#  Pion superfluid phase transition under external magnetic field including
inverse magnetic catalysis effect

Shijun Mao<EMAIL_ADDRESS>Yvming Tian  School of Physics, Xi’an
Jiaotong University, Xi’an, Shaanxi 710049, China

###### Abstract

Pion superfluid phase transition under external magnetic field including the
inverse magnetic catalysis (IMC) effect is investigated by the Pauli-Villars
regularized NJL model. Based on the Goldstone’s theorem, we apply the massless
Goldstone boson ($\pi^{+}$ meson) to determine the onset of pion superfluid
phase. The inverse magnetic catalysis effect is introduced by the magnetic
field dependent coupling $G(eB)$, which is a decreasing function of magnetic
field. At fixed temperature and baryon chemical potential, the critical
isospin chemical potential for pion superfluid phase transition including IMC
effect increases as the magnetic field grows, which is similar as the case
without IMC effect. This demonstrates that magnetic field disfavors the pion
superfluid phase when considering or ignoring IMC effect. The critical isospin
chemical potential at fixed magnetic field, temperature and baryon chemical
potential is shifted to higher value by the IMC effect. Since it is more
difficult to form pion superfluid with weaker coupling.

##  I Introduction

Recently, the magnetic field effect on QCD phase structure attracts much
attention review0 ; review1 ; review2 ; review3 ; review4 ; review5  due to
its close relation to high energy nuclear collisions and compact stars. For
instance, the LQCD simulations performed with physical pion mass observe the
inverse magnetic catalysis (IMC) phenomenon lattice1 ; lattice11 ; lattice2 ;
lattice3 ; lattice4 ; lattice5 ; lattice6 ; lattice7 . Namely, the pseudo-
critical temperature $T_{pc}$ of chiral symmetry restoration and the quark
mass near $T_{pc}$ drop down with increasing magnetic field. On analytical
side, many scenarios are proposed to understand this inverse magnetic
catalysis phenomenon, but the physical mechanism is not clear fukushima ; mao
; kamikado ; bf1 ; bf12 ; bf13 ; bf2 ; bf3 ; bf4 ; bf5 ; bf51 ; bf52 ; bf8 ;
bf9 ; bf11 ; db1 ; db2 ; db3 ; db5 ; db6 ; pnjl1 ; pnjl2 ; pnjl3 ; pnjl4 ; pqm
; ferr1 ; ferr2 ; mhuang ; meijie ; ammmao .

QCD phase structure at finite isospin chemical potential contains the chiral
symmetry restoration and pion superfluid phase transitions. With vanishing
external magnetic field and temperature, when the isospin chemical potential
is higher than the pion mass in vacuum, the $u$ quark and $\bar{d}$ quark form
coherent pairs and condensate. The system enters the pion superfluid phase and
the charged pion becomes massless as the corresponding Goldstone mode lqcd1 ;
lqcd2 ; lqcd3 ; model1 ; model2 ; model3 ; model4 ; model5 ; model6 ; model7 ;
model8 ; model9 ; model10 ; model11 ; model12 ; model13 ; model14 ; model15 ;
model16 ; model17 ; model18 ; model19 ; model20 ; model21 ; model22 ; model23
. With finite magnetic field, the charged pion condensate breaks both isospin
symmetry in the flavor space and translational invariance in the coordinate
space, due to its direct interaction with external magnetic field.
Furthermore, when one introduces a magnetic field into a pion superfluid,
either there is a superconductor or a magnetic vortex, both of which can
change the magnetic field. LQCD simulations exhibit a sign problem at finite
isospin chemical potential and magnetic field. By using a Taylor expansion in
the magnetic field, it is reported that at vanishing temperature, the onset of
pion condensate shifts to higher isospin chemical potential under magnetic
fields lqcdb1 . In the study of pion condensate in effective models, the
interaction between the charged pion condensate and the magnetic field is
simply neglected in Ref. pib1 ; pib2 ; pib3 . To avoid this complication, we
have investigated the magnetic field effect on pion superfluid phase
transition through the Goldstone’s theorem maopionsuperfluid , where, starting
from the normal phase without pion condensate, the phase boundary of pion
superfluid is determined by its massless Goldstone mode ($\pi^{+}$ meson).
Note that the chiral symmetry, which will be restored with increasing isospin
chemical potential, controls the mass of quarks, and will influence the
formation of pion (quark-anti-quark pair) condensate and the pion superfluid
phase transition. However, the previous work on pion superfluid phase
transition under magnetic fields do not consider the IMC effect of chiral
symmetry restoration.

This paper focuses on the IMC effect on pion superfluid phase transition under
magnetic fields. Here we make use of a Pauli-Villars regularized NJL model,
which is inspired by the Bardeen-Cooper-Shrieffer (BCS) theory and describes
remarkably well the quark pairing mechanisms and the Goldstone mode njl1 ;
njl2 ; njl3 ; njl4 ; njl5 ; zhuang . Since the interaction between quarks
determines the symmetry broken and restoration. In our calculations, the IMC
effect is introduced by a magnetic field dependent coupling (see Fig.1). As a
straightforward extension of our previous work maopionsuperfluid , we
investigate pion superfluid phase transition under magnetic fields through its
Goldstone mode ($\pi^{+}$ meson).

The rest of paper is organized as follows. Sec.II describes our theoretical
framework to study the pion superfluid phase transition including the IMC
effect. The numerical results and discussions are presented in Sec.III, which
focus on the comparison between the results with and without the IMC effect.
Finally, we give the summary in Sec.IV.

##  II Framework

The two-flavor NJL model is defined through the Lagrangian density in terms of
quark fields $\psi$ njl1 ; njl2 ; njl3 ; njl4 ; njl5 ; zhuang

${\cal
L}=\bar{\psi}\left(i\gamma_{\nu}D^{\nu}-m_{0}+\gamma_{0}\mu\right)\psi+G\left[\left(\bar{\psi}\psi\right)^{2}+\left(\bar{\psi}i\gamma_{5}{\vec{\tau}}\psi\right)^{2}\right].$
(1)

Here the covariant derivative $D_{\nu}=\partial_{\nu}+iQA_{\nu}$ couples
quarks with electric charge $Q=diag(Q_{u},Q_{d})=diag(2e/3,-e/3)$ to the
external magnetic field ${\bf B}=(0,0,B)$ in $z$-direction through the
potential $A_{\nu}=(0,0,Bx_{1},0)$. $m_{0}$ is the current quark mass. The
quark chemical potential
$\mu=diag\left(\mu_{u},\mu_{d}\right)=diag\left(\mu_{B}/3+\mu_{I}/2,\mu_{B}/3-\mu_{I}/2\right)$
is a matrix in the flavor space, with $\mu_{u}$ and $\mu_{d}$ being the $u$\-
and $d$-quark chemical potentials and $\mu_{B}$ and $\mu_{I}$ being the baryon
and isospin chemical potentials.

$G$ is the coupling constant in scalar and pseudo-scalar channels. In vacuum,
the chiral symmetry $U(1)_{L}\otimes U(1)_{R}\simeq U(1)_{A}\otimes U(1)_{I}$
is spontaneously broken into the isospin symmetry $U(1)_{I}$. In medium with
finite isospin chemical potential, the broken chiral symmetry will be
(partially) restored, which leads to the chiral restoration phase transition,
and meanwhile, the isospin symmetry will be broken, which leads to the pion
superfluid phase transition. Corresponding to the symmetries and their
spontaneous breaking, we have two order parameters, neutral chiral condensate
$\langle\bar{\psi}\psi\rangle$ for chiral restoration phase transition and
charged pion condensate $\langle\bar{\psi}\gamma_{5}\tau^{1}\psi\rangle$ for
pion superfluid phase transition. According to the Goldstone’s theorem, the
pseudo-Goldstone mode of chiral symmetry breaking is the neutral pion
$\pi^{0}$, and the Goldstone mode of isospin symmetry breaking is the charged
pion $\pi^{+}$. Physically, it is equivalent to define the phase transition by
the order parameter and Goldstone mode gold1 ; gold2 .

