# Physics-based nozzle design rules for high-frequency liquid metal jetting

J. Seo<EMAIL_ADDRESS>Palo Alto Research Center, 3333 Coyote Hill Road, Palo
Alto, CA 94304, USA  Department of Mechanical Engineering, Kyung Hee
University, 1732 Deogyeong-daero, Giheung, Yongin, Gyeonggi 17104, Republic of
Korea  C. Somarakis  Palo Alto Research Center, 3333 Coyote Hill Road, Palo
Alto, CA 94304, USA  S. Korneev  Palo Alto Research Center, 3333 Coyote Hill
Road, Palo Alto, CA 94304, USA  M. Behandish  Palo Alto Research Center, 3333
Coyote Hill Road, Palo Alto, CA 94304, USA  A.J. Lew  Palo Alto Research
Center, 3333 Coyote Hill Road, Palo Alto, CA 94304, USA

###### Abstract

We present physics-based nozzle design rules to achieve high-throughput and
stable jetting in drop-on-demand liquid metal 3D printing. The design rules
are based on scaling laws that capture the change of meniscus oscillation
relaxation time with geometric characteristics of the nozzle’s inner profile.
These characteristics include volume, cross-sectional area, and inner surface
area of the nozzle. Using boundary layer theory for a simple geometry, we show
that the meniscus settles faster when the ratio of inner surface area to
volume is increased. High-fidelity multiphase flow simulations verify this
scaling. We use these laws to explore several design concepts with
parameterized classes of shapes that reduce the meniscus relaxation time while
preserving desired droplet specs. Finally, we show that for various nozzle
profile concepts, the optimal performance can be achieved by increasing the
ratio of the circumferential surface area to the bulk volume to the extent
that is allowable by manufacturing constraints.

††preprint: AIP/123-QED

##  I Introduction

Metal additive manufacturing (AM) is emerging as a viable alternative to
traditional methods such as casting to make supply chains more resilient and
cost-effective for small-batch, multi-variety, and spare parts Sukhotskiy  _et
al._ (2018). Drop-on-demand (DoD) liquid metal jetting stands out due to its
high deposition throughput, low porosity, use of off-the-shelf materials
(aluminum wire feed), predictable material properties, and operational
safetySimonelli  _et al._ (2019). The process is characterized by ejecting a
sequence of droplets from a microfluidic nozzle attached to the end of a pump
where the metal is molten and pushed down at frequencies of a few hundred Hz
using mechanisms ranging from magnetohydrodynamicsSukhotskiy  _et al._ (2018)
to pneumaticsLuo  _et al._ (2012). As every droplet is released, the meniscus
(liquid-gas interface) at the tip of the nozzle oscillates due to the dynamic
interplay between surface tension, the inertia of the fluid, and the imposed
pressure at the inlet of the nozzle (Fig. 1). Due to viscous dissipation in
the fluid, the oscillations are damped.

To enable fast and reliable/repeatable builds, it is important to produce
droplets with consistent volume, shape, and velocity distributions, which, in
turn, strongly depend on the rate at which the post-ejection energy in the
liquid in the nozzle dissipates. One way to quantify this rate is by observing
the decay of the oscillations of the meniscus membrane at the tip of the
nozzle. After a brief initial transient, the amplitude of these oscillations
decay exponentially with a characteristic _relaxation time_. Ideally, the
meniscus should settle before the next droplet starts forming.

The most common nozzle concepts are axisymmetric due to their ease of
manufacturing (e.g., via micro-drilling). The resulting circular cross-
sections lead to an inevitable tradeoff between droplet parameters, target
throughput, and relaxation time. A nozzle with a smaller diameter has a
smaller relaxation time, but it is harder to push the liquid through it due to
the need for larger fluid speeds to achieve a target throughput and the
increased viscous resistance. As a result, a higher pressure is required to
meet the target throughput. Reducing the diameter of the nozzle may also lead
to either smaller droplets or, through the use of an increased flow velocity,
to an elongated droplet that breaks apart after the ejection. The critical
question is to design a nozzle that ejects a largely spherical droplet with a
target velocity and mass while minimizing the relaxation time of post-ejection
oscillations.

Analysis of the meniscus damping rate has been known as a challenging problem
in multiphase fluid mechanics Howell  _et al._ (1999); Ting and Perlin (1995)
due to the non-linearity of the physics and of the partial differential
equations associated with multiphase flows. Accurate analysis and prediction
of the decay rate after a droplet ejection requires the solution of the
Navier-Stokes equations through computational fluid dynamics simulations for
extended periods of time. In our study, we identified the sources of viscous
dissipation (both physical and numerical) and considered flows driven by the
oscillatory dynamics induced by the meniscus in which inertial effects form a
thin boundary layer. Previous studies on meniscus damping control for 3D
printing Stachewicz  _et al._ (2009) adopted the one-dimensional harmonic-
oscillator model, having a damper to model the viscous dissipation. These
studies assumed fully-developed viscous flows (Poiseuille solution), but this
assumption is invalid for the types of flow induced in liquid metals, which
often have very low viscosity and for which the dynamics induced by the
meniscus does not allow the boundary layer to develop. To the best of our
knowledge, our work is the first study on the damping of a nozzle’s meniscus
considering transient inertial effects, and on applying these results to
establish design rules.

To demonstrate the application of the design rules, we performed physics
simulations of multiphase flows capturing the oscillatory dynamics of the
liquid-gas interface on several nozzle geometries. We used a multiphase flow
solver in OpenFOAM®, which employs the algebraic volume-of-fluid approach for
tracking gas-liquid interfaces. Our simulation results validated our
hypothesis on the formation of a thin boundary layer near the nozzle wall, and
demonstrated a successful decrease of the relaxation time in nozzles with
increased surface-area-to-volume ratio.

The paper is organized as follows: In Section II, we present the problem
definition including the geometry, quantities of interest, governing
equations, and non-dimensional parameters. In Section III, we present an
analytical investigation of physical mechanisms that determine the nozzle
performance, apply it to a simple cylindrical nozzle, and extend it to more
complex shapes. In Section IV, we describe the simulation setup used in our
study. In Section V, we validate the identified scaling laws applying
numerical simulations to solve the dynamics for a few nozzle concepts. Section
VI summarizes and concludes the current findings. Lastly, in Section VII, we
elaborate on the numerical method we adopted to solve the governing equations
and on the validation of the multiphase flow solver.

##  II Problem Setup

###  II.1 Nozzle Geometry

Figure 1: Time snapshots of the droplet ejection in the liquid metal jetting
printer. The red colored area is filled with liquid metal, and the blue area
corresponds to the area with gas. The nozzle is axisymmetric, and is obtained
by rotating the cross-section shown in the figures around the left edge, or
the axis. (a-b) Droplet ejection by a high pressure pulse on the top surface;
(c-d) meniscus retraction during the pressure pulsing; (e-h) damped
oscillations of the meniscus without external pressure. Time intervals between
snapshots vary.

We consider a nozzle in a domain $V=A\times[0,L]$, where
$A\subset\mathbb{R}^{2}$ is an open planar set that defines the cross-section
of the nozzle, and $L$ is its length. For $A$, we adopt either Cartesian
coordinates $(x,y)$ or cylindrical coordinates $(r,\theta)$. The coordinate
along $[0,L]$ will be denoted by $z$. The nozzle is filled with a viscous
fluid with density $\rho$, dynamic viscosity $\mu$, and kinematic viscosity
$\nu=\mu/\rho$. We assume that the dynamics in the nozzle starts with the
fluid at rest at time $t=0$, and that the next droplet is ejected at time
$t=T$. The nozzle outlet radius and diameter are denoted by $R$ and $d$,
respectively. After a droplet ejection, surface tension drives the meniscus
fluctuations and viscous dissipation dampens its deformation. The vertical
meniscus displacement in the $z$ direction is denoted by $\eta(x,y,t)$, and it
is measured from the planar plane at the nozzle outlet ($z=0$). The contact
line, where the liquid-gas-solid phases meet, is pinned at the sharp nozzle
outlet, so $\eta=0$ at $r=R$, as shown in Fig. 1 (d).

