# Twirling and spontaneous symmetry breaking of domain wall networks in
lattice-reconstructed heterostructures of 2D materials

M. A. Kaliteevsky<EMAIL_ADDRESS>National Graphene
Institute, University of Manchester, Booth St. E. Manchester M13 9PL, United
Kingdom  University of Manchester, School of Physics and Astronomy, Oxford
Road, Manchester M13 9PL, United Kingdom  V. V. Enaldiev  National Graphene
Institute, University of Manchester, Booth St. E. Manchester M13 9PL, United
Kingdom  University of Manchester, School of Physics and Astronomy, Oxford
Road, Manchester M13 9PL, United Kingdom  V. I. Fal’ko
<EMAIL_ADDRESS>National Graphene Institute, University of
Manchester, Booth St. E. Manchester M13 9PL, United Kingdom  University of
Manchester, School of Physics and Astronomy, Oxford Road, Manchester M13 9PL,
United Kingdom  Henry Royce Institute for Advanced Materials, University of
Manchester, Oxford Road, Manchester, M13 9PL, UK

###### Abstract

Lattice relaxation in twistronic bilayers with close lattice parameters and
almost perfect crystallographic alignment of the layers results in the
transformation of moiré pattern into a sequence of preferential stacking
domains and domain wall networks. Here, we show that reconstructed moiré
superlattices of the perfectly aligned heterobilayers of same-chalcogen
transition metal dichalcogenides have broken-symmetry structures featuring
twisted nodes (’twirls’) of domain wall networks. Analysing twist-angle-
dependences of strain characteristics for the broken-symmetry structures we
show that the formation of twirl reduces amount of hydrostatic strain around
the nodes, potentially, reducing their infuence on the band edge energies of
electrons and holes.

This study addresses a detailed analysis of domain wall networks (DWN) which
form in long-period moiré patterns characteristic for highly aligned
heterostructures of same-chalcogen transition metal dichalcogenides MoX2/WX2
(TMDs with X=S or Se). As same chalcogen TMDs have very close lattice
constants, moiré patterns at their interface have long periods offering a
sufficient space for creating preferential stacking areas (domains). That is,
the energy gain due to better adhesion can surmount the cost of intralayer
strain in each of the constituent crystals. The reconstruction of small-angle
twisted bilayers into an array of domains [1] has been observed [2, 3, 4, 5,
6] both in MoS2/WS2 and MoSe2/WSe2 heterostructures. The observed [2, 3, 4, 5,
6] and theoretically modelled [7, 8, 1] structures feature hexagonal for anti-
parallel (AP) orientation of unit cell and triangular for parallel (P)
orientation of unit cells DWN.

Here, we show that lattice relaxation in P/AP-MoX2/WX2 (X=S,Se) bilayers in
the limit of $\theta=0^{\circ}$ twist angles undergoes symmetry breaking of
the domain wall patterns resulting in formation of twirled structures of DWN
nodes (see Fig. 1 for P-MoX2/WX2 bilayers). The emergence of twirled DWN nodes
is specific for heterobilayers, in contrast to previously studied marginally
twisted homobilayers (MX2/MX2) [7, 8, 1], where only symmetric star-like nodes
have been found. This difference stems from the hydrostatic strain component
determined by a small lattice mismatch of the two constituent 2D crystals. By
adjusting lattice constants of MoX2 and WX2 inside the domains, each monolayer
compression/expansion is inflicted onto domain walls creating hot spots of
hydrostatic strain in the DWN nodes with XX (chalcogen over chalcogen)
stacking [9]. An excessive energy costs of a large hydrostatic strain around
XX nodes (as compared to shear deformations dominating DWN structure in
homobilayers) is, then, negotiated by left/right-handed twirling of the domain
walls, leading to a broken-symmetry configuration sketched in Fig. 1.

Figure 1:  Symmetric and broken-symmetry (twirl) structures of DWN in aligned
P-MoX2/WX2 bilayers. Bottom plot shows energy gain ($\Delta$) from the
formation of twirls.

Methods. To study lattice relaxation in MoX2/WX2 bilayers we use multiscale
modelling approach developed in Refs. [1, 10]. This combines mesoscale
elasticity with analytically interpolated description of adhesion energy of
two layers, computed using density functional theory (DFT). The latter is
represented below upon taken into account interlayer distance relaxation (see
SM):

$\begin{split}W_{\rm
P/AP}(\bm{r}_{0})=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\\
\sum_{n,l=1,2,3}\left[w^{(s)}_{n}\cos\left(\bm{G}^{(n)}_{l}\bm{r}_{0}\right)+w^{(a)}_{n}\sin\left(\bm{G}^{(n)}_{l}\bm{r}_{0}+\gamma_{\rm
P/AP}\right)\right].\end{split}$ (1)

Here, $\bm{r}_{0}$ is a locally defined in-plane vector determining stacking
arrangement between layers ($\bm{r}_{0}=\bm{0}$ for XX stacking,
$\bm{r}_{0}=(0,-a/\sqrt{3})$ for XW/2H stacking and
$\bm{r}_{0}=(0,a/\sqrt{3})$ for MoX/MoW stacking in P/AP-heterostructures),
phases $\gamma_{\rm P}=\pi/2$, $\gamma_{\rm AP}=0$ account
$\bm{r}_{0}\to-\bm{r}_{0}$ symmetry of the adhesion energy in P- and AP-
bilayers [1]; $G_{1,2,3}^{(1,2,3)}$ are three sets of the shortest reciprocal
lattice vectors ($|G_{1,2,3}^{(1)}|=G$, $|G_{1,2,3}^{(2)}|=G\sqrt{3}$,
$|G_{1,2,3}^{(3)}|=2G$) of a commensurate MoX2/WX2 heterostructure, related by
120∘-rotations within each set. The values of parameters $w^{(s,a)}_{1,2,3}$
established using DFT [1, 10] are listed in Table 1.

Next, we combine (1) with elasticity theory by substituting local stacking in
the form,

$\bm{r}_{0}(\bm{r})=\delta\cdot\bm{r}+\theta\hat{z}\times\bm{r}+\bm{u}^{\rm
Mo}-\bm{u}^{\rm W},$ (2)

where $\delta\approx 0.2\%$ for MoS2/WS2 bilayers and $\delta\approx 0.4\%$
for MoSe2/WSe2; $\bm{u}^{\rm Mo}(\bm{r})$ and $\bm{u}^{\rm W}(\bm{r})$ are in-
plane displacement fields describing lattice relaxation in MoX2 and WX2
layers, respectively. Then, we minimise total energy of moiré superlattice,

$\mathcal{E}=\int d^{2}\boldsymbol{r}\left\\{W_{\rm
AP/P}\left(\bm{r}_{0}(\bm{r})\right)+\sum_{l={\rm
W,Mo}}\left[\frac{\lambda_{l}+\mu_{l}}{2}\left({\rm
div}\bm{u}^{(l)}\right)^{2}+\frac{\mu_{l}}{2}\left(\left(u_{xx}^{(l)}-u_{yy}^{(l)}\right)^{2}+4u_{xy}^{(l)2}\right)\right]\right\\},$
(3)

with respect to the displacement fields $\bm{u}^{\rm W}$ and $\bm{u}^{\rm
Mo}$. Here, $\lambda_{\rm Mo,W}$, $\mu_{\rm Mo,W}$ are elastic moduli of MoX2
and WX2 monolayers (see Table 1), and
$u_{ij}^{l}=(\partial_{j}u^{l}_{i}+\partial_{i}u^{l}_{j})/2$ are components of
strain tensors.

