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# High-Resolution Scanning Tunneling Microscope and its Adaptation for Local
Thermopower Measurements in 2D Materials

Jose D. Bermúdez-Perez  School of Engineering, Science and Technology,
Universidad del Rosario, Bogotá 111711, Colombia  Facultad de Ingeniería y
Ciencias Básicas, Universidad Central, Bogotá 110311, Colombia  Edwin Herrera-
Vasco  Facultad de Ingeniería y Ciencias Básicas, Universidad Central, Bogotá
110311, Colombia  Laboratorio de Bajas Temperaturas, Departamento de Física de
la Materia Condensada, Instituto de Ciencia de Materiales Nicolás Cabrera,
Condensed Matter Physics Center (IFIMAC), Facultad de Ciencias Universidad
Autónoma de Madrid, 28049 Madrid, Spain  Departamento de Física Aplicada.
Facultad de Ciencias. Universidad Autónoma de Madrid, E-28049 Madrid, Spain
Javier Casas-Salgado  Facultad de Ingeniería y Ciencias Básicas, Universidad
Central, Bogotá 110311, Colombia  Hector A. Castelblanco  Facultad de
Ingeniería y Ciencias Básicas, Universidad Central, Bogotá 110311, Colombia
Karen Vega-Bustos  Department of Physics, Universidad de Los Andes, Bogotá
111711, Colombia  Gabriel Cardenas-Chirivi  Department of Physics, Universidad
de Los Andes, Bogotá 111711, Colombia  Oscar L. Herrera-Sandoval  Facultad de
Ingeniería y Ciencias Básicas, Universidad Central, Bogotá 110311, Colombia
Hermann Suderow  Laboratorio de Bajas Temperaturas, Departamento de Física de
la Materia Condensada, Instituto de Ciencia de Materiales Nicolás Cabrera,
Condensed Matter Physics Center (IFIMAC), Facultad de Ciencias Universidad
Autónoma de Madrid, 28049 Madrid, Spain  Paula Giraldo-Gallo  Department of
Physics, Universidad de Los Andes, Bogotá 111711, Colombia  Jose Augusto
Galvis  [  School of Engineering, Science and Technology, Universidad del
Rosario, Bogotá 111711, Colombia

###### Abstract

We present the design, fabrication and discuss the performance of a new
combined high-resolution Scanning Tunneling and Thermopower Microscope
(STM/SThEM). We also describe the development of the electronic control, the
user interface, the vacuum system, and arrangements to reduce acoustical noise
and vibrations. We demonstrate the microscope’s performance with atomic-
resolution topographic images of highly oriented pyrolytic graphite (HOPG) and
local thermopower measurements in the semimetal $Bi_{2}Te_{3}$. Our system
offers a tool to investigate the relationship between electronic structure and
thermoelectric properties at the nanoscale.

Scanning Tunneling Microscope, Thermopower Microscope, thermoelectric
properties, nanoscale

††preprint: AIP/123-QED

##  I introduction

The invention of the Scanning Tunneling Microscope (STM) by G. Binnig and H.
Rohrer brought about a flurry of research in instrumentationG.Binnig and
H.Rohrer (1983); G.Binnig and D.Smith (1986); G. Binnig (1982); Anirban
(2022). Numerous new microscopes were built on the basis of the STM, which
enabled imaging of many new propertiesBenstetter, Biberger, and Liu (2009).
The STM still has a prominent role, as it probes the tunneling current between
tip and sample, one of the most delicate but also most informative local
measurements. For example, the tunneling current can be related to the local
electronic density of states, which is useful to study metals, semimetals,
semiconductors, and superconductorsTersoff and Hamann (1985, 1983); Crespo
_et al._ (2012); Galvis  _et al._ (2012, 2015, 2017); Willa  _et al._ (2020);
Boi  _et al._ (2021); Li  _et al._ (2021); Herrera  _et al._ (2023).