###  II.2 Quantities of Interest

Our goal when designing a nozzle for high-speed jetting is to reduce the
relaxation time of the meniscus while retaining the ability to eject droplets
with a desired geometry and speed. We will describe the long-time dynamic
behavior of the meniscus deformation, $\eta$, with that of a damped harmonic
oscillator model

$\eta(x,y,t)=\eta_{t=0}(x,y)e^{-\gamma t}\cos(\omega t+\phi),$ (1)

where ${\eta_{t=0}}$ is the initial meniscus displacement, $\gamma>0$ is the
damping rate, $\omega$ is the oscillation frequency of the meniscus and $\phi$
is a phase. The relaxation time is $1/\gamma$. Shortly after a droplet is
ejected, the dynamics of the meniscus is non-linear and cannot be fully
described by a single mode damped harmonic oscillator model. After the initial
transient, in which high wavenumber modes decay fast, we find that the long-
time behavior of the meniscus can be well captured by the damped harmonic
oscillator model with a single wavenumber mode. In the end, we aim to find a
nozzle design geometry for which $\gamma$ is large enough to damp the
oscillations within the ejection period (e.g., $e^{-\gamma T}\sim 10^{-2}$).
In the long-time behavior, the value of $\gamma$ appears to be a feature of
the geometry and the fluid only, and independent of the specific way in which
a droplet is generated.

###  II.3 Governing Equations

The dynamics of liquid and gas in and around the nozzle are governed by the
Navier-Stokes (N-S) equation,

$\begin{split}\frac{\partial\rho_{f}\vec{u}}{\partial
t}+\nabla\cdot(\rho_{f}\vec{u}\otimes\vec{u})=-\nabla
p+\nabla\cdot\mu_{f}(\nabla\vec{u}+\nabla\vec{u}^{T})\\\
+\rho_{f}\vec{g}+\sigma\kappa\vec{n}\hat{\delta}(\vec{x}-\vec{x}_{s}),\end{split}$
(2)

where $\vec{u}$ is the velocity field, $p$ is pressure, $\vec{g}$ is the
acceleration of gravity, and $\sigma$ is the surface tension coefficient.
$\rho_{f}$ is density of fluid either being $\rho$ for liquid metal and
$\rho_{g}$ for gases. $\mu_{f}$ is viscosity of fluid, either being $\mu$ for
liquid metal and $\mu_{g}$ for gases. The locus of the liquid-gas interface is
described by a surface $\vec{x}_{s}$, $\kappa$ is its mean curvature,
$\vec{n}$ is the local surface normal vector, and
$\hat{\delta}(\vec{x}-\vec{x}_{s})$ is the three-dimensional Dirac delta
function. The last term in the right hand side of the equation represents the
surface tension force, which is a concentrated force and non-zero only at the
interface.

We consider the algebraic Volume-of-Fluid (VoF) framework and introduce a
three-dimensional scalar field $\alpha(x,y,z,t)$, defined as the volume
fraction of liquid ranging from 0 to 1, to track the interface location
$\vec{x}_{s}$. The (diffuse) interface between liquid and gas is algebraically
described by setting $\alpha=0$ in the spatial domain filled with gas, and
$\alpha=1$ in the spatial domain filled with liquid. In this way, the
interface curvature and the surface normal vector can be calculated by
relations such as $\kappa=-\nabla\cdot\vec{n}$ and
$\vec{n}=\nabla\alpha/|\nabla\alpha|$. The time evolution of the scalar field
$\alpha$, and hence of the gas-liquid interface, is governed by the phase
transport equation

$\frac{\partial\alpha}{\partial t}+\nabla\cdot(\vec{u}\alpha)=0.$ (3)

Finally, the system of equations is completed by stating the equation for mass
conservation

$\nabla\cdot\vec{u}=0,$ (4)

at all points of the domain. This states the incompressibility of the liquid
metal, and for simplicity, of the surrounding gas.

We assume that the contact line is effectively pinned at the nozzle outlet
rim. The nozzle outlet is macroscopically sharp, but assumed to be rounded at
the microscale, so that the slope of the interface is not limited by the
contact angle imposed by the material properties Seo, Garcia-Mayoral, and Mani
(2015). With the fixed contact line, we assume negligible contribution from
the contact line to the dissipation of energy in the systemSnoeijer and
Andreotti (2013). Our high-speed video recording of the meniscus (not shown
here) confirmed that the meniscus can be considered to be pinned at the nozzle
outlet.

###  II.4 Dimensional Analysis

We proceed to derive dimensionless numbers to estimate the relative importance
of different physical phenomena based on orders of magnitude approximations to
typical material properties for a liquid metal and a gas.

Typical values of mass density of liquid metals considered in metal AM is
$\rho\sim O(10^{3})$kg/m3, and the kinematic viscosity is $\nu\sim
O(10^{-7})\text{m}^{2}/\text{s}$ (e.g. aluminum alloy) Assael  _et al._
(2006). The density of the surrounding gas (e.g. argon) is $\rho_{g}\sim
O(1)$kg/m3, and the kinematic viscosity is $\nu_{g}\sim
O(10^{-6})\text{m}^{2}/\text{s}$. Surface tension at the interface between
aluminum alloyMolina  _et al._ (2007) and the argon gas is $\sigma\sim
O(1)$N/m. We consider the range of nozzle diameter in the order of hundred
micrometers $d\sim O(10^{-4})$m. The droplet size is mainly determined by the
nozzle outlet diameter, so that the diameter of the droplet is
$d_{\text{Droplet}}\sim d$.

The droplet ejection rate for the droplet-on-demand liquid metal jetting is
assumed $f_{\text{Jet}}\sim O(10^{2})$Hz. Based on the capillary flow theory
Landau and Lifshitz (1958), the meniscus oscillation frequency is
$\omega\sim\sqrt{\sigma k^{3}/\rho}\sim O(10^{3})Hz$, where $k$ is the
wavenumber as $k\sim 1/d$ with $d=500\mu m$. Therefore, we expect several
periods of meniscus oscillations in between droplet ejections.

Based on these values, we find that the Reynolds number associated with the
oscillatory dynamics of the meniscus is $Re=\omega d^{2}/\nu\gg O(1)$.
Therefore, we expect the flow inside nozzle to involve inertial effects
forming a thin boundary layer near the nozzle wall. A rough estimate of
boundary layer thickness resulting from oscillating flow is
$\delta\sim\sqrt{\nu/\omega}\sim O(10)\mu m$, which is thinner than the nozzle
radius, $\delta/R\ll O(1)$. The flow inside the nozzle cannot be regarded as
fully developed.

The micro-scale nozzle size leads to capillary-force-dominant meniscus
dynamics, in which the capillary length $l_{c}=\sqrt{\sigma/\Delta\rho
g}=O(10^{-3})m$ is comparable or larger than the nozzle diameter, where
$\Delta\rho$ is the difference in densities in the two fluids. Likewise, the
Bond number is small, $Bo=gd^{2}\Delta\rho/\sigma\ll O(1)$, therefore the
relative importance of the gravitational force on the meniscus deformation is
negligible. The Weber number is $We=\rho U^{2}d/\sigma\sim O(1)$, for
$U\sim\omega d$ a characteristic speed, and the ejected droplet shape is
determined by both inertial effect and the surface tension force.