As a straightforward extension of our previous paper maopionsuperfluid , we
use the Goldstone mode (massless $\pi^{+}$ meson) to determine the pion
superfluid phase transition at finite temperature, chemical potential and
magnetic field,

$\displaystyle m_{\pi^{+}}(eB,T,\mu_{B},\mu_{I})=0.$ (2)

To include the inverse magnetic catalysis effect, in our calculations, we
apply the magnetic field dependent coupling $G(eB)$, which is obtained by
fitting the LQCD reported decreasing pseudo-critical temperature of chiral
symmetry restoration lattice1 . As plotted in Fig.1, the magnetic field
dependent coupling $G(eB)/G(eB=0)$ is a decreasing function of magnetic field,
and it reduces $20\%$ at $eB/m^{2}_{\pi}=35$ (with $m_{\pi}=134$ MeV).

Figure 1: Magnetic field dependent coupling $G(eB)$ fitted from LQCD reported
decreasing pseudo-critical temperature of chiral symmetry restoration. In this
paper, we fix $m_{\pi}=134$ MeV.

In NJL model, mesons are constructed through quark bubble summations in the
frame of random phase approximation njl2 ; njl3 ; njl4 ; njl5 ; zhuang .
Taking into account of the interaction between charged mesons and magnetic
fields, the meson propagator $D_{\pi^{+}}$ can be expressed in terms of the
polarization function $\Pi_{\pi^{+}}$ maopionsuperfluid ; ritus1 ; ritus2 ;
ritus21 ; maopimeson ,

$\displaystyle
D_{\pi^{+}}({\bar{k}})=\frac{2G(eB)}{1-2G(eB)\Pi_{\pi^{+}}({\bar{k}})}.$ (3)

where ${\bar{k}}=(k_{0},0,-\sqrt{(2l+1)eB},k_{3})$ is the conserved Ritus
momentum of $\pi^{+}$ meson under magnetic fields.

The meson pole mass $m_{\pi^{+}}$ is defined through the pole of the
propagator at zero momentum $(l=0,\ k_{3}=0)$,

$\displaystyle 1-2G(eB)\Pi_{\pi^{+}}(k_{0}=m_{\pi^{+}})=0,$ (4)

and

$\displaystyle\Pi_{\pi^{+}}(k_{0})$ $\displaystyle=$ $\displaystyle
J_{1}(m_{q})+J_{2}(k_{0}),$ (5) $\displaystyle J_{1}(m_{q})$ $\displaystyle=$
$\displaystyle
3\sum_{f,n}\alpha_{n}\frac{|Q_{f}B|}{2\pi}\int\frac{dp_{3}}{2\pi}\frac{1}{E_{f}}$
$\displaystyle\times\left[1-f(E_{f}+\mu_{f})-f(E_{f}-\mu_{f})\right],$
$\displaystyle J_{2}(k_{0})$ $\displaystyle=$
$\displaystyle\sum_{n,n^{\prime}}\int\frac{dp_{3}}{2\pi}\frac{j_{n,n^{\prime}}(k_{0})}{4E_{n}E_{n^{\prime}}}$
$\displaystyle\times\big{[}\frac{f(-E_{n^{\prime}}-\mu_{u})-f(E_{n}-\mu_{d})}{k_{0}+\mu_{I}+E_{n^{\prime}}+E_{n}}$
$\displaystyle+\frac{f(E_{n^{\prime}}-\mu_{u})-f(-E_{n}-\mu_{d})}{k_{0}+\mu_{I}-E_{n^{\prime}}-E_{n}}\big{]},$
$\displaystyle j_{n,n^{\prime}}(k_{0})$ $\displaystyle=$
$\displaystyle\left[{(k_{0}+\mu_{I})^{2}/2}-n^{\prime}|Q_{u}B|-n|Q_{d}B|\right]j^{+}_{n,n^{\prime}}$
(8) $\displaystyle-2\sqrt{n^{\prime}|Q_{u}B|n|Q_{d}B|}\ j^{-}_{n,n^{\prime}},$

with flavors $f=u,d$, spin factor $\alpha_{n}=2-\delta_{n0}$, quark energy
$E_{f}=\sqrt{p^{2}_{3}+2n|Q_{f}B|+m_{q}^{2}}$, quark Landau level
$n=0,1,2...$, (dynamical) quark mass
$m_{q}=m_{0}-2G(eB)\langle\bar{\psi}\psi\rangle$ and Fermi-Dirac distribution
function $f(x)=1/(e^{x/T}+1)$ in $J_{1}(m_{q})$, and the $u$-quark energy
$E_{n^{\prime}}=\sqrt{p^{2}_{3}+2n^{\prime}|Q_{u}B|+m_{q}^{2}}$ and $d$-quark
energy $E_{n}=\sqrt{p^{2}_{3}+2n|Q_{d}B|+m_{q}^{2}}$ in $J_{2}(k_{0})$.

The (dynamical) quark mass $m_{q}$ is determined by the gap equation,

$\displaystyle 1-2G(eB)J_{1}(m_{q})$ $\displaystyle=$
$\displaystyle\frac{m_{0}}{m_{q}}.$ (9)

Because of the four-fermion interaction, the NJL model is not a renormalizable
theory and needs regularization. In this work, we make use of the gauge
invariant Pauli-Villars regularization scheme njl1 ; njl2 ; njl3 ; njl4 ; njl5
; mao ; maopionsuperfluid ; maopimeson ; rev6 , where the quark momentum runs
formally from zero to infinity. The three parameters in the Pauli-Villars
regularized NJL model, namely the current quark mass $m_{0}=5$ MeV, the
coupling constant $G(eB=0)=3.44$ GeV-2 and the Pauli-Villars mass parameter
$\Lambda=1127$ MeV are fixed by fitting the chiral condensate
$\langle\bar{\psi}\psi\rangle=-(250\ \text{MeV})^{3}$, pion mass $m_{\pi}=134$
MeV and pion decay constant $f_{\pi}=93$ MeV in vacuum with
$T=\mu_{B}=\mu_{I}=0$ and $eB=0$.

##  III results

Figure 2: (upper panel) Critical isospin chemical potential at $\mu^{\pi}_{I}$
for pion superfluid phase transition at $T=\mu_{B}=0$ (red lines), $\pi^{+}$
meson mass $M_{\pi^{+}}$ at $T=\mu_{B}=\mu_{I}=0$ (black lines) and critical
isospin chemical potential $\mu^{c}_{I}$ for chiral restoration phase
transition at $T=\mu_{B}=0$ (blue lines) as functions of magnetic field with
constant coupling $G(eB=0)$ (dashed lines) and magnetic field dependent
coupling $G(eB)$ (solid lines). (lower panel) The zoom-in figure in the region
$4<eB/m^{2}_{\pi}<7.5$. In this paper, we fix $m_{\pi}=134$ MeV.