Table 1: Adhesion [1] and elastic [11, 12] energy parameters (in eV/nm2) used in Eq. (3).  X | $w_{1}^{(s)}$ | $w_{1}^{(a)}$ | $w_{2}^{(s)}$ | $w_{2}^{(a)}$ | $w_{3}^{(s)}$ | $w_{3}^{(a)}$
---|---|---|---|---|---|---
S, AP | 0.1415 | 0.0269 | -0.0338 | 0 | -0.0166 | -0.0030
S, P | 0.1559 | 0 | -0.0398 | 0 | -0.0199 | 0
Se, AP | 0.1128 | 0.0256 | -0.0201 | 0 | -0.0098 | -0.0024
Se, P | 0.1284 | 0 | -0.0248 | 0 | -0.0124 | 0
X | $\lambda_{\rm W}$ | $\lambda_{\rm Mo}$ | $\mu_{\rm W}$ | $\mu_{\rm Mo}$ |  |
S | 328.2 | 520.1 | 453.2 | 443.0 |  |
Se | 185.5 | 264.3 | 302.6 | 310.3 |  |
Figure 2:  Maps of hydrostatic (a-c) and shear (d-f) components of strain
tensor in principal axes for symmetric (b) and twirled (a,c) structures of
reconstructed moiré superlattice in aligned P-MoX2/WX2 bilayers. Sketch on the
top shows energy gain from formation of twirled structures. Scale bar is 30 nm
and 45 nm for X=Se and X=S, respectively. Dashed rectangles on (c) and (f)
show area displayed in Fig. 3. Figure 3:  Twirl in P-MoSe2/WSe2 for
$\theta=0.01^{\circ},0.1^{\circ},0.2^{\circ}$ and $0.3^{\circ}$. (a-d) show
maps of ${\rm div}\bm{u}_{\rm Mo}$. The arrows indicate direction of stacking
vector $\bm{r}_{0}$ ($|\bm{r}_{0}|=a/\sqrt{3}$) inside domains; (e-h) show
$B^{*}=2\partial_{x}u^{\rm Mo}_{xy}+\partial_{y}(u^{\rm Mo}_{xx}-u^{\rm
Mo}_{yy})$. (i) Values of ${\rm div}\bm{u}_{\rm Mo,W}$ at twirl center for
various twist angles.

To find optimal distributions of displacement fields, $\bm{u}^{\rm W}$ and
$\bm{u}^{\rm Mo}$, we solve a system of Lagrange-Euler equations, with
periodic boundary conditions, implemented via the finite difference method. In
particular, we chose rectangular supercell with $\ell\sqrt{3}/2$ and $\ell$
sides ($\ell=a/\sqrt{\delta^{2}+\theta^{2}}$ is the moiré superlattice period,
$a$ is averaged monolayer lattice parameter), setting the following boundary
conditions: $\bm{u}^{\rm Mo/W}(0,y^{\prime})=\bm{u}^{\rm
Mo/W}(\ell\sqrt{3}/2,y^{\prime}\mp\ell/2)$, at $0\leq\pm
y^{\prime}\leq\ell/2$, and $\bm{u}^{\rm Mo/W}(x^{\prime},\ell/2)=\bm{u}^{\rm
Mo/W}(x^{\prime},-\ell/2)$. Here, prime superscripts indicate that
$x^{\prime}Oy^{\prime}$ reference frame was rotated by an angle
$\phi=\pi/6+{\rm
arctan}\left[(\sqrt{3}\delta-\theta)/(\sqrt{3}\theta+\delta)\right]$ with
respect to the fixed frame for which $Ox$ and $Oy$ axes are along zigzag and
armchair directions in the crystal, respectively. Such a rotation allows us to
fix boundary conditions (note that the Lagrange-Euler equations themselves are
rotational-invariant).

In the numerical analysis we use a sufficiently dense grid (one point per nm)
to provide convergence of the computed displacement fields. This requires
solving $\sim 30000$ coupled nonlinear equations. This solution is obtained in
an annealing-type computational scheme: first, we choose the starting point as
$\bm{u}^{\rm Mo/W}=\bm{0}$ and scale down the adhesion energy parameters
$w^{(s,a)}_{1,2,3}$ by a small factor $\eta=10^{-3}$, finding slightly relaxed
structure. Then, we gradually enlarge this factor, up to $\eta=1$ (at the
final step), using solutions obtained at the previous iterations as starting
points at each next step.

Twirls in P-MoX2/WX2 heterostructures. Lattice relaxation in P-MoX2/WX2
heterostructures results in formation of DWN separating triangular domains
with MoX and XW stackings, similar to those found in 3R-TMD polytypes [2, 3]
(see Fig. 2). Minimising functional (3) we find the following three competing
distributions of strain. One of them (symmetric), shown on the middle panels,
is characterized by straight edge dislocation lines linking XX stacking nodes
of DWN. The other two (left and right panels) are twirled (broken-symmetry)
structures of DWN nodes with a left/right-handed twist. The computed total
energies per supercell, gathered in Table 2, show that the twirled DWN
structure has lower energy than the symmetric one. Note that left/right-handed
twirls have the same energy suggesting that DWN in perfectly aligned
heterostructure ($\theta=0^{\circ}$) undergoes a spontaneous symmetry breaking
into twirled state.

Values of adhesive and elastic energies compared in Table 2 also suggest that
the dominant energy gain for the twirled structures comes from lowering
elastic energy. The latter is determined by hydrostatic strain, ${\rm
div}\bm{u}$, and shear deformations, characterized by a vector, ${\bf
A}=(u_{xx}-u_{yy},-2u_{xy})$. Their distribution across DWN is shown Fig. 2,
in the form of color maps for ${\rm div}\bm{u}$ and $|{\bf A}|$. Here, the
difference between symmetric and twirled structures is such that hydrostatic
strain component, concentrated around nodes, is higher for symmetric one.
Since energy costs of the hydrostatic strain ($\propto\lambda+\mu$) are higher
than those of shear strain ($\propto\mu$), the twirled structures emerge as
energetically favourable.

Table 2: Values of adhesion, elastic, total energies (in eV/supercell) computed for symmetric (sym) and twirled (twirl) moiré superlattice structures and their total energy differences, $\Delta$, in aligned P-MoX2/WX2 bilayers.  X | structure | elastic | adhesion | total | $\Delta$
---|---|---|---|---|---
Se | sym | 223.01 | -2349.68 | -2126.67 | 15.12
twirl | 208.95 | -2350.74 | -2141.79
S | sym | 349.92 | -5321.11 | -4971.19 | 28.26
twirl | 323.33 | -5322.78 | -4999.45

In Fig. 3 we analyse dependence of ${\rm div}\bm{u}$ and ${\bf A}$ on the
twist angle. The data shown in Fig. 3(i) indicate that the maximal value of
${\rm div}\bm{u}$ at the hot spot of strain decreases with the twist, which
also explains why twirling is weaker for larger $\theta$’s. Another
deformation field characteristic which is interesting to consider is
$B^{*}=[{\rm rot}{\bf A}]_{z}=2\partial_{x}u^{\rm
Mo(W)}_{xy}+\partial_{y}(u^{\rm Mo(W)}_{xx}-u^{\rm Mo(W)}_{yy})$. This
characteristic determines the size of piezoelectric charges, generated by
inhomogeneous strain in each layer, and pseudomagnetic field that would be
experienced by charge carriers at the K-valley band edge in the
heterostructure [1, 10]. Note that in P-heterostructures piezocharges
($\propto e_{11}B^{*}$) have opposite signs in MoX2 and WX2 layers, as
$B^{*}_{\rm Mo}\approx-B^{*}_{\rm W}$, and piezocoefficients have the same
signs, $e_{11}^{\rm Mo}\approx e_{11}^{\rm W}$. For the displayed range of
angles, $0^{\circ}\leq\theta\lesssim 0.4^{\circ}$, distributions of $B^{*}$
shows change of sign and zero value in the middle of the domain wall. For
larger twist angles $\theta\gg\delta$ piezocharge and pseudomagnetic field
distributions take the form of those established earlier for homobilayers [1].

Figure 4:  Maps of hydrostatic (a-c) and shear (d-f) components of strain
tensor in principal axes for symmetric (b) and twirled (a,c) structures of
reconstructed moiré superlattice in aligned AP-MoX2/WX2 bilayers. Sketch on
the top shows energy gain from formation of twirled structures. Scale bar is
30 nm and 45 nm for X=Se and X=S, respectively. Dashed rectangles on (c) and
(f) show area displayed in Fig. 5.

Twirls in AP-MoX2/WX2 heterostructures. Lattice relaxation in AP-MoX2/WX2
heterostructures leads to the formation of hexagonal domains of 2H-like
stacking (simultaneous metal-on-chalcogen and chalcogen-on-metal like in bulk
2H TMD crystals). Those domains are separated by a hexagonal DWN with two
inequivalent nodes: one with XX stacking (chalcogen-on-chalcogen) and the
other with two metallic site on the top of each other, MoW. Upon minimising
energy in Eq. (3) for crystallographically aligned bilayers, we identify two
candidates for the lowest energy configuration of DWN: symmetric and
left/right-handed twirled structures, Fig. 4. Twirled structures form around
XX nodes, whereas DWN around MoW nodes is almost unchanged (only rotated as a
whole by $\approx 5^{\circ}$), which is because those nodes host metastable
MoW stacking with much weaker hydrostatic strain component as compared to XX
nodes). Energies of symmetric and twirled structures compared in Table 3 show
that the broken-symmetry structure is energetically favourable.