In a STM a tip is brought to tunneling distance with a flat sample. When a
voltage difference is applied, a tunneling current flows. The tunneling
current depends exponentially on the tip-sample distance $d$ as $I_{T}\sim
e^{-2\kappa d}$, where $\kappa$ is the sample workfunctionTersoff and Hamann
(1985, 1983); Bardeen (1961). A change of only one Angstrom in the tip-to-
sample distance may produce a change of one order of magnitude in the
tunneling current. The tip, typically made of a metal such as gold or
tungsten, is mounted on a piezoelectric material that controls its position
and can be used to scan the tip over a flat sample’s surface.G.Binnig and
D.Smith (1986). There are two main modes for operation of the STM: in the
constant current mode, the tunneling current is maintained constant by a
feedback circuit, and an image is produced with the z-position of the
piezoelement. In the constant height mode, the tip is scanned at a constant
height over the surface and the image is made by plotting the current vs
position.

A STM can be modified to include new measurement modes for the
characterization of physical phenomena not included in conventional
techniques, leading to a better understanding of material propertiesLyeo  _et
al._ (2004); Hwang  _et al._ (2022). For example, one can adapt to measure the
thermovoltage, which is related to the conversion of heat into electrical
energy. This effect can be quantified through the Seebeck coefficient (S) or
thermoelectric power, related to the material figure of merit (ZT) as
$(S^{2}/\sigma\kappa)T$, where $\sigma$ and $\kappa$ are the electrical and
thermal conductivity respectively, and T the temperature. A Scanning
Thermoelectric Microscope (SThEM) can be adapted from STM to measure S at the
local scale, providing a unique technique for the characterization of
topographic, electronic, and thermodynamic propertiesLyeo  _et al._ (2004);
Kim  _et al._ (2022). SThEM has potential in the field of nanoelectronics,
where it has been used to investigate the thermoelectric properties of
individual nanostructuresVanis  _et al._ (2016).

This work presents a high-resolution STM system that operates at room
temperature and high vacuum, which has been adapted as a SThEM to incorporate
thermovoltage measurements. In Section II, we discuss the design of the STM,
including its key components and features. We then describe the modifications
made to the system to enable thermovoltage measurements in Section III. In
Section IV, we detail the development of the electronic control, acquisition,
and data processing software. In Section V, we discuss the strategies and
instruments implemented to minimize mechanical and electrical noise. Finally,
in Section VI, we demonstrate the system’s performance by presenting
topographic and electronic properties of Highly Ordered Pyrolytic Graphite
(HOPG), as well as thermovoltage measurements in $Bi_{2}Te_{3}$ single
crystal.

Figure 1: (a) Home-built STM system set-up. The system is inserted in a vacuum
chamber. Here, the STM and a sample-holder position control motor are firmly
fixed. The assembled STM is shown in (b). As a coarse approach system, we use
a slider prism (A) and a height control piece (B) to adjust the distance
between the slider prism to the sample. The STM bottom part (C) includes the
sample holder chamber and the screws (D) to adjust the initial sample
position. A detailed view of the sample holder assembly is given in (c). It
includes a metal piece (e.g., Chrome, or aluminum) (E) to mount the sample
using silver conductive epoxy, a silicon diode (DT-670-SD-1.4H) temperature
sensor (F), and a heater (G). In (d), we show the STM’s detailed 3D CAD
rendering. The piezoelectric tube (H) is attached to the prism by a small Ti
piece (I), a copper washer (J) sputtered with gold, provides a ground shield
for the current signal, electrical insulation layers (K), the tip holder (L)
and the tip (M). The coarse approach system includes alumina plates covered
with graphite (N), piezo stacks (O), a spring (P) to attach the slider prism
firmly to the STM body, and the sample slider (Q). Finally, the springs are to
fix the slider to the sample holder (R).

##  II Description of the STM

In Fig 1(a), the vacuum chamber cover is shown, which holds the platform with
the STM body (Supplementary Fig S1(a)). The design of the STM (Fig. 1(b)) is
based on previous reports Pan, Hudson, and Davis (1999); Suderow, Guillamon,
and Vieira (2011); Galvis  _et al._ (2015); Fernández-Lomana  _et al._ (2021);
Martín-Vega  _et al._ (2021). The STM is made of titanium, which has one of
the smallest linear thermal expansions ($8.5\times 10^{-6}/^{\circ}C$) among
other metals like stainless steel, chrome, or aluminum, resulting in low
structural distortions due to temperature changesHidnert (1943). This
property, combined with the zero magnetic response of titanium, allows for the
operation of the STM under extreme conditions of very low temperatures and
high magnetic fieldsSuderow, Guillamon, and Vieira (2011); Galvis  _et al._
(2015); Fernández-Lomana  _et al._ (2021); Martín-Vega  _et al._ (2021). These
two characteristics are essential for the study of superconducting materials
using STMCrespo  _et al._ (2012); Galvis  _et al._ (2012, 2017); Willa  _et
al._ (2020).