Fig.2 depicts the critical isospin chemical potential $\mu^{\pi}_{I}$ for pion
superfluid phase transition at $T=\mu_{B}=0$ (in red), critical isospin
chemical potential $\mu^{c}_{I}$ for chiral symmetry restoration at
$T=\mu_{B}=0$ (in blue) and $\pi^{+}$ meson mass $M_{\pi^{+}}$ at
$T=\mu_{B}=\mu_{I}=0$ (in black) as functions of magnetic field with constant
coupling $G(eB=0)$ (dashed lines) and magnetic field dependent coupling
$G(eB)$ (solid lines). Here, $\mu^{\pi}_{I}$ is determined by condition
$m_{\pi^{+}}(eB,T=0,\mu_{B}=0,\mu_{I}=\mu^{\pi}_{I})=0$, and $\mu^{c}_{I}$ is
defined through the quark mass jump. When including the inverse magnetic
catalysis effect by $G(eB)$, the critical isospin chemical potential
$\mu^{c}_{I}$ for chiral symmetry restoration changes from MC phenomenon
(increasing with magnetic field) to IMC phenomenon (decreasing with magnetic
field). However, the critical isospin chemical potential $\mu^{\pi}_{I}$ for
pion superfluid phase transition with magnetic field dependent coupling
$G(eB)$ is similar as the case with constant coupling $G(eB=0)$. It is an
increasing function of magnetic field, which means that magnetic field
disfavors the pion superfluid phase, even including the IMC effect. There
exists some numerical differences. In the regions $eB/m^{2}_{\pi}<4.5$ and
$eB/m^{2}_{\pi}>5$, the $\mu^{\pi}_{I}$ with IMC effect (red solid line) is
higher than that without IMC effect (red dashed line), and at strong magnetic
field, for instance, $eB/m^{2}_{\pi}=35$, the difference increases up to
$10\%$. Physically, it is harder to form the pion (quark-anti-quark pair)
condensate with weaker interaction. Therefore, it is expected to obtain higher
$\mu^{\pi}_{I}$ when including IMC effect. However, with
$4.5<eB/m^{2}_{\pi}<5$, the $\mu^{\pi}_{I}$ with IMC effect (red solid line)
is lower than that without IMC effect (red dashed line). This might be related
to the first-order chiral restoration phase transition, associated with an
abrupt jump of quark mass, which happens at very similar isospin chemical
potential. Note that the crossing point of $\mu_{I}^{c}$ and $\mu_{I}^{\pi}$
located at $eB/m^{2}_{\pi}=4.5$ with IMC effect and $eB/m^{2}_{\pi}=4.75$
without IMC effect, respectively.

In Fig.2, we also make comparison between critical isospin chemical potential
$\mu^{\pi}_{I}$ for pion superfluid phase transition and $\pi^{+}$ meson mass
in vacuum $M_{\pi^{+}}=m_{\pi^{+}}(eB,T=0,\mu_{B}=0,\mu_{I}=0)$. Under weak
magnetic field $eB/m^{2}_{\pi}<4.5$, the $\mu^{\pi}_{I}$ is equal to the
$\pi^{+}$ meson mass in vacuum $M_{\pi^{+}}$, which can be analytically proved
by directly comparing the pole equation (4) and gap equation (9) model9 ;
maopionsuperfluid . With $4.5<eB/m^{2}_{\pi}<7$, we obtain
$\mu^{\pi}_{I}<M_{\pi^{+}}$, and with stronger magnetic field
$eB/m^{2}_{\pi}>7$, we have $\mu^{\pi}_{I}>M_{\pi^{+}}$, which are obtained
numerically. Without IMC effect, the turning points are located at
$eB/m^{2}_{\pi}=4.75$ and $eB/m^{2}_{\pi}=7.5$. The deviation between
$\mu^{\pi}_{I}$ and $M_{\pi^{+}}$ at strong magnetic field is enhanced by the
IMC effect.

What is the situation when turning on the temperature and baryon chemical
potential? Fig.3 is the phase diagram of pion superfluid in $\mu_{I}-T$ (with
$\mu_{B}=0$) and $\mu_{I}-\mu_{B}$ (with $T=0$) planes at $eB/m^{2}_{\pi}=10$
and $eB/m^{2}_{\pi}=20$, where the solid (dashed) lines correspond to the case
with (without) IMC effect. Fixing temperature (upper panel) or baryon chemical
potential (lower panel), the phase transition from normal phase to pion
superfluid phase happens with increasing isospin chemical potential. On the
left side of the phase transition line, it is the normal phase, and on the
right side, it is the pion superfluid phase. With higher temperature, the
thermal motion of quarks are stronger. Hence it is more difficult to form pion
condensate, and the critical isospin chemical potential becomes higher. With
fixed temperature and vanishing baryon chemical potential, the critical
isospin chemical potential increases with magnetic fields. With higher baryon
chemical potential, the mismatch between the Fermi surface of quark and anti-
quark is larger. This also prohibits the pion condensate, and leads to higher
critical isospin chemical potential. With fixed baryon chemical potential and
vanishing temperature, the critical isospin chemical potential also increases
with magnetic fields. Including the IMC effect, at finite magnetic field,
temperature and baryon chemical potential, the pion superfluid phase
transition happens at higher isospin chemical potential, which is caused by
the weaker coupling between quark and anti-quark. The difference of critical
isospin chemical potential is $\delta\mu^{T}_{I}\simeq(8\sim 11)$ MeV with
fixed $T$ and vanishing $\mu_{B}$, and $\delta\mu^{\mu_{B}}_{I}\simeq(10\sim
19)$ MeV with fixed $\mu_{B}$ and vanishing $T$ at $eB/m^{2}_{\pi}=10$, and
$\delta\mu^{T}_{I}\simeq(14\sim 25)$ MeV and
$\delta\mu^{\mu_{B}}_{I}\simeq(14\sim 48)$ MeV at $eB/m^{2}_{\pi}=20$. The
deviation between the phase transition lines with and without IMC is enhanced
by the magnetic field.

Figure 3: (upper panel) Pion superfluid phase diagram in $\mu_{I}-T$ plane
with $\mu_{B}=0$ and fixed magnetic field. (lower panel) Pion superfluid phase
diagram in $\mu_{I}-\mu_{B}$ plane with $T=0$ and fixed magnetic field. The
left lines are for $eB/m^{2}_{\pi}=10$, and right for $eB/m^{2}_{\pi}=20$. In
this paper, we fix $m_{\pi}=134$ MeV.

##  IV summary

Pion superfluid phase transition under external magnetic field including the
inverse magnetic catalysis effect is investigated through the Pauli-Villars
regularized NJL model. Based on the Goldstone’s theorem, we apply the massless
Goldstone boson ($\pi^{+}$ meson) to determine the phase boundary of pion
superfluid. The inverse magnetic catalysis effect is introduced by the
magnetic field dependent coupling $G(eB)$, which is a decreasing function of
magnetic field.

At fixed temperature and baryon chemical potential, including IMC effect, the
critical isospin chemical potential $\mu_{I}^{\pi}$ for pion superfluid phase
transition increases with the magnetic field, which is qualitatively similar
as the case without IMC effect. This indicates that magnetic field disfavors
the pion superfluid phase with and without IMC effect. The deviation of
$\mu_{I}^{\pi}$ with and without IMC effect is enhanced by the magnetic field.

Comparing with the case ignoring IMC effect, the critical isospin chemical
potential for pion superfluid phase transition at fixed magnetic field,
temperature and baryon chemical potential is shifted to higher value by the
IMC effect. There exist different methods to mimic IMC effect in NJL model,
which determine the decreasing coupling with magnetic field but with
quantitatively different value bf8 ; bf9 ; geb1 ; geb2 ; geb3 . Due to the
weakened coupling, it becomes harder to form pion condensate, and the critical
isospin chemical potential for pion superfluid phase transition will become
higher. This conclusion is independent on the specific formula for coupling
$G(eB)$.