In sum, the dynamics of meniscus oscillation will strongly depend on the
interplay between surface tension, inertia, and viscous forces in the liquid
metal. We use the boundary layer theory for oscillatory flows Lamb (1932);
Landau and Lifshitz (1958) to analyze the flow physics.

##  III Physics-based Design Rules for Optimal Nozzle Performance

In this section we present an analytical study of the physical mechanisms that
determine the nozzle performance, and their scaling relations to the geometric
parameters of the nozzle design. After deriving an analytical expression for
the viscous dissipation from the Navier-Stokes equation, we develop scaling
laws for the relaxation time $1/\gamma$ (alternatively, damping rate $\gamma$)
for the cylindrical nozzle geometry, and introduce modified geometries with
faster relaxation times based on these law.

###  III.1 Source of Viscous Dissipation

From the dimensional analysis in Section II.4, we assume that a thin viscous
boundary layer is formed near the nozzle wall responding to the oscillatory
fluid motion after the ejection of a droplet. A standard approach in potential
flow analyses of classical fluid mechanics on multiphase flows Landau and
Lifshitz (1958); Lamb (1932) is to decompose the total damping rate $\gamma$
into two components. One source of dissipation is from the boundary layer,
where the velocity gradient is dominated by the no-slip effect of the wall.
Another source of viscous dissipation is from the bulk of fluid away from the
wall, where the velocity gradient mainly depends on the flow structure. This
approach has been extensively used to estimate the dissipation rate of
sloshing liquid in containers Henderson and Miles (1994); Case and Parkinson
(1956); Howell  _et al._ (1999). Accordingly, we decompose the total damping
rate of the meniscus into two components,

$\gamma=\gamma_{\text{bulk}}+\gamma_{\text{bl}},$ (5)

where $\gamma_{\text{bulk}}$ is the damping rate from the bulk of the fluid
and $\gamma_{\text{bl}}$ is the damping rate from the boundary layer. These
damping rates stem from the contribution in each region to the dissipation of
the kinetic energy of the fluid. To estimate them, we adopt standard flow
structures (velocity fields) for the bulk of the fluid and the boundary layer.

###  III.2 Scaling of Damping Rate

The damping rate can be estimated by the energy dissipation rate over the
total energy of the systemLandau and Lifshitz (1958),

$\gamma\sim\frac{\overline{\epsilon}}{\overline{E}},$ (6)

where $\overline{\epsilon}$ is the mean (in time) viscous dissipation rate and
$\overline{E}$ is the mean (in time) total energy in the system. Multiplying
the N-S equation (2) by $\frac{1}{2}\vec{u}$ and integrating over the volume
(i.e. kinetic energy form), we can derive the energy dissipation rate at a
given time in a viscous fluid as

$\epsilon=\frac{\mu}{2}\iiint\nabla^{s}\vec{u}\colon\nabla^{s}\vec{u}\;dV,$
(7)

where $\nabla^{S}\vec{u}$ is the symmetric gradient of the velocity field, and
$A\colon B$ is the Frobenius inner product between tensors $A$ and $B$. The
total energy of the system is

${E}={\rho}\iiint|\vec{u}|^{2}dV,$ (8)

which is twice the kinetic energy Lamb (1932). The mean values of the
dissipation rate and the total energy can be computed via time integration of
the instantaneous values over a sufficiently long period of time.

We first evaluate the scaling of the dissipation rate in the bulk of the
fluid, away from the wall. Following Eq. (7), the scaling of the dissipation
rate corresponding to a perturbed flow structure at a length scale $\lambda$
is Landau and Lifshitz (1958)

$\epsilon\sim\mu\frac{U^{2}}{{\lambda^{2}}}V,$ (9)

where $U$ is the characteristic speed in the system, in our case, a
characteristic speed of the meniscus, and $V$ the volume of the fluid. Given
that the mean energy scales as

$E\sim\rho U^{2}V,$ (10)

the damping rate in the bulk of the fluid scales

$\gamma_{\text{bulk}}\sim\frac{\nu}{\lambda^{2}}.$ (11)

This indicates that perturbed flow structures with smaller size will decay
faster. Our major focus will be on the largest wavenumber mode, which stays in
the system for the longest time.

For the boundary layer dissipation near the wall, the surface integral form of
Eq. (7) is considered,

$\epsilon\sim\mu\frac{U^{2}}{\delta}S,$ (12)

where $S$ is the surface area exposed to solid surface and $\delta$ is the
boundary layer thickness formed near the solid surface. Under the oscillatory
flow with frequency $\omega$, a dimensional analysis on the boundary layer
thickness $\delta$ estimates Landau and Lifshitz (1958)

$\delta\sim\sqrt{\frac{\nu}{\omega}}.$ (13)

Finally, the damping rate at the boundary layer of the fluid scales as

$\gamma_{\text{bl}}\sim\sqrt{\nu\omega}\frac{S}{V},$ (14)

where Eq. (8) is used to estimate the kinetic energy. The oscillation
frequency of the meniscus can be roughly estimated as
$\omega\sim\sqrt{\sigma/\rho R^{3}}$. The scaling relation Eq. (14) indicates
that the interface deformation decays faster when the surface area to the
volume ratio is large. This provides an important insight on the nozzle design
and we will use this rule to control the dissipation rate.

Potential flow analyses Landau and Lifshitz (1958); Case and Parkinson (1956)
identified the exact relation in Eq. (11) for ideal waves as
$\gamma_{\text{bulk}}=2\nu k^{2}$, where $k$ is the wavenumber of the flow
structure of interest. For example, in a closed brimful cylinder, the lowest
wavenumber is $k=\beta/R$, where $\beta\simeq 3.83$ is the first root of the
derivative of the first Bessel function of the first kindCase and Parkinson
(1956). Furthermore, the boundary layer dissipation rate identified by linear
analyses Case and Parkinson (1956) is
$\gamma_{\text{bl}}\simeq\sqrt{\frac{\nu\omega}{2}}\frac{1}{2R}$, for a
cylinder with radius $R$ and height $L$ ($V=\pi R^{2}L$ and $S=2\pi RL$). We
note that although this linear solution provided an accurate estimate of the
decay rate in small amplitude capillary waves ($L\gg R$)Case and Parkinson
(1956); Henderson and Miles (1994); Howell  _et al._ (1999), it does not
guarantee accurate predictions when the meniscus dynamics is non-linear due to
large interface fluctuationsTing and Perlin (1995). However, we expect the
damping rate to scale similarly with $S$ and $V$ even during the initial, non-
linear stages of the meniscus oscillations.

Using the insights from the scaling, we draw the following nozzle design rules
for high speed droplet jetting:

  * •

Given the inverse scaling of $\gamma_{\text{bulk}}$ and $\gamma_{\text{bl}}$
with the size of the system (with both $\lambda$ and $\frac{V}{S}$), both
damping rates can be controlled by shaping (i.e. constricting) the cross-
section of the nozzle.

  * •

Given that $\gamma_{\text{bl}}\sim S/V$, the damping rate can be controlled by
maximizing the surface area of the nozzle.

In the next subsections, we apply these ideas to estimate the expected scaling
of the damping rate upon a design change of the nozzle.