In Fig. 5 we show twist angle dependences of strain field characteristics
around a single twirl. The data for ${\rm div}\bm{u}$ in Fig. 5(i) indicate
decrease of its magnitude (in each of the layers) with the twist angle. This
trend corresponds to the decay of twirling with growth of layers’
misorientation (Fig. 5(a-d)) as already discussed about P-heterostructures in
the previous section. We also point out that, in contrast to P-bilayers, for
AP-heterostructures shear strain results in the same signs of piezocharge
densities ($\propto e_{11}^{\rm Mo}B^{*}_{\rm Mo}\approx e_{11}^{\rm
W}B^{*}_{\rm W}$) in both layers (opposite signs of piezocoefficients,
$e_{11}^{\rm Mo}\approx-e_{11}^{\rm W}$, due to anti-alignment of layers, are
compensated by signs inversion for $B^{*}_{\rm Mo}\approx-B^{*}_{\rm W}$).
Note that DWN nodes host area of non-zero $B^{*}$, which reverses sign at
$\theta\approx\delta$, with magnitude at MoW nodes substantially higher than
that of twirled nodes (see Fig. 5(j)).

Table 3: Values of adhesion, elastic, total energies (in eV/supercell) computed for symmetric (sym) and twirled (twirl) structures and total energy differences, $\Delta$, for aligned AP-MoX2/WX2 bilayers.  X | structure | elastic | adhesion | total | $\Delta$
---|---|---|---|---|---
Se | sym | 329.46 | -2648.55 | -2319.09 | 6.12
twirl | 323.77 | -2648.98 | -2325.21
S | sym | 496.25 | -5904.35 | -5408.10 | 14.44
twirl | 482.70 | -5905.24 | -5422.54
Figure 5:  Twirl in AP-MoSe2/WSe2 for
$\theta=0.01^{\circ},0.1^{\circ},0.2^{\circ}$ and $0.3^{\circ}$. (a-d) show
maps of ${\rm div}\bm{u}_{\rm Mo}$. The arrows indicate direction of stacking
vector $\bm{r}_{0}$ ($|\bm{r}_{0}|=a/\sqrt{3}$) inside domains; (e-h) show
$B^{*}=2\partial_{x}u^{\rm Mo}_{xy}+\partial_{y}(u^{\rm Mo}_{xx}-u^{\rm
Mo}_{yy})$. (i) Values of ${\rm div}\bm{u}_{\rm Mo,W}$ at twirl center for
various twist angles. (j) $B^{*}$-dependence in centers of XX (twirl) and MoW
nodes.

Conclusion. The main finding of this study consists in the prediction of a
spontaneous symmetry breaking in the form of domain wall network in MoX2/WX2
heterostructures. For almost perfectly aligned MoX2/WX2 bilayers, this
symmetry breaking consists in the formation of twirl-shaped nodes, accompanied
by substantial reduction of a hydrostatic strain component in the vicinity of
those nodes - as compared to earlier-studied symmetric (non-twirled) DWN
structures [9]. The reduction of hydrostatic strain component by twirling
could have a pronounced effect in the energetics of the band edge states of
the electrons and holes, making those less confined at the twirled nodes, as
compared to symmetric nodes. As a result, the earlier studies [9]
overestimated electron, hole and exciton bindings by the XX DWN nodes, so that
the analysis of electron, hole and interlayer excitons localisation at the
twirls requires farther studies.

Acknowledgement. We acknowledge useful discussions with Marek Potemski, Isaac
Ochoa and Fabio Ferreira. This work was supported by EC-FET Core 3 European
Graphene Flagship Project, EC-FET Quantum Flagship Project 2D-SIPC, EPSRC
grants EP/S030719/1 and EP/V007033/1, and the Lloyd Register Foundation
Nanotechnology Grant.

Author Contributions. MAK, VVE, and VIF contributed equally to this paper. MAK
and VVE developed formalism and software and carried out calculaions. VIF
conceived the paper. All authors contributed to the data analysis and
manuscript writing.

## References

  * Enaldiev  _et al._ [2020] V. V. Enaldiev, V. Zólyomi, C. Yelgel, S. J. Magorrian, and V. I. Fal’ko, Stacking domains and dislocation networks in marginally twisted bilayers of transition metal dichalcogenides, Phys. Rev. Lett. 124, 206101 (2020).
  * Weston  _et al._ [2020] A. Weston, Y. Zou, V. Enaldiev, A. Summerfield, N. Clark, V. Zólyomi, A. Graham, C. Yelgel, S. Magorrian, M. Zhou, J. Zultak, D. Hopkinson, A. Barinov, T. H. Bointon, A. Kretinin, N. R. Wilson, P. H. Beton, V. I. Fal’ko, S. J. Haigh, and R. Gorbachev, Atomic reconstruction in twisted bilayers of transition metal dichalcogenides, Nature Nanotechnology 15, 592 (2020).
  * Rosenberger  _et al._ [2020] M. R. Rosenberger, H.-J. Chuang, M. Phillips, V. P. Oleshko, K. M. McCreary, S. V. Sivaram, C. S. Hellberg, and B. T. Jonker, Twist angle-dependent atomic reconstruction and moiré patterns in transition metal dichalcogenide heterostructures, ACS Nano 14, 4550 (2020).
  * Sung  _et al._ [2020] J. Sung, Y. Zhou, G. Scuri, V. Zólyomi, T. I. Andersen, H. Yoo, D. S. Wild, A. Y. Joe, R. J. Gelly, H. Heo, S. J. Magorrian, D. Bérubé, A. M. M. Valdivia, T. Taniguchi, K. Watanabe, M. D. Lukin, P. Kim, V. I. Fal’ko, and H. Park, Broken mirror symmetry in excitonic response of reconstructed domains in twisted MoSe2/MoSe2 bilayers, Nature Nanotechnology 15, 750 (2020).
  * McGilly  _et al._ [2020] L. J. McGilly, A. Kerelsky, N. R. Finney, K. Shapovalov, E.-M. Shih, A. Ghiotto, Y. Zeng, S. L. Moore, W. Wu, Y. Bai, K. Watanabe, T. Taniguchi, M. Stengel, L. Zhou, J. Hone, X. Zhu, D. N. Basov, C. Dean, C. E. Dreyer, and A. N. Pasupathy, Visualization of moiré superlattices, Nature Nanotechnology 15, 580 (2020).
  * Shabani  _et al._ [2021] S. Shabani, D. Halbertal, W. Wu, M. Chen, S. Liu, J. Hone, W. Yao, D. N. Basov, X. Zhu, and A. N. Pasupathy, Deep moiré potentials in twisted transition metal dichalcogenide bilayers, Nature Physics 17, 720 (2021).
  * Naik and Jain [2018] M. H. Naik and M. Jain, Ultraflatbands and shear solitons in moiré patterns of twisted bilayer transition metal dichalcogenides, Phys. Rev. Lett. 121, 266401 (2018).
  * Carr  _et al._ [2018] S. Carr, D. Massatt, S. B. Torrisi, P. Cazeaux, M. Luskin, and E. Kaxiras, Relaxation and domain formation in incommensurate two-dimensional heterostructures, Phys. Rev. B 98, 224102 (2018).
  * Enaldiev  _et al._ [2022] V. V. Enaldiev, F. Ferreira, J. G. McHugh, and V. I. Fal’ko, Self-organized quantum dots in marginally twisted MoSe2/WSe2 and MoS2/WS2 bilayers, npj 2D Materials and Applications 6, 74 (2022).
  * Enaldiev  _et al._ [2021] V. V. Enaldiev, F. Ferreira, S. J. Magorrian, and V. I. Fal’ko, Piezoelectric networks and ferroelectric domains in twistronic superlattices in WS2/MoS2 and WSe2/MoSe2 bilayers, 2D Materials 8, 025030 (2021).
  * Iguiñiz  _et al._ [2019] N. Iguiñiz, R. Frisenda, R. Bratschitsch, and A. Castellanos-Gomez, Revisiting the buckling metrology method to determine the young’s modulus of 2d materials, Advanced Materials 31, 1807150 (2019).
  * Androulidakis  _et al._ [2018] C. Androulidakis, K. Zhang, M. Robertson, and S. Tawfick, Tailoring the mechanical properties of 2d materials and heterostructures, 2D Materials 5, 032005 (2018).