The STM tip is attached to the end of a piezo-tube, which allows for
approaching the sample, which is mounted on the sample holder (shown as (E) in
Fig.1(c)). In Fig. 1(d), the tip (M) is soldered to a small screw with indium,
which is then screwed into a small nut (L). To screen the induction from the
high voltage signals of the piezo tube electrodes, a grounded copper washer
(J) with 10 nm of gold sputtered onto it is used. We also glued two non-
conductive epoxies (Stycast 2850FT) washers (K) to provide insulation layers
between the tip, the grounded washer, and the piezo tube electrodes. The piezo
tube (H) is mounted on the slider prism (A). The slider prism is tightly
clamped to the titanium body through a spring (P) and four shear
piezoelectrics (O) on each side. The shear piezoelectrics consist of a stack
of 4 shear piezoplates glued with epoxy. The stack is 5 mm high and 10 mm
wide. The piezo stacks and the prism are joined together using alumina plates
(N). These alumina plates are carefully covered with a thin layer of graphite
as a lubricant to promote the sliding of the slider prismFernández-Lomana  _et
al._ (2021). This approach system allows the tip movement towards the sample
in a maximum range of 10 mm when voltage pulses of a sawtooth signal (up to
$60V_{pp}$) are applied in the piezoelectric shears, producing a global
displacement of the piezoelectric tube through a stick-slip motionPan, Hudson,
and Davis (1999); Suderow, Guillamon, and Vieira (2011); Galvis  _et al._
(2015); Battisti  _et al._ (2018); Fernández-Lomana  _et al._ (2021).

Figure 2:  (a) Schematic representation of the STM/SThEM system showing how to
change between tunnel current to thermovoltage measurements. A local
temperature gradient is generated in the sample when a tip-sample nanocontact
is made (b) Infrared pyrometer schematically configuration, where the focus of
the sensor is adjusted and calibrated. Figure 3: Block diagram of the
STM/SThEM electronic hardware set-up, showing the relationship with the
AtomicView (AV) user interface (See the section IV.3). In the dark blue box,
the four AtomicView capabilities are shown as "STM topography",
"Spectroscopy", "Thermopower measurements" and "Data visualization". In STM
operation, The "STM topography" menu allows the STM control parameters (such
as Field of view, number of pixels, scanning direction, and time per pixel) to
resolve the sample surface in topography mode, and the "Spectroscopy" menu
allows the control parameters (as resolution, timer per point and Bias voltage
range), which allow the user to obtain sample LDOS information. In SThEM
operation, the "Thermopower Measurements" menu is used to measure the Seebeck
coefficient. The last AV menu, "Data Visualization" offer visualization
feedback as a time function of all the STM/SThEM control parameters. The
electronic hardware for STM and SThEM operation are shown in the light blue
and red boxes, respectively. The STM/SThEM is shown in the middle of the blank
box. Finally, the instrumentation in green boxes shows commercial instruments,
and yellow boxes indicate electronic modules, used for the home-built
electronic control.

The STM tips are fabricated from Au following the method described in Refs
Agrait, Yeyati, and van Ruitenbeek (2003); Rodrigo  _et al._ (2004). Firstly,
a cross-section of a 0.008" diameter gold wire is cut using surgical scissors
in a glove box (see Section 7). Secondly, the tip is mounted in the piezo
tube. Then, under High Vacuum (HV) conditions, we use the STM to crash the tip
in a controlled manner into a clean Au substrate repeatedly until it causes
the plastic deformation between the tip and substrate. Repetitive nanocontact
indentation between the tip and substrate causes plastic deformation of both,
resulting in a narrow connective neck. After several repetitions, the final
step involves retracting the tip once quantum transport is established,
resulting in the breaking of the neck and the formation of an atomically sharp
tipRubio, Agraït, and Vieira (1996).