###  III.3 Dissipation Control by Nozzle Constriction

Figure 2: Geometric configurations of nozzles with constricted cylinders. The
nozzle is axisymmetric, and is obtained by rotating the cross-section shown in
the figures around the left edge, or the axis, indicated by a dashed line.
Here $R$ is the radius of the nozzle and $R_{i}$ is the radius of the
constricted part. (Left) Standard cylindrical nozzle, showing a deformed
liquid-gas interface; (right) constricted nozzle with $R_{i}$=$0.8R$ and a
flat interface. The upper section with diameter $R_{i}$ is termed the
dissipative section, while the lower section with diameter $R$ is referred to
as the shaping section.

As expected, the scaling of the damping rate identifies the diameter of the
nozzle as the parameter that most significantly defines the damping rate for a
cylindrical nozzle. This naturally motivates the design of a nozzle with a
smaller diameter, for damping control. However, the diameter of the nozzle is
limited by the targeted droplet size. To account for this constraint, we
consider nozzles with two distinct sections: a constricted section destined to
control the damping rate, separated from another section that controls the
droplet shape. An example of a nozzle with these two sections is found in Fig.
2. The part of the nozzle with a smaller diameter $R_{i}<R$ is referred to as
the dissipative section, whereas the section near the nozzle outlet with
radius $R$, whose role is to control the the droplet size, is referred to as
the shaping section (Fig. 2).

###  III.4 Dissipation Control by Increasing Surface Area to Volume

Figure 3: Geometric configurations of nozzles with concentric cylinders. (Top)
vertical slice of the nozzles with concentric cylinders. The computational
domain is designed for axisymmetric simulations. The inner cylinders are
located above the nozzle outlet by a distance $d$. The thickness of the inner
cylinders is denoted as $t$. All inner cylinders have the same thickeness.
Dashed lines indicate axes of rotation. (Bottom) Horizontal slices of nozzles
with concentric cylinders. Dark gray regions are the inner walls of the
cylinder. From left to right, n=2,3,4.

Based on the scaling relation, Eq. (14), the second method to reduce the
relaxation time is to maximize the surface area to volume ratio ($S/V$). The
main advantage of this method is that we can control the dissipation without
changing the cross-sectional area, and hence can circumvent large changes of
the mean speed of the fluid in the nozzle.

To show the proof-of-concept of this design rule, we considered an (idealized)
cylindrical nozzle with $n$ total concentric cylinders identically spaced in
the radial direction, as shown in Fig. 3. This idealized geometry is a great
example of design that demonstrates larger surface area compared to a typical
cylinder nozzle, while keeping the cross-sectional area essentially the same.
We consider the thickness of the inner cylinders in the radial direction, $t$,
much smaller than the radius of the outer cylinder, to keep the cross-
sectional area largely constant. In this way, we control the dissipation while
not changing the dynamics of the meniscus, which is mainly controlled by the
outlet diameter.

The total surface area of nozzle with $n\geq 1$ number of cylinders111Here
$n=1$ is the standard nozzle, i.e. with zero internal concentric cylinders. is
$S_{\text{n}}=2\pi RLn$, therefore, the resulting dissipation rate scales

$\gamma_{\text{bl,n}}\sim\sqrt{\nu\omega}\frac{n}{R}.$ (15)

We expect that the bulk dissipation will not change significantly if the axial
component of the velocity is the dominant flow feature associated with the
damping.

##  IV Simulation Setup

To demonstrate the effectiveness of our design rule, we employed the open-
source CFD software OpenFOAM® and conducted multiphase flow simulations of the
meniscus dynamics in the nozzle with several different designs. We studied the
relaxation time of the meniscus suspending at the outlet of the nozzle
responding to a pressure pulse at the nozzle inlet. The meniscus was initially
flat and pinned at the nozzle outlet.

We chose material properties and nozzle dimensions inspired by 3D printing
scenarios. We set the material properties to those of a liquid aluminum alloy
Assael  _et al._ (2006) $\rho=2435kg/m^{3}$, $\nu=4.16\times 10^{-7}m^{2}/s$,
and of argon gas $\rho_{g}=1.6kg/m^{3}$, $\nu_{g}=2.6\times 10^{-5}m^{2}/s$,
and surface tensionMolina  _et al._ (2007) $\sigma=0.85N/m$.

In this study, we considered $R=250\mu m$ as the nominal radius of the outlet
of the nozzle. For the constricted nozzle (Fig. 2), we set $R_{i}=200\mu m$
and $R_{i}=150\mu m$. The connecting area between the dissipative section and
the shaping section had smooth curves with a finite curvature to minimize the
formation of vortices at sharp corners, although flow separation can still be
observed. For concentric cylinders, we placed equally spaced inner cylinders
along the radial direction (Fig. 3). The thickness of each inner cylinder is
$t=4\mu m$. The inner cylinders were pushed inward by a distance $d=50\mu m$
so that the meniscus did not directly touch the inner cylinders during
oscillations. This last part effectively acts as the shaping section of this
nozzle.

The OpenFOAM thin wedge geometry was employed to represent planes in the swirl
direction for 2-D rotationally symmetric cases. The axi-symmetric wedge
geometries (shown in Fig. 2 and Fig. 3) were spatially discretized into
approximately 300K-400K hexaderal mesh elements (400 elements in the radial
direction, 900 elements in the vertical direction inside the liquid domain).
We used a non-uniform grid and applied extra grid refinement in the region the
liquid-gas interface sweeps during oscillations. Our multi-step grid
refinement study confirmed the convergence of the quantities of interest.

The simulations were performed until 4.5 millisecond after the initial
transident time period. We conducted high-performance parallel computing
simulations with 48 processors on the Amazon AWS clusters.

##  V Proof of the Design Rule

###  V.1 Standard Cylindrical Nozzle

Figure 4: Time snapshots of velocity profiles generated by the meniscus
oscillation inside the cylindrical nozzle after a pressure pulse is applied on
top. The cylindrical nozzle radius is $R=250\mu m$. From brown to yellow
($t_{1}-t_{8}$), the vertical velocity $u_{z}$ is plotted in the increasing
order in time in every 0.1 ms. The velocity profiles are measured at $z=200\mu
m$ above the nozzle outlet.

Figure 4 shows instantaneous time snapshots of the vertical velocity profiles
every 0.1 milliseconds at $z=200\mu m$ above the nozzle outlet. From the
velocity profiles near the nozzle wall, it is evident that our hypothesis that
the meniscus dynamics forms a thin oscillatory boundary layer near the wall is
valid. The steep velocity gradient near the wall is confined to the near wall
region up to $R-\delta<r<R$ with $\delta\approx 20\mu m$. The observed
boundary layer thickness is consistent with the estimate from the oscillatory
viscous flow theory, $\delta\sim\sqrt{\nu/\omega}\approx 18\mu m$.

While the velocity gradient inside the boundary layer is governed by the no-
slip condition on the wall, the bulk region of the fluid has almost flat
velocity profiles with negligible velocity gradients. The plug-like
oscillatory flow is driven by the motion of the meniscus. This clear
difference in velocity gradient in the bulk and the boundary layer indicates
that the boundary layer dissipation is the major source of damping.

