes small.Due to the smaller value of α 1 compared to α 2 , the speed profile in the log-log plot exhibits a change in slope from 1 − 1 α1 to 1 − 1 α2 .This transition is clearly observed in the case of N = 512 near R = 0.5.As a result, the second-stage expansion displays a distinctive hip curve, rapidly decreasing to zero as R approaches one.
The crossover between the first and second stages of expansion can be studied by equating the expansion speed given by Eq. (S8) and Eq.(S9), which defines the chain size R * at crossover.The R * is then used to determine the corresponding crossover time t * on the plot of R versus t, as shown in Figure S13, where the open circle symbols denotes the crossover points found.We can see that these crossover points align basically on a straight line on the log-log plot, exhibiting a power-law behavior as R * ∼ t 0.326(19) * .On the left of the line is the region for the occurrence of the first expansion stage, while on the right is the region for the one of the second stage.The methods and procedures described in this study are utilized to reanalyze the expansion data of polymers released from a spherical cavity in a three-dimensional space (original data published in Polymers 15 (2023) 198).The reanalysis is conducted to verify the applicability of the theory and the methods across different spatial dimensions.
The new calculations for the collapses of the size evolution curves in the first and in the second expansion stages for the 3D simulations are given in Figure S14.Four confining volume fractions, ϕ 0 = 0.4, 0.2, 0.1, and 0.05, in the three .The chain length N can be found in the legend of Panel (a1) and (a2).The optimal values of ω 1 and ω 2 for achieving the best collapse, as well as the fit parameters α 1 , τ 1 , α 2 , and τ 2 , are reported in the respective panels.The fitting curves for the first and second expansion stages are represented by gray and magenta dashed lines, respectively.dimensional space are investigated.The fits for the collapse of the first-stage curves are plotted as gray dashed lines atop of the data, while the ones for the second-stage curves are plotted as magenta dashed lines.The fit parameters are reported in the corresponding panels in the figure .Figure S15 presents the summarized study, concerning the variations of the characteristic times in the two expansion stages, τ 1 and τ 2 , and the associated exponents, α 1 and α 2 , with respect to the confining volume fraction ϕ 0 .As for ν b = 0.5 given in Table 1 in the paper.The second characteristic time τ 2 is of several orders of magnitude larger than τ 1 and appears to be insensitive to the volume fraction ϕ 0 of the confinement, as predicted by the theory.Panel (b) shows that α 2 is roughly constant against ϕ 0 , adopting about the predicted 3D value of 0.2, which is distinguishable from the 2D value, 0.25.On the other hand, the exponent α 1 displays a slight increasing trend from 0.11 to 0.13 as ϕ 0 decreases.The exponents χ 1 and χ 2 account for the additional dependence on N in the characteristic time beyond the predictions, resulting from the utilization of the single global speed to estimate the energy dissipation in Eq. (1) of the paper.They can be obtained by calculating χ 1 = log 2 (ω 1 )− 2 3 and χ 2 = log 2 (ω 2 )−2.The value of χ 1 increases as ϕ 0 decreases, showing a more significant change compared to χ 2 (see Figure S15(c)).
The kinetic equations are newly investigated from the 3D simulations for the case with ϕ 0 = 0.4.The variations of the dimensionless speeds, d R d t vs. R and d R d t vs. R, are presented in Panel (a1) and (a2) of Figure S16.We can see that the three-dimensional study recovers the two featured universal behaviors, represented by the gray dashed and magenta dashed lines, respectively.On the logarithmic scales, the curves exhibit a slope of −8.51(8) in the first stage.They then transition to show an asymptotic slope of about −4 when the expansion process enters the second stage (refer to Panel (a2)).Towards the end of the process, the speed decreases sharply to zero.Figures S16(b1 FIGURE S2: (a) Snapshots showing how a chain is loaded into a disk-shaped cavity.The chain is pumped downward through a small tube (not shown) perpendicular to the disk.The yellow beads represent the monomers and the blue circle indicates the bottom wall of the cavity.The top disk wall and the cylindrical side wall are not shown for clear visualization of the chain.(b) Snapshot of a chain equilibrated inside the disk-shaped cavity.The chain length is N = 512 and the confinement has ϕ 0 = 0.6.