Acknowledgement: The work is supported by the NSFC Grant 11775165 and
Fundamental Research Funds for the Central Universities.

## References

  * (1) I.A. Shovkovy, Lect. Notes Phys. 871, 13(2013).
  * (2) F. Preis, A. Rebhan and A. Schmitt, Lect. Notes Phys. 871, 51(2013).
  * (3) R. Gatto and M. Ruggieri, Lect. Notes Phys. 871, 87(2013).
  * (4) M. D’Elia, Lect. Notes Phys. 871, 181(2013).
  * (5) V.A. Miransky and I.A. Shovkovy, Phys. Rep. 576, 1(2015).
  * (6) J.O. Anderson and W.R. Naylor, Rev. Mod. Phys. 88, 025001(2016).
  * (7) G.S. Bali, F. Bruckmann, G. Endr$\ddot{o}$di, Z. Fodor, S.D. Katz, S. Krieg, A. Sch$\ddot{a}$fer and K.K. Szab$\acute{o}$, J. High Energy Phys. 02, 044(2012); Phys. Rev. D86, 071502(2012).
  * (8) G.S. Bali, F. Bruckmann, G. Endr$\ddot{o}$di, F. Gruber and A. Sch$\ddot{a}$fer, J. High Energy Phys. 04, 130(2013).
  * (9) V. Bornyakov, P.V. Buividovich, N. Cundy, O.A. Kochetkov and A. Sch$\ddot{a}$fer, Phys. Rev. D90, 034501(2014).
  * (10) M. D’Elia, F. Manigrasso, F. Negro and F. Sanfilippo, Phys. Rev. D98, 054509(2018).
  * (11) G. Endr$\ddot{o}$di, J. High Energy Phys. 07, 173(2015).
  * (12) G. Endr$\ddot{o}$di, M. Giordano, S. D. Katz, T. G. Kovacs and F. Pittler, J. High Energy Phys. 07, 009(2019).
  * (13) A. Tomiya, H.T. Ding, X.D. Wang, Y. Zhang, S. Mukherjee and C. Schmidt, PoS Lattice2018, 163(2019).
  * (14) E.M. Ilgenfritz, M. Muller-Preussker, B. Petersson and A. Schreiber, Phys. Rev. D89, 054512(2014).
  * (15) K. Fukushima and Y. Hidaka, Phys. Rev. Lett 110, 031601(2013).
  * (16) S.J. Mao, Phys. Lett. B758, 195(2016); Phys. Rev. D94, 036007(2016); Phys. Rev. D97, 011501(R)(2018); Chin. Phys. C45, 021004(2021).
  * (17) K. Kamikado and T. Kanazawa, J. High Energy Phys. 03, 009(2014).
  * (18) J.Y. Chao, P.C. Chu and M. Huang, Phys. Rev. D88, 054009(2013).
  * (19) F. Bruckmann, G. Endr$\ddot{o}$di and T.G. Kovacs, arXiv: 1311.3178.
  * (20) J. Braun, W.A. Mian and S. Rechenberger, Phys. Lett. B755, 265(2016).
  * (21) N. Mueller and J.M. Pawlowski, Phys. Rev. D91, 116010(2015).
  * (22) T. Kojo and N. Su, Phys. Lett. B720, 192(2013).
  * (23) F. Bruckmann, G. Endrodi and T.G. Kovacs, J. High Energy Phys. 04, 112(2013).
  * (24) A. Ayala, M. Loewe, A. J$\acute{u}$lia Mizher and R. Zamora, Phys. Rev. D90, 036001(2014).
  * (25) A. Ayala, L. Alberto Hern$\acute{a}$ndez, A. J$\acute{u}$lia Mizher, J. Crist$\acute{o}$bal Rojas and C. Villavicencio, Phys. Rev. D89, 116017(2014).
  * (26) A. Ayala, C.A. Dominguez, L.A. Hern$\acute{a}$ndez, M. Loewe, and R. Zamora, Phys. Rev. D92, 096011(2015).
  * (27) R.L.S. Farias, K.P. Gomes, G. Krein and M.B. Pinto, Phys. Rev. C90, 025203(2014).
  * (28) M. Ferreira, P. Costa, O. Lourenco, T. Frederico and C. Provid$\hat{e}$ncia, Phys. Rev. D89, 116011(2014).
  * (29) F. Preis, A. Rebhan and A. Schmitt, J. High Energy Phys. 03, 033(2011).
  * (30) E.S. Fraga and A.J. Mizher, Phys. Rev. D78, 025016(2008), Nucl. Phys. A820, 103C(2009).
  * (31) K. Fukushima, M. Ruggieri and R. Gatto, Phys. Rev. D81, 114031(2010).
  * (32) C.V. Johnson and A. Kundu, J. High Energy Phys. 12, 053(2008).
  * (33) V. Skokov, Phys. Rev. D85, 034026(2012).
  * (34) E.S. Fraga, J. Noronha and L.F. Palhares, Phys. Rev. D87, 114014(2013).
  * (35) R. Gatto and M. Ruggieri, Phys. Rev. D82, 054027(2010); Phys. Rev. D83, 034016(2011).
  * (36) M. Ferreira, P. Costa and C. Provid$\hat{e}$ncia, Phys. Rev. D89, 036006(2014).
  * (37) M. Ferreira, P. Costa, D.P. Menezes, C. Provid$\hat{e}$ncia and N.N. Scoccola, Phys. Rev. D89, 016002(2014).
  * (38) P. Costa, M. Ferreira, H. Hansen, D.P. Menezes and C. Provid$\hat{e}$ncia, Phys. Rev. D89, 056013(2014).
  * (39) A.J. Mizher, M.N. Chernodub and E.S. Fraga, Phys. Rev. D82, 105016(2010).
  * (40) E.J. Ferrer, V.de la Incera, I. Portillo and M.Quiroz, Phys. Rev. D89, 085034(2014).
  * (41) E.J. Ferrer, V.de la Incera and X.J. Wen, Phys. Rev. D91, 054006(2015).
  * (42) K. Xu, J.Y. Chao and M. Huang, Phys. Rev. D103, 076015(2021).
  * (43) J. Mei and S.J. Mao, Phys. Rev. D102, 114035(2020).
  * (44) S.J. Mao, Phys. Rev. D106, 034018(2022).
  * (45) J.B. Kogut and D.K. Sinclair, Phys. Rev. D66, 034505(2002); D66 014508(2002); D70, 094501(2004).
  * (46) P. Scior, L. Smekal and D. Smith, arXiv: 1710.06314.
  * (47) B.B. Brandt, G.Endr$\ddot{o}$di and S.Schmalzbauer, Phys. Rev. D97, 054514(2018).
  * (48) D.T. Son and M.A. Stephanov, Phys. Rev. Lett. 86, 592(2001); Phys. At. Nucl. 64, 834(2001).
  * (49) J.B. Kogut and D. Toublan, Phys. Rev. D64, 034007(2001).
  * (50) K. Splittorff, D.T. Son and M.A. Stephanov, Phys. Rev. D64, 016003(2001).
  * (51) M.C. Birse, T.D. Cohen and J.A. McGovern, Phys. Lett. B516, 27(2001).
  * (52) M. Loewe and C. Villavicencio, Phys. Rev. D67, 074034(2003); D70, 074005(2004).
  * (53) D. Toublan and J.B. Kogut, Phys. Lett. B564, 212(2003).
  * (54) M. Frank, M. Buballa and M. Oertel, Phys. Lett. B562, 221(2003).
  * (55) A. Barducci, R. Casalbuoni, G. Pettini and L. Ravagli, Phys. Rev. D69, 096004(2004); D71, 016011(2005).
  * (56) L.Y. He, M. Jin and P.F. Zhuang, Phys. Lett. B615, 93 (2005); Phys. Rev. D71, 116001(2005).
  * (57) S. Mukherjee, M.G. Mustafa and R. Ray, Phys. Rev. D75, 094015(2007).
  * (58) C.F. Mu, L.Y. He and Y.X. Liu, Phys. Rev. D82, 056006(2010).
  * (59) S.J. Mao and P.F. Zhuang, Phys. Rev. D86, 097502(2012).
  * (60) T. Xia, L.Y. He and P.F. Zhuang, Phys. Rev. D88, 056013(2013).
  * (61) L.Y. He, S.J. Mao and P.F. Zhuang, Int. J. Mod. Phys. A28, 1330054(2013).
  * (62) H. Ueda, T.Z. Nakano, A. Ohnishi, M. Ruggieri and K. Sumiyoshi, Phys. Rev. D88, 074006(2013).
  * (63) K. Kamikado, N. Strodthoff, L. Smekal and J. Wambach, Phys. Lett. B718, 1044(2013).
  * (64) R. Stiele, E.S. Fraga and J.S. Bielich, Phys. Lett. B729, 72(2014).
  * (65) S.J. Mao, Phys. Rev. D89, 116006(2014).
  * (66) P. Adhikari, J.O. Andersen and P. Kneschke, Phys. Rev. D98, 074016(2018).
  * (67) M. Mannanelli, Particles 2, 411(2019).
  * (68) T. Xia, J. Hu and S.J. Mao, Chin. Phys. C43, 054103(2019).
  * (69) P. Adhikaria and J.O. Andersen, Phys. Lett. B804, 135352(2020).
  * (70) S.S. Avancini, A. Bandyopadhyay, D.C. Duarte and R.L.S. Farias, Phys. Rev. Research 2, 013007(2020).
  * (71) G. Endr$\ddot{o}$di, Phys. Rev. D90, 094501(2014).
  * (72) X. Kang, M. Jin, J. Xiong and J.R. Li, arXiv:1310.3012.
  * (73) M. Loewe, C. Villavicencio and R. Zamora, Phys. Rev. D89, 016004(2014).
  * (74) G.Q. Cao and P.F. Zhuang, Phys. Rev. D92, 105030(2015).
  * (75) S.J. Mao, Phys. Rev. D102, 114006(2020).
  * (76) Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345(1961) and 124, 246(1961).
  * (77) S.P. Klevansky, Rev. Mod. Phys. 64, 649(1992).
  * (78) M.K. Volkov, Phys. Part. Nucl. 24, 35(1993).
  * (79) T. Hatsuda and T. Kunihiro, Phys. Rep. 247, 221(1994).
  * (80) M. Buballa, Phys. Rep. 407, 205(2005).
  * (81) P. Zhuang, J. H$\ddot{u}$fner, S.P. Klevansky, Nucl. Phys. A567, 525(1994).
  * (82) J. Goldstone, Nuovo Cim. 19, 154-164(1961).
  * (83) J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965-970(1962).
  * (84) V. Ritus, Annals Phys. 69, 555(1972).
  * (85) E. Elizalde, E. Ferrer, V. Incera, Ann. Phys. (N.Y.) 295, 33(2002).
  * (86) C. Leung and S. Wang, Nucl. Phys. B747, 266(2006).
  * (87) S.J. Mao, Phys. Rev. D99, 056005(2019).
  * (88) S.J. Mao and Y.X. Wang, Phys. Rev. D96, 034004(2017).
  * (89) H. Liu, L. Yu, M. Chernodub and M. Huang, Phys. Rev. D94, 113006(2016).
  * (90) S. Avancini, R. Farias, M. Pinto, W. Tavares and V. Tim$\acute{o}$teo, Phys. Lett. B767, 247(2017).
  * (91) T.H. Moreira and F.L. Braghin, Phys. Rev. D105, 114009(2022).