Our main quantity of interest is the damping rate of the amplitude of the
meniscus motion, $\gamma$. We post-processed the time history of the interface
location at the center of the nozzle, $\eta_{0}(t)=\eta(t,r=0)$, in Fig. 5. At
each time $t$, the interface apex location was obtained via fitting
$[1+\tanh(z-\eta_{0}(t))]/2$ to the continuous volume fraction field along the
axis, to find the value of $\eta_{0}(t)$. On the time history of the
interface, we calculated $\gamma$ and $\omega$ by fitting a damped sinusoidal
function $f(\gamma,\omega,\phi,A)=A\text{exp}(-\gamma t)\text{cos}(\omega
t+\phi)$ using the curve fit tool provided in the Scipy.optimize. To disregard
initial transient effects, only data after 0.5 milliseconds were used in the
fit. As a result, the decay rate of the meniscus for the cylindrical nozzle
(shown in Fig. 2) with $R=250\mu m$ is obtained as

$\gamma=167/\text{sec},$ (16)

and $\omega$=1269/sec. The variance associated with the fitted parameters are
$\pm 5$/sec for $\gamma$, and $\pm 1$/sec for $\omega$ with 95$\%$ confidence
interval. All 95$\%$ confidence bounds for $\gamma$ and $\omega$ in this
paper, although not specified, have less than 2.6 percent of their reported
values.

Figure 5: Time history of the meniscus displacement at the center of the
channel, $\eta_{0}(t)=\eta(r=0,t)$, for the standard cylindrical nozzle with
$R=250\mu m$ and the nozzle with a constricted section with radius
$R_{i}=200\mu$ and $R_{i}=150\mu m$.

###  V.2 Nozzle with a Constriction

The time evolution of the meniscus amplitude for standard and constricted
nozzles is plotted in Fig. 5. The nozzle with a constricted section
successfully leads to faster dissipation of the meniscus oscillation. The
decay rates of the meniscus displacement with the constricted nozzle are
significantly larger than the standard nozzle as shown in Table 1.

Nozzle type | Radius at constriction | $\gamma(\text{sec}^{-1})$ | $\omega(\text{sec}^{-1})$
---|---|---|---
Standard | $250\mu m$ | 167 | 1269
Constricted | $200\mu m$ | 358 | 1318
Constricted | $150\mu m$ | 1176 | 1655
Table 1: Damping rate $\gamma$ and oscillation frequency $\omega$ of the
meniscus motion for the standard nozzle and the nozzles with $n$ concentric
cylinders.

This result demonstrates that only a fraction reduction of the nozzle radius
(here 20$\%$-40$\%$) was effective to increase the decay rate by approximately
factor of 2-7. The higher dissipation rate at more constricted nozzle can be
explained by the trends from both $\gamma_{\text{bl}}$ and
$\gamma_{\text{bulk}}$. The constricted cylinder has higher boundary
dissipation rate due to larger surface area to the volume ratio as well as
higher bulk dissipation rate due to smaller diameter against the standard
cylindrical nozzle. The relationship between $\gamma$ and $R_{i}$ is non-
linear due to combination of these two factors and the contribution from the
intermediate section between the shaping and the dissipative section. The
direct proportionality between the surface area to volume ratio and
$\gamma_{\text{bl}}$ is investigated via the concentric cylinders in the
following section.

###  V.3 Nozzle with Concentric Cylinders

Figure 6: Time history of the meniscus displacement at the center of the
channel, $\eta_{0}(t)=\eta(r=0,t)$, for the standard cylindrical nozzle with
$R=250\mu m$ and a nozzle of the same radius with concentric inner cylinders.
Here $n$ is the number of concentric cylinders in the nozzle.

The time evolution of the meniscus amplitude for nozzles with different number
of concentric cylinders inside is plotted in Fig. 6. The amplitude of the
meniscus motion decays faster in nozzles with larger surface area, therefore
the concentric cylinders are effective at reducing the relaxation time. We
summarized the damping rate and the oscillation frequency for $n=1,2,3,4$ in
Table 2. The results show that the damping rate increases linearly (with a
slope of $\approx 200$/sec) and confirms the proposed scaling of the damping
rate dominated by the boundary layer, $\gamma_{\text{bl}}\sim n$ in Eq. (15).
Notice that the oscillation frequency remains largely fixed, since the outlet
diameter and total mass inside the nozzle are unchanged.

Nozzle type | $\gamma(\text{sec}^{-1})$ | $\omega(\text{sec}^{-1})$
---|---|---
Standard nozzle, n=1 | 167 | 1269
Concentric cylinders, n=2 | 343 | 1256
Concentric cylinders, n=3 | 550 | 1237
Concentric cylinders, n=4 | 768 | 1205
Table 2: Damping rate $\gamma$ and oscillation frequency $\omega$ of the
meniscus motion for the standard nozzle and the nozzles with $n$ concentric
cylinders.

In Fig. 7 and Fig. 8 we plot the vertical velocity contours and profiles
inside the nozzle with concentric cylinders. They show the formation of thin
boundary layers near the surface of the inner cylinder(s) and the standard
cylinder wall. In Fig. 7, we observe that, as the number of inner cylinders is
increased, more layers of boundary layers contribute to increasing the energy
dissipation in the liquid. The boundary layer thickness at the inner cylinder
(Fig. 8) is similar to that in the standard nozzle wall, $\delta\approx 20\mu
m$, since the oscillation frequency stays nearly constant.

Figure 7: Instantaneous snapshots of the contours of the axial velocity
component in the liquid domain, $u_{z}$, at $t=1$ msec for the nozzle with
concentric inner cylinders. The countour plots show boundary layers near the
solid wall surfaces. Figure 8: Time snapshots of velocity profiles generated
by the meniscus oscillation inside a nozzle with a concentric cylinder ($n=1$)
after a pressure pulse is applied on top. The cylindrical nozzle radius is
$R=250\mu m$. From brown to yellow ($t_{1}-t_{8}$), the axial velocity $u_{z}$
is plotted in the increasing order in time every 0.1 ms. The velocity profiles
are measures at $z=200\mu m$ above the nozzle outlet.

##  VI Discussion

The nozzle geometry with equally spaced concentric cylinders is a successful
example of dissipation control without introducing a cross-sectional area
constriction. However, the concentric cylinders are a concept geometry and not
a favorable geometry for fabrication. Practical nozzle design should be able
to consider ease of manufacturing.

In Fig. 9, we have listed several examples of manufacturable nozzle designs
inspired by the present analysis. The constricted nozzle (Fig. 9 (a)) can be
easily fabricated using 3D printing or machine tools. A drawback of this
geometry, when the constriction ratio $R/R_{i}$ is large enough, is the flow
separation and formation of vortices at the intersection between dissipative
and shaping section. This separation can lead to a very inhomogeneous velocity
field in the shaping section that affects the formation of the droplet. For
liquid metals, these vortices can easily form for values of $R/R_{i}$ that are
not too far from $1$, albeit the droplet shape may not be affected at such low
ratios.

The other class of nozzle geometry has a cross-section extruded along the flow
direction (e.g., the star- or cross-shaped geometry shown in Fig. 9 (b) and
(c)). These geometries are expected to promote viscous dissipation because
both the cross-sectional area and the volume-to-surface area ratio are smaller
than in the standard cylindrical nozzle. While flow separation can also occur
in these geometries in the transition between the dissipative and shaping
section, they better "distribute" the faster fluid leaving the dissipative
section. This helps forming a more homogeneous velocity field in the ejected
droplet, and hence to more regular jetting. This type of nozzle geometries can
be manufactured by drilling, or potentially 3D printing.