# Polymer translocation: Effects of confinement

Manish Dwivedi1, Sumitra Rudra1 and Sanjay Kumar1 1Department of Physics,
Banaras Hindu University, Varanasi 221005, India

###### Abstract

We investigate the influence of varying confinement on the dynamics of polymer
translocation through a cone shaped channel. For this, a linear polymer chain
is modeled by a self avoiding walks (SAWs) on a square lattice. The cis-side
of a cone-shaped channel has a finite volume, whereas the trans-side has a
semi-infinite space. The confining environment is varied either by changing
the position of the back wall while keeping the apex angle fixed or altering
the apex angle while keeping the position of the back wall fixed. In both
cases, the effective space $\phi$, which represents the number of monomers in
a chain relative to the total number of accessible sites within the cone, is
reduced due to the imposed confinement. Consequently, the translocation
dynamics are affected. We analyze the entropy of the confined system as a
function $\phi$, which exhibits non-monotonic behaviour. We also calculate the
free energy associated with the confinement as a function of a virtual
coordinate for different positions of the back wall (base of the cone) along
the conical axis for various apex angles. Employing the Fokker-Planck
equation, we calculate the translocation time as a function of $\phi$ for
different solvent conditions across the pore. Our findings indicate that the
translocation time decreases as $\phi$ increases, but it eventually reaches a
saturation point at a certain value of $\phi$. Moreover, we highlight the
possibility to control the translocation dynamics by manipulating the solvent
quality across the pore. Furthermore, our investigation delves into the
intricacies of polymer translocation through a cone-shaped channel,
considering both repulsive and neutral interactions with the pore wall. This
exploration unveils nuanced dynamics and sheds light on the factors that
significantly impact translocation within confined channels.

##  I Introduction

The translocation of a bio-polymers through a pore is a fundamental step in
many cellular processes e.g. passage of DNA and RNA through nuclear poresVella
(1994), protein transport through membrane channels Nakielny and Dreyfuss
(1999), packaging of genome in bacteriophages, ejection of virus from capsid
etcMolineux and Panja (2013); Marenduzzo  _et al._ (2013); Berndsen  _et al._
(2014). Apart from it’s biological relevance, interest has also been fueled to
its biotechnological applications such as gene therapy, drug delivery, DNA
sequencing etc Kasianowicz  _et al._ (1996); Schneider  _et al._ (2010). In
fact, recent studies revealed that polymer translocation through pores is a
complex process, and is governed by the combined effects of various factors,
such as the nature of the pore-polymer interaction Luo  _et al._ (2007); Chen
_et al._ (2009); Luo and Cao (2012), the geometry of the pore Polson and
Heckbert (2019); Mohan  _et al._ (2010), external driving forcesLuo  _et al._
(2006), salt concentration Dabhade  _et al._ (2022); Hsiao (2020) across the
pore etc. A detailed understanding of the effect of various system parameters
on the translocation process, is prerequisite to understand the transport
dynamics of the translocating molecule.

For the unforced translocation, the mean passage time $\tau$ scales as
$N^{\alpha}$ with $\alpha=1+2\nu$, where $\nu$ is the Flory exponent. However,
recent studies suggest that the value of scaling exponent $\alpha$ depends on
the pore size, apex angle, quality of solvents etc. Luo  _et al._ (2008);
Edmonds  _et al._ (2012); Ikonen  _et al._ (2012); Bhattacharya  _et al._
(2009). Significant amount of work have been carried out now to understand the
role of driving force on the translocation. In case of driven translocation,
there exist several possibilities: the driving force could be in the form of a
chaperon binding Abdolvahab  _et al._ (2011); Suhonen and Linna (2016); Yu and
Luo (2011), confinement Muthukumar (2001); Cacciuto and Luijten (2006); Wong
and Muthukumar (2008), crowders Polson and Heckbert (2019); Khalilian  _et
al._ (2021); Tan  _et al._ (2021) or an external fieldLuo  _et al._ (2006) .
In case of charged polymers, e.g., DNA, the translocation is driven by an
electric field acting across the channel. Kasianowicz et al. Kasianowicz  _et
al._ (1996) studied the translocation of single molecules by measurement of
the ionic current passing through pores. The potential difference across the
pore drives the ionic current, which also drives charged biopolymers through
the pore. The amount of the variation in the ionic current has been used to
extract the information about the sequence and length of DNA.

Figure 1:  Schematic representation of of a polymer threading through a
conical channel (pore) on a square lattice. The impenetrable rigid side walls
are depicted as thick brown walls, while the impenetrable rigid back wall is
shown as a thick blue line. The conical channel separates the cis and trans
sides, with infinitely extended rigid walls. The confinement within the pore
can be adjusted by either changing the position of the back wall, indicated by
the dashed blue line, or by varying the angle of the cone, represented by the
dashed brown line. The polymer within the pore undergoes a random walk,
starting from various positions on the principal axis, denoted by the dotted
line.

Confinement arising due to the geometry of the pore-interface (e.g.
Mycobacterium smegmatis porin A (MspA), HIV-1 capsid) also drives the
translocation jin Jeon and Muthukumar (2016). The geometry of the pore looks
similar to a cone-shaped channel Fig.(1), where driving force may be due to
the different solvents across the pore or entropy gradient because of the
shape of the channel Nikoofard and Fazli (2012); Sharma  _et al._ (2022). The
polymer translocation through the cone-shaped channel has been studied
experimentally Lan  _et al._ (2011); Sexton  _et al._ (2007); Siwy and
Fuliński (2002) and theoretically in recent years. The focus was to obtain
scaling associated with translocation Sung and Park (1996); Muthukumar (1999,
2003); Wong and Muthukumar (2008); Muthukumar (2001). On the other hand,
theoretical understanding of equilibrium properties of a polymer inside the
cone has been achieved through the conformal invariance technique and the
exact enumeration technique followed by the series analysis Guttmann and
Torrie (1984) to calculate various exponents. Using exact enumeration
technique followed by the Langevin Dynamics simulations, Kumar et alKumar  _et
al._ (2020) showed that the confinement leads to the non-monotonic behaviour
in the $\theta-$temperature, where a polymer chain acquires the globule state
from the coil state. Interestingly, by varying the apex angle $\alpha$, they
showed that $\theta-$ temperature can be shifted.

In this study, we embark on a comprehensive exploration of polymer
translocation dynamics with a distinct focus on the influence of the confining
environment. What sets our research apart are three pivotal elements: firstly,
our model embraces a realistic depiction by featuring a finite volume on the
cis side and semi-infinite space on the trans side, closely mirroring real-
world systems. Secondly, our model allows for dynamic adjustments in the apex
angle and the position of the back wall, potentially inducing shifts in the
polymer’s center of mass due to changes in geometry. Thirdly, we delve into
the entropy variation concerning the back wall position and the apex angle and
unveil the free energy profile of the system, offering insights into the
thermodynamics of polymer translocation. Finally, we employ the Fokker-Planck
equation to calculate the translocation time as a function of the back wall
position and the effective space available to the polymer. These novel aspects
of our research not only deepen our understanding of polymer translocation but
also illuminate the hitherto uncharted territory of confining environment
effects on this dynamic process, highlighting the unexplored areas that demand
attention within this field. The outline of the paper is as follows:

In Sec.(II) , we provide a brief overview of the model and elucidate the
employed techniques. Sec. (III) delves into the calculation of the system’s
entropy as a function of $\phi$ and the free energy. Since, the exact
information of density of states is known, we employ the Fokker Planck
equation to calculate the time required to cross the pore and termed it as a
translocation time ($\tau$). In Sec.(IV), we explore the impact of both
repulsive and neutral interactions with the pore wall, revealing nuanced
dynamics that significantly influence translocation within confined channels.
The paper ends with a brief discussion and future perspective of the present
study in Sec.(V).