# Isolating a Single Microtubule in Nanofluidic Device

Ssu-Ying Chen<EMAIL_ADDRESS>Department of Physics, New Jersey Institute of
Technology, Newark, NJ, USA  Arooj Aslam<EMAIL_ADDRESS>Department of
Physics, New Jersey Institute of Technology, Newark, NJ, USA  David J. Apigo
<EMAIL_ADDRESS>Department of Physics, New Jersey Institute of
Technology, Newark, NJ, USA  Sagnik Basuray<EMAIL_ADDRESS>Department of
Biomedical Engineering, New Jersey Institute of Technology, Newark, NJ, USA
Camelia Prodan<EMAIL_ADDRESS>Department of Physics, New Jersey Institute
of Technology, Newark, NJ, USA

###### Abstract

Biological systems have been theoretically predicted to support topological
wave-modes, similar to the ones existing in meta-materials. The existing
methods to measure these modes are not implementable to biological systems;
new techniques have to be developed to accommodate measurements in life
science. Motivated by this perspective, we report a nanofluidic device for
studying one microtubule at a time. Micro-channels etched into fused-silica
using reactive ion etching were interfaced with nanochannels milled with
electron beam lithography, and sealed with a PDMS-coated glass coverslip. The
microchannels are 1 $\mu$m deep and 100 $\mu$m wide, and the nanochannels are
150 nm deep and 750 nm wide, they are tested to be effective for isolating
microtubules. The methods presented here are for an adaptable nanofluidic
platform for phonon measurements in biopolymers made of proteins or DNA.

##  I Introduction

The field of topological mechanics and metamaterials is expanding to
incorporate soft topological materials, such as micro and nano scale colloids
and polymers, including biopolymers, suspended in water based media[1, 2]
(CITE Vitelly). Phenomena similar to the one seen in topological insulators,
like Thouless pumping, are theoretically proposed to exist in polymers
suspended in fluids. In particular the the microtubules and actin have been
proposed to support topological phonon modes[1], and has been inspiring the
development of other lattices[2]. While the bending modes of the microtubules
have been experimentally measured, the rest of the modes requires confinement
and precise actuation.[3]. Previous experimental methods to obtain of the
density of state of colloidal micro-particles in suspension require accurate
tracking of particle for a long period of time. (INSERT CITATION ISLAM
colloids) To adapt such methods to microtubules, one needs to confine the
microtubules to stay in the field of view of the microscope for a longer time,
while allowing them to exibit dynamic instability. The development of
techniques in nanofabrication has become an increasingly useful tool in the
life sciences. [7, 8, 9, 10] .[4, 5, 6]Methods that are developed for
nanofluidic channels are adaptable for microscopy and can integrate
electronics for sensing and actuation of single proteins or
macromolecules.[11, 12, 13] Single proteins and macromolecules are typically
imaged at high magnification using fluorescence or bright-field microscopy.
This requires imaging through a thin optically transparent material, typically
glass or fused-silica. Furthermore, the ability to adapt nanofluidic devices
for electrodes to incorporate manipulation and sensing requires the
nanochannels to be in an insulating material, and this is another benefit to
fabricating in glass or fused-silica.

In this paper we present a method for confining a single microtubules in a
nanochannel and a highly repeatable and detailed method for interfacing
microchannels that are 50 $\mu$m apart with 750 nm wide by 150 nm deep
nanochannels and capping them with a glass coverslip using PDMS as a bonding
medium. The bonding is done outside the cleanroom and does not require
expensive equipment. [20] The nanochannels are written into fused silica using
electron beam lithography (E-beam lithography or EBL). This device can be used
for transmission and fluorescence microscopy at high magnification with low
fluorescence background. This method can be adapted to include electrodes in
the nanochannels for sensing and actuation with dimensions that promote high
signal to noise ratio.