FIGURE S3 :
FIGURE S3: Snapshots of a chain in the expanding phase at (a) t = 0.0, (b) t = 4.5 × 10 4 , (c) t = 2.3 × 10 5 , (d) t = 6.3 × 10 5 , (e) t = 2.3 × 10 6 , and (f) t = 6.5 × 10 6 .The upper wide pictures in the panels present the top-view images of the system, while the lower narrow ones show the side-view images.The monomers are represented by yellow beads.The blue circle in a upper picture indicates the location of the cavity which confines the chain prior to the release.The chain length is N = 512 and the confining condition is ϕ 0 = 0.6.
FIGURE S4: (a) Probability density distribution p(b) of bond length plotted for ϕ 0 = 0.6 while varying the chain length N .(b) Distribution p(b) for N = 512 plotted by varying ϕ 0 .
FIGURE S5: (a) Evolution of chain size, R g , in five single runs represented by colored thin lines.For comparison, the average curve over 1000 independent single runs is plotted using the black thick line.(b) The average curves for different chain lengths released from the confining condition of ϕ 0 = 0.6.
FIGURE S6: Time evolution curves averaged from 10, 100, and 1000 independent runs for N = 512 released from ϕ 0 = 0.6, plotted using (a) a linear and (b) a logarithmic time scale.
FIGURE S7: (a) Collapse of the R curves plotted against t ω1 = t × ω 9−g 1 using three different ω 1 values, 1.53, 2.03, and 2.53.The mean collapse width ⟨W 1 ⟩ is calculated within the light-gray region, t ω1 < 10τ 1 .(b) Collapse of the R curves plotted against t ω2 = t × ω 9−g 2 using three different ω 2 values, 5.76, 6.36, and 6.96.The mean collapse width ⟨W 2 ⟩ is calculated within the light-gray region, t ω2 > 0.5τ 2 .Here g = log 2 N is the exponent associated with the change of chain length in the study.

FIGUREFIGURE S9 :
FIGURE S8: (a) Mean distribution width ⟨W 1 ⟩ of the collapse curves in the first expansion region as a function of the parameter ω 1 .(b) Mean distribution width ⟨W 2 ⟩ of the collapse curves in the second expansion region as a function of the parameter ω 2 .
FIGURE S11: (a) Evolution curves R(t) of chain size are plotted at a given N for various ϕ 0 , as indicated in the legend of the panel.The value of N is provided near the set of the curves corresponding to this chain length.(b)The same plot as in panel (a), but with the chain size replaced by the normalized chain size R = R/R F .The magenta dash-dotted line represents the predicted curve for the case of N = 512, by using Eq.(S7) (or Eq. (3) in the paper) with a c = 1.
FIGURE S12: (a) d R/dt (or equivalently V /R 0 ) vs. R.(b) d R/dt (or equiv.V /R F ) vs. R.The confining condition before release is ϕ 0 = 0.6.The chain length N can be read from the legend in Panel (a).The gray dashed line shows the power-law behavior in the 1st expansion stage while the magenta dashed lines indicate the predicted behaviors in the 2nd stage.
Figure S12(b) presents the calculation results for d R dt versus R. The theoretical curves, d R dt = α2 τ2
) and S16(b2) present the variation curves of d R dt vs. R and d R dt vs. R, respectively.The two plots provide a clear visualization of the individual kinetics in the first and second expansion stages.
FIGURE S10: R vs. t ω1 and R vs. t ω2 for (a1,a2) ϕ 0 = 0.15 and (b1,b2) ϕ 0 = 0.075.Recall the definitions: R = R R0 Expansion chain size R vs. t for the case of ϕ 0 = 0.6.The open circle symbols indicate the crossover points between the first and the second expansion stages for a given chain lengths N .These points exhibit a scaling relation R * ∼ t