##  II Model and method

We investigate a self avoiding walk model of a linear polymer chain on a
square lattice, which is confined in a cone shaped channel. The cone shaped
channel composed of two perfectly reflecting hard walls that are inclined at
an apex angle $\alpha$ (Fig. 1). A single monomer is only allowed to pass
through a pore, which prohibits the formation of a fork or hairpin at pore. To
explore the process of translocation resulting from reduced entropy due to the
closed cone shaped channel, we employ an exact enumeration technique to
systematically generate all possible conformations of the polymer, and examine
whether the polymer resides in the cis-side (within a channel) or trans-side
(out side the channel) or in a state of threading (from the cis-side to the
trans-side). The inherent fourfold symmetry of the square lattice naturally
produces angles that are multiples of $\pi/4$, such as $\pi/4$ and $\pi/2$. To
explore angles beyond these specific multiples, we employed an innovative
technique outlined in Ref Chauhan and Kumar (2021). Specifically, we consider
$n_{x}$ bonds along the $\pm x$ direction, followed by $+n_{y}$ bonds in the
$y$ direction. Distinguishing our study from previous study Chauhan and Kumar
(2021), where the cis and trans\- sides were characterized by semi-infinite
space. The model developed here represents a significant departure from
previous approaches, reflecting a fundamental shift towards a more realistic
representation of real-world systems. The presence of a perfectly reflecting
hard back wall in the conical channel (Fig. (1)) alters the translocation
dynamics. This marked departure from the conventional semi-infinite space
assumption may impact our understanding of polymer translocation within
confined environments. The position of the back wall can be placed at
different positions of the principal axis (say $X=-18,-17,....$) away from the
origin. Moreover, the trans-side is demarcated by a reflecting vertical
(perpendicular to the principal axis) hard wall with a channel (Fig. (1)).
It’s worth emphasizing that alterations to the apex angle or the back wall’s
position lead to changes in the available space within the cis-side. However,
in stark contrast, the available space within the trans-side remains constant
and unaffected by variations in the confining environment.

We have attached one end of a SAW at the middle of the back wall placed at a
distance $-X$ away from the pore and generated all possible conformations of
$N=28$ steps walk with the imposed boundary conditions. We systematically
varied the position of the back wall ($X=-18$ to $X=-1$), and evaluated the
partition function as a simply sum over all possible conformations for
different apex angles.

$Z_{X}=C_{N}z^{N},$ (1)

where $\it C_{N}$ is the total number of all possible conformations of a
polymer chain whose one end is attached on the back wall (Fig. (1)) and $z$ is
the fugacity of each step of the walk. For the large $\it N$, $\it
C_{N}\sim\mu^{N}N^{\gamma^{\prime}-1}$ Vanderzande (1998); Kumar and Li
(2010). Here $\mu$ is the connectivity constant of the lattice and
$\gamma^{\prime}$ is the critical exponent associated with the SAW confined in
a cone of the apex angle $\alpha$. It’s important to note that the techniques
previously developed and successfully applied in Ref. Chauhan and Kumar (2021)
have established their efficacy in reproducing earlier findings. Building upon
this well-established foundation, we have extended these techniques to a more
realistic model. This extension allows us to elucidate the fundamental physics
governing translocation in scenarios characterized by diverse geometric
considerations and confinement factors.

Figure 2: Figures show non-monotonic variation of entropy $S$ as a function of
(a) back wall position $X$ and b) effective space $\phi$ for different apex
angles $\alpha$. A nice collapse of data on a single line up to a certain
value of $\phi$ is a apparent from the plot.

##  III Results

###  III.1 Entropy in good solvent

While calculating entropy exactly remains a formidable challenge in most of
theoretical studies, the exact enumeration technique offers a distinct
advantage. With exact knowledge of the density of states, one can compute
entropy by exhaustively enumerating all possible configurations within a given
system’s complexity. For instance, in the context of polymer translocation,
exact enumeration method systematically generates all polymer conformations
within the cis and trans-side of the cone-shaped-channel providing an exact
measure of configurational entropy. This approach serves as a powerful tool
for understanding complex systems where analytical solutions are elusive,
offering insights into the fine-grained thermodynamics that govern
translocation behavior. The entropy of the system is calculated as

$S=k_{B}lnZ_{X},$ (2)

where $k_{B}$ is the Boltzmann constant.

From here on wards, we work in the reduced energy system by setting $k_{B}=1$.
Since there is no external force on the monomer other than the confinement
(due to the back-wall and conical shape of the channel), a polymer comes out
from the pore as a result of the entropy difference across the pore. Using Eq.
(2), we calculate entropy associated with a polymer chain confined in the
conical pore. In Fig. (2), we show the variation of entropy as a function of
the position of the back wall for various apex angles. As the back wall
approaches the pore, the entropy of the system decreases. This decline is
attributed to the reduction in the number of configurations available to a
finite-size polymer within the cone. However, beyond a certain position, we
observe a reversal, with entropy increasing once more. This may be attributed
to the fact that a part of the chain moves towards the trans-side, which has
more configurational space, as a result there is increase in the entropy. To
emphasize this behavior, we also plot entropy as a function of the effective
space $\phi$, representing the fraction of accessible sites within the cone.
Remarkably, we observe a collapse of entropy onto a single curve, particularly
evident up to a fraction $\phi\approx 0.3$, as shown in Fig. (2)b.

###  III.2 Free Energy Profile

Figure 3: Free energy as a function of virtual coordinate $X_{V}$ (in case of
apex angle $\alpha=45^{o}$) for various solvent conditions: (a) good solvent
on both the $\it cis$-side and $\it trans$-side
($\epsilon_{c}=0.0=\epsilon_{t}$), (b) poor solvent on both the $\it cis$-side
and $\it trans$-side ($\epsilon_{c}=-1.0=\epsilon_{t}$), (c) the solvent on
the $\it cis$-side ($\epsilon_{c}=-1.0$) is poorer compared to the $\it
trans-$ side ($\epsilon_{t}=-0.8$), and (d) the solvent on the $\it
trans$-side is poorer compared to the $\it cis$\- side
($\epsilon_{c}=-0.8,\epsilon_{t}=-1.0$)

A comprehensive understanding of the translocation process hinges on a
complete description of the associated free energy within different regions of
the channel. In Ref. Chauhan and Kumar (2021), we have studied the free-energy
as a function polymer-pore interaction. However, to unravel the intricate
interplay between entropic confinement (arising due to the back wall position
or apex angle) and translocation dynamics, we introduce variations in solvent
conditions, including both good and poor solvents, across the pore. These
varying solvent conditions lead to distinct polymer chain states, from swollen
coils in good solvents to collapsed globules in poor solvents. To investigate
polymer translocation in these diverse solvents, it is imperative to analyze
the time required for each stage of the process, spanning from initiation to
completion. Consequently, we divide the translocation into three pivotal
stages, each holding unique significance. The initial stage involves the time
taken to reach the pore, the second stage is dedicated to the threading
process, and the third stage marks the full translocation to the trans-side.

Throughout these stages, the free energy is tied to a ”virtual coordinate”
denoted as $X_{V}$, which ranges from $-X$ to $2L$. During stage 1, when the
back wall remains fixed at a specific position $X$ on the principal axis,
$X_{V}$ corresponds to the physical distance $x$ from the pore at origin.
Here, one end of the polymer is anchored at various $x$ positions on the
principal axis, while the other end is free to be any where in the $\it
cis$-side of the pore. During this stage, $X_{V}$ assumes the values within
the bounds of the actual coordinates $-X\leq X_{V}<0$.