##  II Fabrication of Nanofluidic Devices in Fused Silica

Figure 1: Cross-sectional nanochannel process flow. Directions of steps
indicated by arrows. First, start with an HMDS vapor primed fused-silica wafer
coated in 1.8 $\mu$m AZ1518 photoresist. Next, perform UV photolithography
exposure with indicated photomask. RIE exposed pattern to etch it into the
substrate. Strip wafer in microposit remover 1165 photoresist stripper at
70$\degree$ C. This allows microchannels of width, $w=50\mu$m, and depth,
$d=1.5\mu$m to be present on substrate. Prepare wafer with 15 nm gold for
E-beam writing. Gold etch after the nanochannel pattern is written.

###  II.1 Fabrication of Microchannels

The devices were fabricated on 500 $\mu$m thick fused-silica wafers with a
diameter of $d=76.2$ mm. The wafers were cleaned in a Piranha solution 3:1
H2SO4:H2O2 for 5 minutes, followed by a DI water bath for 1 mintue. The wafers
were rinsed with DI water and dried with N2. Next, the wafers were placed in
an isopropyl alcohol (IPA) bath for one minute, followed by a DI water bath
for another minute. Afterwards, the wafers were rinsed with DI water and dried
with N2.

Next, the wafers were primed with hexamethyldisilazane (HMDS) in a Yield
Engineering Systems (YES) HMDS prime oven at $148\degree$C. The procedure
dehydrates the wafers and then vapor primes them with HMDS. Next, AZ1518
positive photoresist was spun on the wafers at 4000 RPM for 40 s. The wafers
are baked at $110\degree$C for 2 minutes and then cooled on a cooling plate.
Standard photolithography was performed for 30 s using a EVG620 mask aligner
to expose the sample with the microchannel pattern shown in Figure 2. The
wafers were manually developed in AZ300 MIF for 60 s.

Wafers were measured with a Bruker Dektak-XT profilometer to verify that the
photoresist was properly removed from the patterning area. Next, the wafers
were placed in an Oxford PlasmaPro 80 Reactive Ion Etch (RIE) and an O2 descum
was run for 2 minutes to remove any residual resist from the microchannel
structure due to exposure and development. Next, an SiO2 etch was performed
for 15 minutes. The etch is a combination of CF4 and O2 gasses at 35 sscm and
3 sscm, respectively. The set forward power was 300 W and the DC bias was
between 520 and 530 V. The wafers were measured with the profilometer to
verify the etch depth. The photoresist was stripped for 1 hour in Dow
Microposit Remover 1165, and the final depth of each channel was measured once
more in the profilometer. The final depth of microchannels was around 1
$\mu$m.

In order to prep the fused-silica substrates for E-beam lithography, E-beam
resist was spun on the wafers before they were coated in 15 nm of gold in an
Angstrom Nexdep E-beam evaporator.

Figure 2: Rendered CAD model of the microchannel device integrated with the
nanochannels. Nanochannels are highlighted by the zoomed view insert. Imaging
is done through the glass coverslip with an inverted microscope.

###  II.2 Electron-Beam Written Nanochannels

The pattern of nanochannels were designed to be cascading to guide MTs
in(Figure 3a, 3b). Nanochannels were written between the microchannels using
the Elionix ELS-G100, an electron-beam lithography system with high speed and
ultra high precision thermal field emission. The ELS-G100 is capable of
writing and generating patterns with a line width of 6 nm. One wafer was
loaded into the FIB chamber at a time. The SEM with a beam current 500 pA was
used to focus on the sample surface to determine the z-axis working distance
and locate the writing area on the sample. After focusing the beams on the
wafer, the pattern of nanochannels were aligned and then the wafer was ready
to be exposed.

After E-beam lithography, wafers were developed in 70% cold IPA and then
inspected under microscopy to confirm that the pattern was well-written. A
gold etch was then performed on the samples to remove the Au coating. Next,
the wafers were placed in RIE again to etch the nanochannels with calculated
parameters until the depth reached 150 nm(Figure 3d). The E-beam resist was
stripped for 1 hour in Dow Microposit Remover 1165. Next, the wafers were
again coated with AZ1512 photoresist to be diced into 20 mm x 20 mm devices,
and transported to the lab for bonding.

Figure 3: a) SEM image of microchannels integrated with nanochannels. b)
10000x magnification of the cascading junction between the micron-sized and
nano-sized channels. c) AFM image of one nanochannel. d) The measured depth of
the nanochannel is around 150 nm.

###  II.3 Packaging Fluidic Devices

Port holes were made at the ends of the microchannels using a BNP 220 Suction
Blast Cabinet. The metal mask was made beforehand and the devices were fixed
and protected by 3M Scotch Magic Tape.

To seal the devices, a modified PDMS bonding protocol was performed. To
prepare for the sealing of the nanofluidic devices, it is critical that the
bench top, fume hood, and laminar flow hood be as clean and dust-free as
possible. They are all cleaned with 70$\%$ ethanol and again with IPA. A Ni-Lo
4 XL vacuum holder digital spin coater is cleaned, lined with fresh foil and
set to 3 stages of spinning: 500 RPM for 10 s, 3000RPM for 30 s, and 5000 RPM
for 20 s. The speed was reached with 500 RPM ramp speed.

Photoresist is stripped from the diced devices with 1165 microposit remover
for 15 minutes at $70\degree$C before the bonding procedure begins. The
devices and 22 mm x 22 mm coverslips are cleaned by placing them in a
chemically resistant coverslip rack and sonicating them in acetone for 5
minutes. Devices and coverslips are rinsed in the rack sequentially with
acetone, IPA, ethanol, and DI water.

PDMS(SYLGARD 184 Silicone Elastomer Kit, Dow Corning) was mixed with its
curing agent in the volume ratio of 10:1, then the mixture was placed in a
desiccator for an hour to remove the bubbles. Next, 500 $\mu$l bubble-free
PDMS was spun on per coverslip, coated coverslips were then moved to a pre-
heated $120\degree$C hot plate, and cured for at least an hour.

Both the device(patterns face up) and coverslip(PDMS side up) were placed side
by side in laminar flow hood, and treated with BD-20AC laboratory corona
treater for 5 minutes. After corona treatment, the device was flipped and
carefully placed on the coverslip, the bonding should happen consequently, if
not, use the blunt side of a tweezers to gently tap the device on the corner.
After the devices were sealed, 4 resin reservoirs were adhered to 4 ports
using Loctite Epoxy instant glue, after the Epoxy was cured, the devices were
ready for use.

###  II.4 Flow of Microtubule into Nanochannels

All the steps below have to be done in a humidity chamber that can be made
from an empty pipette tip rack with 2 Kimwipes soaked with DI water inside.
The devices should sit on the rack and the lid should be closed during waiting
to prevent the device from dehydration.

The channel has to be first rinsed with PEM buffer (80 mM
piperazine-N,N’-bis2-ethanesulfonic acid(PIPES), 1 mM ethylene glycol
tetraacetic acid (EGTA), 1mM MgCl2, pH 6.9). 15-20 $\mu$l PEM buffer was
pipetted into one port on the left microchannel, the nanochannels began
filling up with liquid within 15 minutes; the excess solution from the port
was pipetted out, then another 15-20 $\mu$l was placed at the port of the
right microchannel. This step is to create fluid pressure for the liquid to
flow in. The device was let sit in humidity chamber for another 15 minutes
until all channels were filled up.