Advancing the concept of the cross or the star cross-sections further, cross-
sections with $n$ straight channels of width $w$ separated by the same angle
stemming from the center of the cross-section can be considered (Fig. 9 (c)).
This makes it possible to control the surface area $S$ while while keeping the
volume $V$ roughly constant, by increasing $n$ and decreasing $w$. The width
$w$ should be chosen based on the estimation of the thickness of the boundary
layer. When $w\lesssim 2\delta$, the velocity profile inside the branch will
be close to parabolic. In this case, the scaling of viscous dissipation will
be modified, since $\delta\sim w$, so that
$\gamma_{\text{bl}}\sim(\nu/w)(S/V)\sim\nu/w^{2}$, assuming $w\sim\delta\ll
R$. The value of $n$ should be chosen as a compromise between
manufacturability, volume of the dissipative section, and droplet shape.

Figure 9: Example of nozzle designs with a dissipative section potentially
simple to manufacture. The inset figures are cross-sectional views of the
nozzle outlet. (a) A nozzle with a dissipative section with a constriction;
(b) A nozzle with a dissipative section formed by an extruded star-shaped
cross-section (sharp edges may be smoothed); (c) A nozzle with a dissipative
section formed by an extruded cross-shaped cross-section. Here $w$ is the
width of a branch of the cross.

##  VII Conclusion and Summary

Drop-on-demand liquid metal jetting is an emerging technology that enables the
creation of quality products and provides a safe printing process. To enable
reliable high-speed printing, it is required to develop a nozzle that promotes
fast decay of the meniscus oscillation after a droplet ejection. Design
guidelines for high-speed jetting need a physical understanding of the energy
dissipation mechanisms that controls the relaxation time of the fluid in the
nozzle. Dimensional analysis reveals that the oscillatory flow inside the
nozzle, driven by the meniscus motion, is in the inertial flow regime, and
that it forms a thin boundary layer near the solid wall. In this paper, we
propose physics-based design rules to speed up the decay of the meniscus
fluctuations, based on scaling relations derived from the Navier-Stokes
equation and the viscous boundary layer theory. These scaling relations
informs that nozzles with smaller diameters and nozzles that maximize the
surface area to volume ratio have faster damping rates. These dimensional
considerations need to be balanced by the need to obtain droplets of a given
shape and size, so we focus on the design of nozzles with an outlet of a given
cross sectional area. To demonstrate these ideas, we performed multiphase flow
simulations for several nozzle designs. We used a class of idealized
geometries that include multiple equally spaced concentric cylinders
perpendicular to the nozzle meniscus movement. The simulation results showed
that the damping rate increases as the surface area of the nozzle does, and
displays the linear growth with the number of concentric cylinders predicted
by the scaling relations. Probing the vertical velocity profiles inside the
nozzle confirmed the formation of a thin boundary layer near the wall.

Finally, we discussed a class of manufacturable nozzle designs inspired by the
discussion herein. To the best of our knowledge, our study is the first
investigation of the damping rate in a liquid metal jetting printer nozzle
using oscillatory viscous boundary layer theory. Also, our study is the first
demonstration of dissipation control by introducing design changes in the
nozzle shape. We note that an accurate estimation of the meniscus damping in
numerical simulations is challenging due to the multiple sources of
dissipation, including the boundary layer, bulk, contact line, and numerical
dissipation. A systematic and rigorous analysis of the sources of viscous
dissipation, and a demanding numerical convergence study, were crucial
prerequisites for the results herein.

This study also suggests a number of questions for future investigation.
First, guidelines for the design of the shaping section are needed, in which
its shape should promote the homogenization of the velocity field prior to
droplet formation. This would minimize the likelihood of appearance of
multiple droplets as a result of a single pressure pulse. Second, experimental
validation of the ideas herein is pending. Finally, accurately computing the
damping rate of non-axisymmetric nozzles (fully three-dimensional geometries)
is very computationally demanding, and hence was left for future studies.

##  VIII Numerical Methods and Validation

We describe the numerical tools we used to discretize the governing equations
and to perform the simulations of multiphase flows shown here. All of them
were computed with the interfoam solver in the OpenFOAM version 2106.

###  VIII.1 Interface Capturing Scheme

In interfoam, the algebraic Volume-of-Fluid (VoF) interface tracking method is
applied to track the dynamics of the gas-liquid interface. In a discretized
domain with VOF, the fraction of volume of liquid, $\alpha$, can take a value
between 0 and 1 near the interface. As a result, in numerical simulations the
gas-liquid interface has a finite thickness, instead of the sharp interface
described as a Dirac delta function in Eq. (2). Using the Continuum Surface
Force model (CSF)Brackbill, Kothe, and Zemach (1992), the surface tension
force in the momentum Eq. (2) is calculated as

$\sigma\kappa\vec{n}\hat{\delta}(\vec{x}-\vec{x}_{s})\approx\sigma\kappa\nabla\alpha.$
(17)

Given the volume fraction of fluid, $\alpha(\vec{x},t)$, computed at each time
$t$ and position $\vec{x}$, the viscosity and the density fields are updated
as

$\mu_{f}(\vec{x},t)=\alpha(\vec{x},t)\mu+(1-\alpha(\vec{x},t))\mu_{g},$ (18)

and

$\rho_{f}(\vec{x},t)=\alpha(\vec{x},t)\rho+(1-\alpha(\vec{x},t))\rho_{g},$
(19)

and used to solve the N-S equation. We refer to Desphande et al. Deshpande,
Anumolu, and Trujillo (2012) for a description of detailed algorithms and
finite volume formulations to solve the N-S equation and the volume fraction
field.

###  VIII.2 Temporal and Spatial Discretization

OpenFOAM employs a cell-centered finite volume discretization. The spatial
domain is discretized with hexahedral meshes generated by the built-in meshing
tool, blockMesh and snappyHexMesh. We chose the linear scheme for cell center-
face interpolation, and the Gauss linear scheme for divergence, gradient, and
Laplacian operators.

The time discretization was performed via the explicit Euler time scheme. A
variable time-step method was chosen to ensure numerical stability, and
controlled by the time step restrictions based on a Courant-Freidich-
Lewis(CFL) number derived from the momentum equation and the surface
tensionDeshpande, Anumolu, and Trujillo (2012). We enforced CFL numbers below
0.2 for both momentum and interface driven time-step restrictions.

The solution of the momentum equation was obtained by constructing a predicted
velocity field and correcting it using the Pressure Implicit with Splitting of
Operators (PISO) algorithm Issa (1986). For the Poisson equation in this
pressure correction, we used the preconditioned conjugate gradient scheme with
the diagonal-based incomplete Cholesky preconditioner.

We turned off the interface compression method in interfoam by setting
cAlpha=0 in fvSolution. Instead, we created extra refined meshes near the
region of the interface to avoid numerical diffusion of the interface
thickness.

###  VIII.3 Validation

The interfoam solver has been validated in previous studies Deshpande,
Anumolu, and Trujillo (2012), and it showed nice agreement in tracking
interfaces for highly inertial flows and surface-tension-driven capillary
flows. Standard tests showed that the CSF formulation in interfoam led to a
proper discrete balance between pressure and surface tension Deshpande,
Anumolu, and Trujillo (2012).