In stage 2, the polymer commences its translocation from the $\it cis$-side to
the $\it trans$-side, heralding the threading stage. Here, $X_{V}=m$
represents the number of monomers that have successfully translocated to the
$\it trans$-side, while $(N-m-1)$ monomers remain within the $\it cis$-side of
the pore, in addition to a monomer residing at the pore. The final stage 3,
manifests as $X_{V}$ ranging from $L+1$ to $2L$, signifying that the polymer
has fully emerged from the pore and now exists entirely on the $\it
trans$-side. Analogous to stage 1, here $X_{V}=L+|x|$, where $x$ denotes the
physical distance of the polymer’s anchored end on the $\it trans$-side. It is
worth noting that while distances from the origin are denoted by $x$ along the
real axis, the free energy is expressed here as a function of the ”virtual
coordinate” $X_{V}$. The relation between these two coordinates is given by
Sun  _et al._ (2009)

$X_{V}=\begin{cases}-|x_{i}|,x_{i}=...,-X,..,-2,-1&\text{Stage1}\\\
m=i,i=0,1,2,3.......L&\text{Stage2}\\\
|L+x_{i}|,x_{i}=1,2,...,L,...&\text{stage3}\end{cases}$ (3)

The partition function is summed over all the walks on a square lattice for a
virtual coordinate $X_{V}$ with a fixed back wall position $X$ is given by the
following equation:

$Z_{X}(X_{V})=\sum_{N_{c},N_{t}}C^{X_{V}}(N_{c},N_{t})u^{N_{c}}v^{N_{t}}$ (4)

where $C^{X_{V}}(N_{c},N_{t})$ is the number of conformations of respective
stages as defined by Eq. (3). $N_{c}$ and $N_{t}$ are the number of nearest
neighbour contacts of the $cis$ and $trans$ sides, respectively.
$u=exp(-\beta\epsilon_{c})$ and $v=exp(-\beta\epsilon_{t})$ are the Boltzmann
weights of nearest-neighbour interactions $\epsilon_{c}$(cis-side) and
$\epsilon_{t}$(trans-side), respectively. The free energy of the different
stages may be obtained from following equation:

$F(X_{V})=-TlnZ_{X}(X_{V}).$ (5)

$T$ is the temperature and it is set equal to 0.5. First, we consider a
situation where both cis and trans sides have a good solvent i.e.
$\epsilon_{c}=\epsilon_{t}=0$. In a good solvent, the polymer adopts a coil-
like conformation on both sides of the cone. In Fig. (3)a, we have shown the
free-energy as a function of virtual coordinate $X_{V}$ for different wall
positions $X$. It is evident from the plots that when one end of the polymer
is anchored at a distance far from the pore, the number of conformations of a
polymer remain close to its bulk value and the confinement by the cone-shaped
channel does not affect the free-energy up to a certain value of $X_{V}$.
However, when the anchored site shifts towards the pore, the polymer
conformations get constrain by the cone and as a result its configurational
properties changes relative to its bulk behavior. Consequently, one observes a
change in the free energy at the vicinity of the pore. Once the threading
process starts, the number of conformations of polymer at trans-side increases
and one observes a gradual decrease in the free energy followed by a sharp
fall indicating that entire polymer is translocated from the cis-side to the
trans-side. As anchored end of a polymer moves away from the pore, the polymer
does not experience any confinement and thus approaches its bulk behaviour.

In the presence of a poor solvent on either side of the pore (Fig. (3)b), we
observe a slight reduction in the free energy as the anchored end of a polymer
chain approaches the pore. This results due to an increase in the number of
nearest neighbor contacts because of the confinement imposed by the channel.
However, as the anchored site moves closer to the pore, the number of
accessible conformations decreases, leading to an overall increase in the free
energy. In contrast to the case of a good solvent, here during threading, the
free energy remains nearly constant. One of the interesting observations is
that the free energy almost remains the same for the different wall positions.
This behavior can be attributed to the polymer adopting a globule state on
both sides, resulting in a sustained level of free energy. Consequently, when
a monomer from the globule state on the cis-side translocates to the trans-
side, it reintegrates into the globule state, thereby maintaining a constant
free energy. Similar to the case with a good solvent, in the final Stage there
is a sharp decrease in the free energy as the polymer assumes its bulk
configuration. However, the magnitude of the free energy differs considerably
compared to the good solvent.

So far we have focused on discussing the solvent quality remaining consistent
across the pore. It would be interesting to observe how the free energy
profile changes when the solvent quality differs across the pore. To
investigate the impact of different solvents, we consider two situations: (i)
the solvent quality of cis-side is relatively poorer compared to the trans-
side ($\epsilon_{c}<\epsilon_{t}$), and (ii) the solvent quality on trans-side
is relatively poorer than the cis-side ($\epsilon_{c}>\epsilon_{t}$). In Fig.
(3)c, we present the free energy profile, where the cis- and trans sides have
different interaction strengths i.e. $\epsilon_{c}=-1.0$ and
$\epsilon_{t}=-0.8$. While the other features remain the same as of Fig. (3)b,
here one observes the free energy has an upward tilt during the threading
stage. This indicates that polymer prefers to stay inside the pore.
Conversely, Fig. (3)d depicts the reverse situation, where the trans-side has
the lower free energy indicating that the polymer prefers to stay out side the
pore. This indicates that there is a subtle competition between entropy and
the internal energy, which modifys the free energy profile. Since, the apex
angle is constant ($\alpha=45^{o}$), the entropic contribution arising due to
different conformations also remains constant. However, by varying the
difference of ($\epsilon_{c}-\epsilon_{t}$), one can modify the barrier
height.

Figure 4: Figures (a-d) show the variation of translocation time as a function
of the back wall position $X$, and (e-f) as a function of effective space
$\phi$ for different apex angles $\alpha$. The solvent quality is varied on
both the $\it cis$ and $\it trans$ side. (a) and (e) represent the case where
there is a good solvent on both sides ($\epsilon_{c}=0.0=\epsilon_{t}$), (b)
and (f) correspond to poor solvent conditions on either side
($\epsilon_{c}=-1.0=\epsilon_{t}$), c) and (g) depict a situation where the
solvent on the cis-side  is poorer compared to the trans-side
$\epsilon_{c}=-1.0,\epsilon_{t}=-0.8$ and, (d) and (h) present the reverse
case where solvent on $\it trans$\- side is poorer compared to the $\it
cis$-side with $\epsilon_{c}=-0.8,\epsilon_{t}=-1.0$.

Another important observation is when a back wall moves toward the pore,
initially the free energy of the cis-side remains almost constant, but show a
significant change after $X=-12.0$. This is because that there is not enough
space to accommodate a polymer at cis-side and therefore, a significant loss
of configurational entropy. Since, a flat wall demarcates the trans-side, the
number of conformations remain same irrespective of the back wall position.
Therefore, the free energy after the translocation of a polymer remain almost
the same for all back wall positions Figs. (3)(a-d). When $X>-6.0$, the space
available to the polymer inside the pore is significantly less, as a result
there is a significant change in the free energy, which is apparent from Figs.
(3)(a-d). Interestingly, for $X=-3.0$, there is not enough space to
accommodate even a single conformation, thus here one can see the contribution
of free energy during threading and after the translocation of entire chain
only. It is pertinent to mention here that the qualitative nature of the free
energy profile remains the same when we change the apex angle $\alpha$.

###  III.3 Translocation Dynamics

The phenomenon of polymer translocation can be well described by considering
the average time required for the polymer to move from the initial state (cis-
side) to the final state (trans-side). This temporal parameter finds
applicability in a diverse range of biological processes e.g DNA melting,
polymer folding, charge hopping etc. The knowledge of this time scale provides
a better understanding of these processes at the cellular level. The diffusion
of the chain can be effectively described by the Fokker-Planck equation,
enabling the calculation of the time required for the chain to travel from the
initial stage say $x_{0}$ to a specific point $x_{t}$, as expressed by the
following equation Sung and Park (1996)

$\mathcal{T}_{o\rightarrow
t}=\frac{1}{D}\int_{x_{o}}^{x_{t}}e^{F(x^{\prime})}dx^{\prime}\int_{x_{o}}^{x^{\prime}}e^{-F(x^{\prime\prime})}dx^{\prime\prime}.$
(6)

The chain diffusivity, denoted as $D$, is a variable that exhibits an inverse
relationship with the chain length Muthukumar (2001). In line with the
approach outlined in the Ref. Sun  _et al._ (2009), we assign a value of
$D=1$. Given the implementation of a lattice model, Eq. (6) can be represented
in a discrete form as

$\mathcal{T}_{o\rightarrow
t}=\sum_{x^{\prime}=x_{o}}^{x_{t}}e^{F(x^{\prime})}\Delta
x^{\prime}\sum_{x^{\prime\prime}=x_{o}}^{x^{\prime}}e^{-F(x^{\prime\prime})}\Delta
x^{\prime\prime},$ (7)

where $\Delta x^{\prime}=\Delta x^{\prime\prime}=1$ because free energy is
calculated at equi-spaced lattice. The time ($\tau$) required for the polymer
to cross the energy barrier imparted by the pore is highly dependent on the
nature of pore geometry and solvent present on either sides of the pore. We
first consider a good solvent on both side of the pore
(($\epsilon_{c}=\epsilon_{t}=0$)) where there is no attractive interaction
among monomers. The Figs. (4)(a) & (e) show the decay of translocation time
($\tau$) for various angles as a function of back wall position $X$ and
$\phi$, respectively. One can notice that for the apex angle $\alpha=45^{o}$,
the value of $\tau$ is higher compared to $\alpha=18.43^{o}$ for the lower
values of $X$ and there is a significant difference in $\tau$ for different
values of $\alpha$. Thus, one can infer that confinement promotes the
translocation. However, when the back wall is closed to pore ($X>-6.0$), the
value of $\tau$ tends to a constant value. It is interesting to note that when
we plot variation of $\tau$ as a function of $\phi$, all the data collapse
nearly on a single curve indicating that for the good solvent, the
translocation time $\tau$ depends only on the fraction of space available to a
polymer chain.