The channels then have to be passivated with BSA to prevent the adhesion of
microtubules to the surface. 10 $\mu$g/ml bovine serum albumin (BSA) diluted
in PEM buffer was flowed into the device. 15-20 $\mu$l BSA was pipetted into
one port on the left microchannel, then waited 30 minutes for BSA to fill up
the left part. The excess solution was pipetted out from the port. Another
15-20 $\mu$l was added at the port of the right microchannel. After adding the
solution, the device was let sit for another 30 minutes for the channels to be
filled up.

To test the quality of the seal, 1 mM fluorescein solution diluted in PEM
buffer was flowed into the device. Fluorescein is sensitive to blue light, so
solutions were made in with red or yellow light and foil was used to cover any
containers to prevent bleaching of the sample before imaging. The nanochannels
were imaged at 100x magnification with a 1.6x optovar to increase
magnification. The fluorescein solution flowed into the nanochannels by
placing 15-20 $\mu$l solution at one port on the left microchannel, the
nanochannels began filling up with liquid within 15 minutes; the excess
solution from the port was pipetted out, then another 15-20 $\mu$l was placed
at the port of the right microchannel, and another 15 minutes was give for the
device to sit in the humidity chamber until all channels were filled up.

Pure double cycled tubulin and tubulin labeled with Alexa594(Red) were stored
in -80∘C in PEM buffer. To grow MTs, the tubulin was thawed on ice and grown
in a GPEM buffer (1 mM guanonine triphosphate (GTP), 80 mM PIPES, 1 mM EGTA,
1mM MgCl2, pH 6.9) with 10% dimethyl sulfoxide (DMSO). The solution was
incubated on ice for 5 minutes and then incubated at 37∘C for one hour. The
tubulin was grown at 2 mg/ml with 10% to 50% labeled tubulin. After
incubation, warm Taxol was added to a final concentration of 10 M. The
stabilized MTs were diluted 100 times in an anti-bleaching solution (250 nM
glucose oxidase, 64 nM catalase, 40 mM D-glucose, 1 beta-mercaptethanol (BME),
10 $\mu$M Taxol) before flowing into fluidic devices.

We found that electrophoresys works best at guiding microtubules from the
ports into the nanochannel. Electric field of 0.9 V/m was applied using a DC
power supply, it took 15-30 minutes until the microtubules approached the
nanochannels.

Figure 4: Image of one fluorescence labeled microtubule(red) moving in the
nanochannel filled with 1 mM fluorescein solution(green) at 100x with
fluorescent microscope. b) was taken 50 millisecond after a).

The cleanliness of the wafer is critical for each stage of the process. Any
particulates on the surface can block the fluidic channels or cause an uneven
photoresist deposition. Wafers that were not cleaned properly at each step did
not have precise patterning of the microchannels due to uneven photoresist,
and this caused jagged edges on the microchannels. Photoresist also does not
adhere well to fused-silica. Priming the wafers with HMDS was a necessary step
to promote the adhesion of the photoresist to the wafers.

The profilometer measurements are important to verify proper photoresist
development, the etch rate of the RIE, and the final depth of the
microchannels. Photoresist AZ1518 is 1.8$\mu$m in depth, and the profilometer
measurement after the photoresist was developed verified that the photoresist
is removed from the microchannels and the wafer is set for etching. The RIE
etch process etches both the fused-silica and the photoresist with different
selectivity, so the etch time for a specific depth must be determined. The
wafers are etched for different times, and profilimeter measurements are
performed for each time to determine the etch rate. An etch time of 1 hour and
15 minutes resulted in a 1.2 $\mu$m deep microchannel.

The gold coating on the samples is necessary for SEM imaging. Fused-silica is
non-conductive, and therefore charging effects due to the SEM makes it
difficult to image. Charging occurs when the electrons do not have a
conductive path to guide them away from the sample surface and a layer of
electrons is generated. The cloud of electrons prevents clear projectile of
the secondary electrons from the surface to the electron detector, resulting
in poor image quality. The gold coating provides a conductive path the the
electrons to move away from surface during imaging. However, long exposure to
the electron beam still resulted in charging effects.

The E-beam lithography was done on one wafer(12-21 devices depends on their
size) at a time. The process is achieved by shooting a focused electron-beam
at the surface of the material. The system provides a stable 1.8 nm electron-
beam using high beam current at 100 kV, it is accurate and highly efficient.
After exposure, the depth of nanochannels can be decided by how long they are
etched in RIE.

The bonding process is robust, repeatable, and easily done outside of the
clean room. A successful bonding can be seen by eye by a change in contrast
between the coverslip and the fused-silica. The drawback of this method is
that the sagging of PDMS material tends to partially or completely seal off
the channels that do not have a high aspect ratio due to the rubber-like
behavior of PDMS. However, it was noticed that if PDMS coated coverslips were
let cool down after one hour heat treat and wait overnight, the chance of PDMS
collapse and block the channels becomes slim.

The port holes used to be made by drilling into the ends of the microchannels
using a drill press at 3600 rpm using a diamond coated drill bit of 0.5 mm
diameter. The drilling of the port holes can be difficult as well as time
consuming, and still result in cracking the device. Sandblasted-through
portholes have cleaner edges, and the sizes of holes are well-controlled by
the metal masks, also portholes of a dozen of devices can be done within a few
minutes.

##  III Results and Discussion

Previous devices for studying microtubule are microns in size. [27, 28]. The
depth of our channels is 150nm and was chosen to keep the microtubules in the
field of view of a microtscope for longer while still allowing space for
dynamic instability. The width of the nanochannel is 750nm and was chosen to
allow future actuation of the microtubule in the nanochannel for the purpose
of exiting different types of vibrational modes in the microtubules lattice.
Microtubules have a persistance length of 1mm making them much stiffer and
longer than DNA. We found that electrophoresis was the best way to guide the
microtubules into the nanochannels. We previously tried flow, and kinesin, a
motor protein, but without success. For this step to be successful is very
important that the bonding be very tight such that the electric field goes
primary through the nanochannel. The width of the microchannel has to be large
to allow the microtubules to align with the electric field prior to the
entrance in the nanochannel.

In figure 4 we present a microtubules flowing, under electric field through
our nano channels. Prior to the flow we test the bonding and wether the
channel is free by flowing flourescein (green) into the nanochannel. The
microtubule (red) is flown through the nanochannel using electric field. The
position of the microtubule can be controlled. This method can be further
adapted to embed electrodes on two sides of the nanochannels for sensing and
actuation.

##  IV Conclusion

In conclusion, flow a microtubule through a 150 nm deep channel is reported.
The channel is well characterized using SEM and AFM. The method to seal the
fluidic channels allows for easy imaging with a microscope. This methodology
allows for an adaptable tool to isolate a single microtubule to perform direct
and quantitative measurements. The reported device is further adaptable to
fabricating electrodes along the channel as an added sensor to detect and
measure in the nanochannel. The small dimensions of the nanochannel are
crucial for the addition of electrodes because to increase the signal to noise
ratio the electrodes need to be close enough to the object of study to
minimize electrical signal from the solution itself.

The use of nanofluidic platforms has already benefited the study of DNA’s
unique structural properties. The adaptability and repeatability of the
fabrication of the devices presented here makes it an ideal platform to open
the doors to isolate and study not only DNA, but also other proteins and
protein assemblies. For example, the nanochannels can be milled to an ideal
height and width to isolate single actin or microtubule proteins. These
biopolymers have unique polymerization and depolymerization properties that
need to be quantitatively measured at the protein level to elucidate the
dynamic mechanics of these structures. This platform allows for variations in
size that can be optimized for the object of study. Furthermore, this platform
affords the ability to isolate and manipulate macromolecules and proteins for
measurements, while keeping their naturally aqueous environment.