Although details are not presented here, we independently performed validation
studies to assess the quantitative predictive ability of the interfoam solver
on two-phase problems with oscillating liquid-gas interfaces. We evaluated
extensive simulation results from interfoam against reference solutions in
canonical multiphase flow test cases in the high capillarity limit, obtained
by analytical investigations Lamb (1932); Case and Parkinson (1956); Miles and
Henderson (1998); Henderson and Miles (1994) with matched experimental
investigations Howell  _et al._ (1999); Becker, Hiller, and Kowaleswski
(1991), or numerical solutions obtained by other numerical methods on highly
resolved grids (level set Herrmann (2008) or phase fieldMirjalili, Ivey, and
Mani (2019)). These canonical validation test cases involve damped
oscillations of initially perturbed two-dimensional circular (cylinder) and
spherical droplets Mirjalili, Ivey, and Mani (2019); Herrmann (2008), of
capillary waves in a periodic domain Herrmann (2008), and of the brimful
cylindrical tank with water Howell  _et al._ (1999); Case and Parkinson
(1956); Henderson and Miles (1994). These simulations were performed with
material properties of either the water-air or the liquid metal-argon systems.
We confirmed that the oscillation frequency and the damping rate on perturbed
capillary flows calculated by the interfoam solver showed nice agreement
against the reference solutions within $\sim 10\%$ errors. We note that the
numerical dissipation of interfoam, especially on some coarse grids,
contributed to the overestimation of the damping rate, and we acknowledged
that the grid convergence study was a critical component of the validation. We
tested three-step grid refinements and demonstrated the grid convergence on
both damping rate and oscillation frequency.

## References

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# Spectrality and monoids

Amartya Goswami  [1] Department of Mathematics and Applied Mathematics,
University of Johannesburg, South Africa;
[2] National Institute for Theoretical and Computational Sciences (NITheCS),
South Africa<EMAIL_ADDRESS>
###### Abstract.

We prove that the set of proper ideals of a monoid endowed with coarse lower
topology is a spectral space.

###### Key words and phrases:

monoid; spectral space

###### 1991 Mathematics Subject Classification:

20M12, 20M14, 54F65

##  1\. Introduction

The aim of this section is to highlight on the study conducted on spectral
spaces in relation to diverse structures and a variety of topologies. The
concept of a spectral space was first introduced in [17]. In this framework,
spectral spaces are defined as the spectra of prime ideals of rings, equipped
with the Zariski topology. Since its inception, the study of spectral spaces
has expanded across various contexts. Several instances illustrate this
diversification. Apart from the Zariski topology, multiple other topologies
such as Lawson, Scott, lower, upper, ultra-filter, patch, inverse, and others
have been explored for prime ideals (for comprehensive insights, we refer our
reader to [11]).

Among distinct classes of ideals, the characterization of the set of all
ideals under the lower topology as a spectral space was established in [21].
Furthermore, [12] demonstrated that spaces of all ideals, proper ideals, and
radical ideals of a ring, endowed with a hull-kernel-type topology, are also
spectral. It is noteworthy that the Zariski-Riemann space of valuation domains
within a given field stands as an early example of a specialized spectral
space (see [8, 9, 14]).

In [13], investigation into semistar operations (of finite type) led to the
identification of their space, equipped with a topology reminiscent of the
Zariski topology, as a spectral space. Additionally, [23] established that the
collection of continuous valuations on a topological monoid, with the topology
determined by any finitely generated ideal, forms a spectral space.

In the context of module, [4] introduced conditions under which prime spectra
of modules qualify as spectral spaces. Another pertinent avenue involving
spectral spaces is the realm of Stone duality. The prime filters of a
distributive lattice, equipped with the Stone topology, constitute a spectral
space (see [11, Chapter 3]). Similarly, the proper filters of a Boolean
algebra or, more generally, an ortholattice, endowed with a Stone-like
topology, exhibit spectral space properties (for references, see [5, 20]).

Although there are intensive studies of various classes of ideals for monoids
(or semigroups with identity) have been done (see [1, 2, 3, 6, 7, 10, 15, 16,
18, 19, 22, 24, 25]), to the best of the author’s knowledge, the concept of
spectrality within the context of monoids has not previously been studied.
This endeavour seeks to furnish an exemplification of a spectral space linked
to monoids. The proof of our main result is self-contained and remains
constructible topology-independent of any specific topology. In establishing
the notion of spectrality, we have employed a technique (see Lemma 3.2) that
circumvents the need to verify existence condition of certain type of basis of
a spectral space. It is worth mentioning that the extension of the algebraic
structure to semigroups does not offer a viable generalization when exploring
spectrality. Notably, for the fulfilment of quasi-compactness–a condition
integral to spectrality–the presence of a multiplicative identity assumes
necessary (see §3, Proof (2)).

To ensure the self-contained nature of this paper, all essential definitions
and proofs have been incorporated.

##  2\. Preliminaries

A (multiplicative) _commutative monoid_ is a system $(M,\cdot,1)$ such that
$(M,\cdot)$ is a commutative (multiplicative) semigroup with $1$ as the
multiplicative identity. For simplicity, we shall write $ab$ for $a\cdot b$,
for all $a,$ $b\in M$. An _ideal of_ $M$ is a nonempty subset $I$ of $M$ such
that $im\in I$, whenever $i\in I$ and $m\in M$. If
$\\{I_{\lambda}\\}_{\lambda\in\Lambda}$ is a family of ideals of $M$, then so
are their intersection $\bigcap_{\lambda\in\Lambda}I_{\lambda}$ and their
_sum_ $\sum_{\lambda\in\Lambda}I_{\lambda}$(that is, the ideal generated by
$\bigcup_{\lambda\in\Lambda}I_{\lambda}$). The notation $\mathcal{I}_{M}$ will
denote the set of all ideals of $M$. It is easy to see that $\mathcal{I}_{M}$
is a complete lattice. An ideal $I$ of $M$ is called _proper_ if $I\neq M$,
and by $\mathcal{I}^{+}_{M}$ we shall denote the set of all proper ideals of
$M$. We say a proper ideal $J$ of a monoid $M$ is _maximal_ if there is no
proper ideal of $M$ that properly contains $J$.

Let $L$ be a complete lattice and $x\in L$. A _cover of_ $x$ is a family
$\\{y_{\lambda}\\}_{\lambda\in\Lambda}$ of elements of $L$ such that
$x\leqslant\bigvee_{\lambda\in\Lambda}y_{\lambda}$. An element $x$ of $L$ is
called _compact_ if every cover of $x$ has a finite subcover. A lattice is
_algebraic_ if it is complete and every element is a least upper bound of
compact elements. It is easy to see that the complete lattice $\mathcal{I}(M)$
is algebraic, for every ideal is a sum of finitely generated (and hence
compact) ideals of $M$. This property of $\mathcal{I}(M)$ is going to play a
crucial role in our proof of the main result.

Let us now recall a few relevant terminologies from topology. A space is
called _quasi-compact_ if every open cover of it has a finite subcover, or
equivalently, the space satisfies the finite intersection property. In this
definition of quasi-compactness, we do not assume the space is $T_{2}.$ A
closed subset $S$ of a space $X$ is called _irreducible_ if $S$ is not the
union of two properly smaller closed subsets of $X$. A space $X$ is called
_sober_ if every non-empty irreducible closed subset $\mathcal{K}$ of $X$ is
of the form: $\mathcal{K}=\mathcal{C}_{\\{x\\}}$, the closure of an unique
singleton set $\\{x\\}$. By [17], a _spectral space_ is a topological space
that is quasi-compact, sober, admitting a basis of quasi-compact open
subspaces that is closed under finite intersections.

Let $M$ be a monoid. The coarse lower topology on $\mathcal{I}^{+}_{M}$ is the
topology for which the sets of the type:

$\mathcal{V}_{J}=\left\\{I\in\mathcal{I}^{+}_{M}\mid J\subseteq
I\right\\},\qquad(J\in\mathcal{I}_{M});$

form a subbasis of closed sets, and by $\mathcal{I}^{+}_{M}$, we shall also
denote the topological space.