If the solvent quality is poor across the
pore($\epsilon_{c}=\epsilon_{t}=-1.0$), we find that the translocation time
decay up to a certain value of $X$ and then approaches to a constant value
(Fig. (4)(b)). In this case, polymer acquires the conformation of globule
state of relatively smaller size compared to the swollen state. Therefore, if
the wall is far away from the pore, translocation time remains constant for
all apex angles studied here as polymer does not see the confinement. As the
back wall approaches towards the pore, we observe a clear difference in
$\tau$. Like the good solvent case, we also find that confinement advances the
translocation, however the value of $\tau$ is almost an order higher than the
one observed for the good solvent. The rise in the $\tau$ may be understood
from the Fig. (4)(b), where we observed that the free energy remains constant
during threading. Unlike the good solvent, in this case the translocation
times are well separated and show a linear decrease with $\phi$ for all
angles. For $\phi=1$, the translocation times for all angles tend to be nearly
equal as there is no space in the $\it cis-$side and the $\it trans-$side has
equal space for all values of $\alpha$.

Interestingly, when the solvent quality across the pore are different
($\epsilon_{c}=-1.0$ $\epsilon_{t}=-0.8$) and ($\epsilon_{c}=-0.8$
$\epsilon_{t}=-1$), we do not see any qualitative change in the translocation
behaviour except the value of $\tau$. If the $\it cis-$side has relatively
poorer solvent, the free energy during threading has up-hill trends and thus
hinders the polymer’s ability to thread through the pore, resulting in
increased translocation times compared to the similar solvent
($\epsilon_{c}=\epsilon_{t}=-1.0$) across the pore. Finally, for the reverse
solvent conditions, ($\epsilon_{c}=-0.8$ and $\epsilon_{t}=-1.0$), the free
energy has down-hill trends during the threading and hence polymer would
prefer to be on the $\it trans-$side resulting a net decrease in $\tau$.

Figure 5: The variation of translocation time as a function of the back wall
position $X$. Four distinct cases are investigated. $\epsilon_{mm}$ and
$\epsilon_{wm}$ are the interactions among monomers and between monomers and
channel walls respectively.

##  IV Influence of Repulsive Forces Among Monomers and Pore Wall in Polymer
Translocation

We now extend our study where the pore wall is either repulsive or neutral for
a polymer. If we consider repulsive interaction in addition to excluded volume
in polymer, the model may mimic the case of charged DNA. For this we choose
apex angle $\alpha=45^{o}$ and vary the back-wall position. We systematically
investigate the intricate dynamics of polymer translocation for four distinct
cases, shown in Fig. (5), each offering insights into the specific molecular
interactions that play a crucial role in the translocation of charged DNA
through confined geometries. Firstly, we show the results where no
interactions were considered among monomers, and between monomers and the
channel walls ($\epsilon_{mm}=0.0$ and $\epsilon_{wm}=0.0$) to establish a
fundamental baseline scenario of translocation dynamics. This setting provides
a crucial reference point against which subsequent cases could be compared.
Next, we consider a scenario with no monomer-wall interaction but included a
repulsive force among monomers ($\epsilon_{wm}=0.0$ and $\epsilon_{mm}=1.0$).
To our surprise, this configuration resulted in a minimization of
translocation time. This may be understood as the repulsive interaction among
monomers serves to prevent entanglements (end-to end distance increases),
reducing the likelihood of collisions during the translocation process. In the
absence of interactions with the channel walls, monomers experience enhanced
freedom of movement, allowing for a smoother and more efficient passage
through the cone-shaped channel. This reduced hindrance to motion, facilitated
by repulsive forces among monomers, contributes to their efficient navigation
within the channel. Moreover, the repulsive interactions foster an optimal
exploration of the available space, minimizing unnecessary delays and
promoting a streamlined progression through the channel.

Interestingly, in the scenario where the only repulsive interaction existed
between the monomer and the pore wall, excluding interactions among monomers
($\epsilon_{wm}=1.0$ and $\epsilon_{mm}=0.0$), we observed an unexpected
increase in translocation time compared to the non-interaction scenario. This
unexpected delay can be attributed to the influential role of the repulsive
forces in the cone-shaped channel, leading to reduced available space for the
polymer. This confinement effect results in a reduction in polymer size due to
increased repulsive interaction with the channel walls. Consequently, we
observed a delay in translocation time when the monomer-wall interaction was
in play, highlighting the significant impact of the repulsive forces in the
cone-shaped channel on the translocation dynamics.

The investigation into the combined effects of monomer-monomer and monomer-
wall repulsive interactions ($\epsilon_{mm}=1.0$ and $\epsilon_{wm}=1.0$)
revealed intriguing dynamics during translocation. When both types of
repulsive forces are in play, the polymer encounters a nuanced environment
that influences its movement through the cone-shaped channel. The repulsive
interaction among monomers prevents excessive clustering, minimizing
collisions and entanglements during translocation. Simultaneously, the
repulsive force between monomers and the channel wall adds another layer of
complexity. The channel, by nature, constrains the available space for the
polymer, leading to a reduction in polymer size due to confinement, ultimately
affecting the end-to-end distance. This reduction in available space and
polymer size contributes to a delayed translocation time compared to scenarios
with no interactions among monomers and walls. However, intriguingly, the
translocation in this combined scenario is faster than when only the wall
exerts a repulsive force. This suggests a delicate interplay between various
forces, emphasizing the importance of understanding the nuanced effects of
multiple interactions in the translocation process.

These findings not only contribute to our understanding of polymer
translocation dynamics through cone-shaped channels but also offer valuable
insights into the specific challenges and optimizations associated with the
translocation of polymer through confined geometries. The unexpected trends
observed in certain scenarios open avenues for further exploration and the
potential refinement of translocation processes, particularly in the context
of charged DNA.

##  V conclusions

In this study, we have investigated the influence of confinement on polymer
translocation through a cone-shaped pore using the exact enumeration
technique. In contrast to previous investigations, we introduce a finite space
on the cis-side that can be varied, while the trans-side remains semi-
infinite. The space available at cis-side can be varied by placing a back wall
at different positions away from the pore. Additionally, the confinement is
also varied by changing the apex angle $\alpha$, allowing us to observe its
effect on polymer translocation. One end of the polymer is anchored on the
principal axis, while the other end is free to be anywhere. We systematically
vary the position of anchored sites from the cis-side to the trans-side, and
obtain the exact free energy of the system for a given value of back wall
position $X$ and the apex angle of the pore $\alpha$. Using the Fokker-Planck
equation, we obtained translocation times from the free energy of the system
having different solvent qualities across the pore.

We place particular emphasis on the novel contribution made in this work is
the calculation of system entropy exactly, which adds the understanding to the
translocation process. Our results, depicted in Fig. (2)(a)), reveal a clear
trend: as the back wall position $X$ shifts, the system’s entropy decreases.
What’s more, this effect is amplified when the apex angle $\alpha$ is reduced.
Strikingly, when plotting entropy as a function of $\phi$ for all $\alpha$
values, interestingly all the data points collapse onto a single line. This
intriguing finding suggests that, up to a certain $\phi$ threshold, entropy
primarily hinges on the space accessible to the polymer within the cis-side.
As the anchored site approaches the pore, the system’s entropy rises,
influenced by the portion of the chain that translocates to the trans-side,
boasting semi-infinite space. This intriguing phenomenon ultimately leads to
the system’s entropy reaching a constant value, irrespective of values of
$\alpha$.