##  V Acknowledgments

The authors acknowledge support from the W.M. Keck Foundation. The Basuray lab
acknowledges funding from the NSF grant, 1751759 , CAREER:”ASSURED”
electrochemical platform for multiplexed detection of Cancer Biomarker Panel
using Shear-Enhanced Nanoporous-Capacitive Electrodes. This research used
resources of the Center for Functional Nanomaterials, which is a U.S. DOE
Office of Science Facility, at Brookhaven National Laboratory under Contract
No. DE-SC0012704. This work was performed in part at the Princeton Institute
of the Science and Technology of Materials (PRISM) Cleanroom at Princeton
University and at the Advanced Science Research Center NanoFabrication
Facility of the Graduate Center at the City University of New York. The
authors thank Fernando Camino, Quan Wang, and Milan Begliarbekov for their
beneficial guidance and conversations.

## References

  * [1] E. Prodan and C. Prodan., “Topological phonon modes and their role in dynamic instability of microtubules”. Phys. Rev. Lett. 103, 248101 (2009).
  * [2] E. Prodan, K. Dobiszewski, A. Kanwal, J, Palmieri, and C. Prodan., “Dynamical Majorana edge modes in a broad class of topological mechanical systems”. Nature Communications, 8, 14587 (2017).
  * [3] A. Aslam and C. Prodan., “Experimentally measured phonon spectrum of microtubules”. J. Phys. D: Appl. Phys. 53, 025401 (2019).
  * [4] H. Cao, J. O. Tegenfeldt, R. H. Austin, and S. Y. Chou.,“Gradient nanostructures for interfacing microﬂuidics and nanoﬂuidics”. Applied Physics Letters, 81(16), 3058-3060 (2002).
  * [5] R. ¨Oz, S. Kk, and F. Westerlund. “A nanoﬂuidic device for real-time visualization of DNA-protein interactions on the single DNA molecule level”. Nanoscale, 11(4), 2071–2078 (2019).
  * [6] J. W. Yeh, K. K. Sriram, A. Taloni, Y. L. Chen, and C. F. Chou. “Quantitative analysis of reptation of partially extended DNA in sub-30 nm nanoslits”. arXiv, 1–24 (2015).
  * [7] V. G. Kutchoukov, F. Laugere, W. van der Vlist, L. Pakula, Y. Garnini, A. Bossche., “Fabrication of nanofluidic devices using glass-to-glass anodic bonding”. Sensors and Actuators A: Physical, 114, 521-527 (2004).
  * [8] C. Iliescu, H. Taylor, M. Avram, J. Miao, S. Franssila., “A practical guide for the fabrication of microﬂuidic devices using glass and silicon”. Biomicrofluidics, 6, 016505 (2012).
  * [9] D. M. Cannon, Jr., B. R. Flachsbart, M. A. Shannon, J. V. Sweedler, and P. W. Bohn., “Fabrication of Single Nanofluidic Channels in Poly(methylmetacrylate) films via Focused-Ion Beam Milling for use as Molecular Gates”. Applied Physics Letters, 85, 1241-1243 (2004).
  * [10] Q. Xia, K. j. Morton, R. H. Austin, S. Y. Chou., “Sub-10 nm Self-Enclosed Self-Limited Nanoﬂuidic Channel Arrays”. Nanoletters, 8, 3830-3833 (2008).
  * [11] w. Reisner, J. Pedersen, R. H. Austin., “DNA conﬁnement in nanochannels: physics and biological applications”. Reports on Progress in Physics, 75, 106601 (2012).
  * [12] N. Kaji, Y. Tezuka, Y. Takamura, M. Ueda, T. Nishimoto, H. Nakanishi, Y. Horiike, Y. Baba.,“Separation of Long DNA Molecules by Quartz Nanopillar Chips under a Direct Current Electric Field”. Analytical Chemistry, 76(1), 15-22 (2004).
  * [13] Y. M. Wang, J. O. Tegenfeldt, W. Reisner, X, Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, R. H. Austin., “Single-molecule studies of repressor–DNA interactions show long-range interactions”. PNAS, 102, 9796-9801 (2005).
  * [14] M. A. Lake, C. E. Naciso, K. R. Cowdrick, T. J. Storey, S. Zhang, J. J. Zartman, D. J. Hoelzle., “Microfluidic device design, fabrication, and testing protocols”. Protocol Exchange, (2014) DOI: 10.1038/protex.2015.069.
  * [15] G. A. C. M. Spierings, J. Haisma, T. M. Michelsen., “Surface-related phenomena in the direct bonding of silicon and fused-silica wafer pairs”. Philips JOurnal of Research, 49, 47 - 63 (1995).
  * [16] P. Rangsten, Ö. Vallin, K. Hermansson, Y. Bäcklund., “Quartz-to-Quartz Direct Bonding”. Journal of the Electrochemical Society, 146(3), 1104 - 1105 (1999).
  * [17] J. O. Tegenfeldt, C. Prinz, H. Cao, S. Chou, W. W. Reisner, R. Reihn, Y. M Wang, E. C. Cox, J. C. Sturm, P. Silberzan, R. H. Austin., “The dynamics of genomic-length DNA molecules in 100-nm channels”. PNAS, 101(30), 10979-10983 (2004).
  * [18] P. Mao and J. Han., “Fabrication and characterization of 20 nm planar nanofluidic channels by glass–glass and glass–silicon bonding”. Lab on a Chip, 5, 837-844 (2005).
  * [19] W. W. Y. Chow, K. F. Lei, G. Shi, W. J. Li, and Q. Huang., “Microfluidic channel fabrication by PDMS-interface bonding”. Smart Mater. Struc., 15, S112-S116 (2006).
  * [20] K. Haubert, T. Drier, and D. Beebe., “PDMS bonding by means of a portable, low-cost corona system”. Lab on a Chip, 6, 1548-1549 (2006).
  * [21] M. Eichler, B. Michel, P. Hennecke, M. Gabriel, C. P. Klages., “Low-Temperature Direct Bonding of Borosilicate, Fused Silica, and Functional Coatings”. ECS Transactions, 33, 339 - 348 (2010).
  * [22] G. Kalkowski, U. Zeitner, T. Benkenstein, J Fuchs, C. Rothhardt, R. Eberhardt., “Direct wafer bonding for encapsulation of fused silica optical gratings”. Microelectric Engineering, 97, 177 - 180 (2012).
  * [23] L. D. Menard, J. M. Ramsey., “The Fabrication of Sub-5-nm Nanochannels in Insulating Substrates using Focused Ion Beam Milling”. Nanoletters, 11(2), 512-517 (2011).
  * [24] W. MoberlyChan, D. Adams, M. Aziz, G. Hobler, T. Schenkel., “Fundamentals of Focused Ion Beam Nanostructural Processing: Below, At, and Above the Surface”. MRS Bulletin, 32(5), 424-432 (2007).
  * [25] H.Y. Wang, et al., “Low temperature bonding for microfabrication of chemical analysis devices”. Sensors and Actuators B: Chemical, 45, 199-207 (1997).
  * [26] J. Khandurina, et al., “Integrated system for rapid PCR-based DNA analysis in microfluidic devices”. Anal. Chem., 72, 2995-3000 (2000).
  * [27] R. Yokokowa, et. al. Unidirectional Transport of a Bead on a Single Microtubule Immobolized in a Submicrometre Channel. Nanotechnology, 17, 289-294 (2006).
  * [28] M.G.L. van den Heuvel, M.P. de Graff, S.G. Lemay, C. Dekker Electrophoresis of Individual Microtubules in Microchannels. PNAS, 104, 7770-7775 (2007).