##  3\. The Main Theorem

We are now prepared to present and proof our main result.

######  Theorem 3.1.

Let $M$ be a monoid. Then the set $\mathcal{I}^{+}_{M}$ of proper ideals of
$M$ endowed with coarse lower topology is a spectral space.

To prove Theorem 3.1, we need to show the following:

  1. (1)

The space $\mathcal{I}^{+}_{M}$ is quasi-compact.

  2. (2)

The space $\mathcal{I}^{+}_{M}$ is sober.

  3. (3)

The coarse lower topology on $\mathcal{I}^{+}_{M}$ admits a basis of quasi-
compact open subspaces that is closed under finite intersections.

Our strategy here is slightly different. Although (1) and (2) are
comparatively easy to verify, however, condition (3) is more cumbersome.
Thanks to the next lemma, the checking of (3) can be avoided with an expense
of showing two additional requirements as we shall see after the proof this
lemma.

######  Lemma 3.2.

A quasi-compact, sober, open subspace of a spectral space is spectral.

###### Proof.

— Suppose that $S$ is a quasi-compact, sober, open subspace of a spectral
space $X$. Since $S$ is quasi-compact and sober, it is sufficient to prove
that the set $\mathcal{O}_{S}$ of compact open subsets of $S$ forms a basis of
a topology that is closed under finite intersections. It is obvious that a
subset $T$ of $S$ is open in $S$ if and only if $T$ is open in $X$, and hence
a subset $T$ of $S$ belongs to $\mathcal{O}_{S}$ if and only if $T$ belongs to
$\mathcal{O}_{X}.$ Now using these facts, we argue as follows.

Let $U$ be an open subset of $S$. Since $U$ is also open in $X$, we have
$U=\cup\,\mathcal{U},$ for some subset $\mathcal{U}$ of $\mathcal{O}_{X}.$ But
each element of $\mathcal{U}$ being a subset of $U$ is a subset of $S$, and it
belongs to $\mathcal{O}_{S}.$ Therefore, every open subset of $S$ can be
presented as a union of compact open subsets of $S$. Now it remains to prove
that $\mathcal{O}_{S}$ is closed under finite intersections, but this
immediately follows from the fact that $\mathcal{O}_{X}$ is closed under
finite intersections. ∎

Proof of Theorem 3.1 — It is now clear that to prove the theorem, it is
sufficient to check the conditions in Lemma 3.2 by taking $X=\mathcal{I}_{M}$
and $S=\mathcal{I}^{+}_{M}$. Therefore, our objective is now to verify the
following:

  1. (1)

$\mathcal{I}_{M}$ is a spectral space;

  2. (2)

$\mathcal{I}^{+}_{M}$ is quasi-compact;

  3. (3)

$\mathcal{I}^{+}_{M}$ is sober.

  4. (4)

$\mathcal{I}^{+}_{M}$ is an open subspace of the space $\mathcal{I}_{M}$.

(1) Since $\mathcal{I}_{M}$ is an algebraic lattice (see §2), the desired
spectrality follows from [21, Theorem 4.2].

(2) Let $\\{K_{\lambda}\\}_{\lambda\in\Lambda}$ be a family of subbasic closed
sets of a normal structure space $\mathcal{I}^{+}_{M}$ such that
$\bigcap_{\lambda\in\Lambda}K_{\lambda}=\emptyset.$ Let
$\\{I_{\lambda}\\}_{\lambda\in\Lambda}$ be a family of ideals of
$\mathcal{I}_{M}$ such that $\forall\lambda\in\Lambda,$
$K_{\lambda}=\mathcal{V}_{I_{\lambda}}.$ Since
$\bigcap_{\lambda\in\Lambda}\mathcal{V}_{I_{\lambda}}=\mathcal{V}_{\bigcup_{\lambda\in\Lambda}I_{\lambda}},$
we get $\mathcal{V}_{\bigcup_{\lambda\in\Lambda}I_{\lambda}}=\emptyset.$ This
implies that the ideal $\sum_{\lambda\in\Lambda}I_{\lambda}$ must be equal to
$M$. Then, in particular, we obtain $1=n_{\lambda_{1}}\cdots n_{\lambda_{n}},$
where $n_{\lambda_{i}}\in I_{\lambda_{i}}$, for $i=1,\ldots,n$. This implies
$M=\sum_{i\,=1}^{n}I_{\lambda_{i}}.$ Hence,
$\bigcap_{i\,=1}^{n}K_{\lambda_{i}}=\emptyset,$ and $\mathcal{I}^{+}_{M}$ is
quasi-compact by Alexander Subbasis Theorem.

(3) To show the existence of generic points of irreducible closed subsets of
$\mathcal{I}^{+}_{M}$, it is sufficient to show that
$\mathcal{V}_{I}=\mathcal{C}_{I}$, whenever $I\in\mathcal{V}_{I}$. Since
$\mathcal{C}_{I}$ is the smallest closed set containing $I$, and since
$\mathcal{V}_{I}$ is a closed set containing $I$, obviously then
$\mathcal{C}_{I}\subseteq\mathcal{V}_{I}$. For the reverse inclusion, if
$\mathcal{C}_{I}=\mathcal{I}^{+}_{M}$, then

$\mathcal{I}^{+}_{M}=\mathcal{C}_{I}\subseteq\mathcal{V}_{I}\subseteq\mathcal{I}^{+}_{M}.$

This proves that $\mathcal{V}_{I}=\mathcal{C}_{I}$. Suppose that
$\mathcal{C}_{I}\neq\mathcal{I}^{+}_{M}$. Since $\mathcal{C}_{I}$ is a closed
set, there exists an index set, $\Lambda$, such that, for each
$\lambda\in\Lambda$, there is a positive integer $n_{\lambda}$ and ideals
$I_{\lambda 1},\dots,I_{\lambda n_{\lambda}}$ of $M$ such that

$\mathcal{C}_{I}={\bigcap_{\lambda\in\Lambda}}\left({\bigcup_{i\,=1}^{n_{\lambda}}}\mathcal{V}_{I_{\lambda
i}}\right).$

Since $\mathcal{C}_{I}\neq\mathcal{I}^{+}_{M},$ we assume that
${\bigcup_{i\,=1}^{n_{\lambda}}}\mathcal{V}_{I_{\lambda i}}$ is non-empty for
each $\lambda$. Therefore,
$I\in{\bigcup_{i\,=1}^{n_{\lambda}}}\mathcal{V}_{I_{\lambda i}}$ for each
$\lambda$, and hence

$\mathcal{V}_{I}\subseteq{\bigcup_{i=1}^{n_{\lambda}}}\mathcal{V}_{I_{\lambda
i}},$

that is, $\mathcal{V}_{I}\subseteq\mathcal{C}_{I}$ as desired. To obtain the
uniqueness of the generic point, it is sufficient to prove that
$\mathcal{I}^{+}_{M}$ is a $T_{0}$-space. Let $I$ and $I^{\prime}$ be two
distinct elements of $\mathcal{I}^{+}_{M}$. Then, without loss of generality,
we may assume that $I\nsubseteq I^{\prime}$. Therefore $\mathcal{V}_{I}$ is a
closed set containing $I$ and missing $I^{\prime}$.

(4) By considering coarse lower topology on $\mathcal{I}_{M}$, we immediately
obtain

$M=\mathcal{V}(M)=\mathcal{C}_{M},$

and hence $\mathcal{I}_{M}\setminus\mathcal{I}^{+}_{M}$ is closed. This
implies $\mathcal{I}^{+}_{M}$ is open as required. ∎

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