The translocation time ($\tau$) obtained from the free energy analysis
provides valuable insights into the translocation process within a confined
environment. In the case of a good solvent, $\tau$ decreases as the level of
confinement increases either due to the position of the back wall or the
confining angle. Remarkably, the most significant finding is that the values
of $\tau$ collapse onto a single curve when plotted against $\phi$. This
suggests that the translocation time is primarily influenced by the confining
space, regardless of its specific shape. Based on this observation, we propose
that the inclusion of molecular crowders or similar entities within the pore
can potentially expedite the translocation process.

Another interesting finding is that by manipulating the difference in solvent
quality across the pore, one can exert control over the translocation process.
The linear decrease in $\tau$ with $\phi$ for different values of $\alpha$
indicates the need for further investigation into scaling phenomena. This
feature holds potential applications in various fields such as cell
metabolism, DNA-RNA sorting and sequencing, and drug delivery mechanisms.
These applications involve the transport of biomolecules, which is influenced
by the chemical potential gradient of the solvent. We hope that future
measurements of translocation times under variable confining environments will
contribute to a better understanding of the translocation process.

We also explored polymer translocation through a cone-shaped channel with
either repulsive or neutral interactions with the pore wall, mimicking
scenarios relevant to charged DNA translocation. Investigating four distinct
cases, we uncover intricate dynamics shaping the translocation of charged DNA
through confined geometries. The absence of interactions among monomers and
channel walls establishes a fundamental baseline, while the introduction of
repulsive forces among monomers surprisingly minimizes translocation time,
preventing entanglements and enhancing freedom of movement. Unexpected delays
arise when only the pore wall exerts a repulsive force, reducing available
space and polymer size. Combined monomer-monomer and monomer-wall repulsive
interactions introduce nuanced dynamics, emphasizing the intricate interplay
between forces and contributing to a delayed but faster translocation compared
to scenarios with only wall repulsion. This study provides comprehensive
insights into the complex factors influencing polymer translocation through
confined channels.

Financial assistance from the SERB, India, UGC, India, SPARC scheme of MoE,
and IoE scheme, MoE, India are gratefully acknowledged.

## References

  * Vella (1994) F. Vella, Biochemical Education 22, 164 (1994).
  * Nakielny and Dreyfuss (1999) S. Nakielny and G. Dreyfuss, Cell 99, 677 (1999).
  * Molineux and Panja (2013) I. J. Molineux and D. Panja, Nature Reviews Microbiology 11, 194 (2013).
  * Marenduzzo  _et al._ (2013) D. Marenduzzo, C. Micheletti, E. Orlandini, and D. W. Sumners, Proceedings of the National Academy of Sciences 110, 20081 (2013).
  * Berndsen  _et al._ (2014) Z. T. Berndsen, N. Keller, S. Grimes, P. J. Jardine, and D. E. Smith, Proceedings of the National Academy of Sciences 111, 8345 (2014).
  * Kasianowicz  _et al._ (1996) J. Kasianowicz, E. Brandin, D. Branton, and D. Deamer, Proceedings of the National Academy of Sciences 93, 13770 (1996).
  * Schneider  _et al._ (2010) G. F. Schneider, S. W. Kowalczyk, V. E. Calado, G. Pandraud, H. W. Zandbergen, L. M. K. Vandersypen, and C. Dekker, Nano Letters 10, 3163 (2010).
  * Luo  _et al._ (2007) K. Luo, T. Ala-Nissila, S.-C. Ying, and A. Bhattacharya, Physical Review Letters 99 (2007).
  * Chen  _et al._ (2009) Y.-C. Chen, C. Wang, Y.-L. Zhou, and M.-B. Luo, The Journal of Chemical Physics 130, 054902 (2009).
  * Luo and Cao (2012) M.-B. Luo and W.-P. Cao, Physical Review E 86 (2012).
  * Polson and Heckbert (2019) J. M. Polson and D. R. Heckbert, Physical Review E 100 (2019).
  * Mohan  _et al._ (2010) A. Mohan, A. B. Kolomeisky, and M. Pasquali, The Journal of Chemical Physics 133, 024902 (2010).
  * Luo  _et al._ (2006) K. Luo, I. Huopaniemi, T. Ala-Nissila, and S.-C. Ying, The Journal of Chemical Physics 124, 114704 (2006).
  * Dabhade  _et al._ (2022) A. Dabhade, A. Chauhan, and S. Chaudhury, ChemPhysChem 24 (2022).
  * Hsiao (2020) P.-Y. Hsiao, ACS Omega 5, 19805 (2020).
  * Luo  _et al._ (2008) K. Luo, S. T. T. Ollila, I. Huopaniemi, T. Ala-Nissila, P. Pomorski, M. Karttunen, S.-C. Ying, and A. Bhattacharya, Physical Review E 78 (2008).
  * Edmonds  _et al._ (2012) C. M. Edmonds, Y. C. Hudiono, A. G. Ahmadi, P. J. Hesketh, and S. Nair, The Journal of Chemical Physics 136, 065105 (2012).
  * Ikonen  _et al._ (2012) T. Ikonen, A. Bhattacharya, T. Ala-Nissila, and W. Sung, The Journal of Chemical Physics 137, 085101 (2012).
  * Bhattacharya  _et al._ (2009) A. Bhattacharya, W. H. Morrison, K. Luo, T. Ala-Nissila, S. C. Ying, A. Milchev, and K. Binder, The European Physical Journal E 29, 423 (2009).
  * Abdolvahab  _et al._ (2011) R. H. Abdolvahab, M. R. Ejtehadi, and R. Metzler, Physical Review E 83 (2011).
  * Suhonen and Linna (2016) P. M. Suhonen and R. P. Linna, Physical Review E 93 (2016).
  * Yu and Luo (2011) W. Yu and K. Luo, Journal of the American Chemical Society 133, 13565 (2011).
  * Muthukumar (2001) M. Muthukumar, Physical Review Letters 86, 3188 (2001).
  * Cacciuto and Luijten (2006) A. Cacciuto and E. Luijten, Physical Review Letters 96 (2006).
  * Wong and Muthukumar (2008) C. T. A. Wong and M. Muthukumar, Biophysical Journal 95, 3619 (2008).
  * Khalilian  _et al._ (2021) H. Khalilian, J. Sarabadani, and T. Ala-Nissila, Physical Review Research 3 (2021).
  * Tan  _et al._ (2021) F. Tan, Y. Chen, and N. Zhao, Soft Matter 17, 1940 (2021).
  * jin Jeon and Muthukumar (2016) B. jin Jeon and M. Muthukumar, Macromolecules 49, 9132 (2016).
  * Nikoofard and Fazli (2012) N. Nikoofard and H. Fazli, Physical Review E 85 (2012).
  * Sharma  _et al._ (2022) A. Sharma, R. Kapri, and A. Chaudhuri, Scientific Reports 12 (2022).
  * Lan  _et al._ (2011) W.-J. Lan, D. A. Holden, B. Zhang, and H. S. White, Analytical Chemistry 83, 3840 (2011).
  * Sexton  _et al._ (2007) L. T. Sexton, L. P. Horne, and C. R. Martin, Molecular BioSystems 3, 667 (2007).
  * Siwy and Fuliński (2002) Z. Siwy and A. Fuliński, Physical Review Letters 89 (2002).
  * Sung and Park (1996) W. Sung and P. J. Park, Physical Review Letters 77, 783 (1996).
  * Muthukumar (1999) M. Muthukumar, The Journal of Chemical Physics 111, 10371 (1999).
  * Muthukumar (2003) M. Muthukumar, The Journal of Chemical Physics 118, 5174 (2003).
  * Guttmann and Torrie (1984) A. J. Guttmann and G. M. Torrie, Journal of Physics A: Mathematical and General 17, 3539 (1984).
  * Kumar  _et al._ (2020) S. Kumar, K. Chauhan, S. Singh, and D. Foster, Physical Review E 101 (2020).
  * Chauhan and Kumar (2021) K. Chauhan and S. Kumar, Physical Review E 103 (2021).
  * Vanderzande (1998) C. Vanderzande,  _Lattice Models of Polymers_ (Cambridge University Press, 1998).
  * Kumar and Li (2010) S. Kumar and M. S. Li, Physics Reports 486, 1 (2010).
  * Sun  _et al._ (2009) L.-Z. Sun, W.-P. Cao, and M.-B. Luo, The Journal of Chemical Physics 131, 194904 (2009).