The piezo tube is attached to a small titanium piece (marked as "H" in Fig.
1(d)) using non-conductive epoxy (Stycast 2850FT), which is then securely
fixed inside the slider prism. We use a piezo tube provided by EBLEBL , which
has five electrodes: four for X-Y scanning and one for Z-axis elongation. To
generate displacement in both the plane and the axis, we polarize the piezo
tube by applying a maximum voltage of $\pm$ 125 V. This generates a maximum
deformation of 1 $\mu$m in the X-Y plane and an elongation of 350 nm along the
Z-axis. The sample is located facing towards the coarse approach system on a
metallic sample-holder (E in Fig. 1(c)). The sample holder is screwed at a
slider (Q in Fig. 1(d)), which can be moved back and forth through a sample
holder position control motor (Fig. 1(a)) and a system of springs (R in Fig.
1(c)) with a minimum step size of 100$\mu$m (the sample-holder position
control system is described in detailed in Supplementary Fig S1(b)).

For the electrical connections in the STM, we use vacuum feed-through
connectors and low-noise cables. The wires are soldered to the body of the STM
via copper tracks, taking care to position it to separate high-voltage signals
from tunnel-current signals, avoiding sources of interference and electrical
noise in the more sensitive signal.

##  III Description of SThEM

To operate the STM as a SThEM, we have built a temperature control system
capable of operating at temperatures ranging from room temperature up to
$\sim$ 500 K. The heater consists of a Bakelite piece that has a cylindrical
gap with a radius of 4 mm and a height of 3 mm, with 0.4 m of Nichrome wire
into the gap in a spiral, which is covered with a thermally conductive epoxy
(832TC-A and 832TC-B) (G in Fig. 1(c)). We use a silicon diode thermometer
glued onto a copper piece with thermal conductive varnish to measure the
sample’s temperature (F in Fig. 1(c)). The sample holder is mechanically
attached to a copper piece and the heater using screws (E in Fig. 1(c)). To
improve thermal conduction, the three pieces are connected through mechanical
contact on their flat surfaces, which are coated with thermally conductive
epoxy.

To measure the thermopower (S) we use the switch system shown in fig 2(a),
which allows to change between STM mode to SThEM mode. Once the SThEM mode is
activated, we establish a controlled temperature difference between the tip
and the sample by elevating the sample temperature under HV conditions. When
the room-temperature tip makes a nanocontact with the heated sample, a
temperature gradient appears in the region surrounding the sample-tip
contactLyeo  _et al._ (2004); Lee  _et al._ (2012). If the sample is a
thermoelectric material, this localized $\Delta T$ generates a thermovoltage
($\Delta V$), which is directly related to the S of the sample. After $\Delta
V$ reaches its maximum value, the tip is retracted away from the heated
sample. S can be found by $-\Delta V/\Delta T$ Lyeo  _et al._ (2004).

We measure the tip temperature using an HV micro-epsilon®infrared pyrometer
(see section IV.2)Pyr . The infrared pyrometer is located close to the STM
using aluminum support, which enables us to adjust the vertical and horizontal
position of the sensor’s focus (Fig. 1(a)). The maximum distance between the
sensor and the tip is determined by the tip size and the optical resolution of
the infrared thermometer. To prevent measurement errors and temperature
contributions from other objects when the SThEM is operating, the pyrometer
position is adjusted and calibrated before closing the chamber.

In our case, we set the pyrometer position at 5.4 mm from the slider prism
(Fig. 2(b)) to fill out the field of view of the radiation focus point
completely. The emissivity ($\epsilon$) of the tip surface is another relevant
parameter for a correct temperature measurement. We calibrate this factor by
using a calibrated silicon diode thermometer in three steps: first, we glue a
silicon diode sensor (DT-670-SD-1.4H) onto the tip holder. Second, we measure
the tip holder temperature when the tip is away from and when it is in contact
with the heated sample. Third, we use these silicon diode temperature
measurements to modify $\epsilon$ until the pyrometer measuring value
corresponds to the diode temperature.

##  IV Description of the STM/SThEM control Hardware and software

Figure 4: Screenshots of AV windows. (a) The main interface header shows the
different control windows in separate menus. (b) Closed-loop analog and
digital current controller window. The analog control parameters are handled
externally except for the tunnel current setpoint. The digital part is
operated on AV and in the background. (c) STM mode windows. (d) Real-time plot
window of the voltages at the piezo tube electrodes. (e) SThEM mode.  Figure
5: Screenshots of AV in scanning tunneling mode. (a) X-Y positioning window
for topography scanning. The red circle and the white grid mark the tip’s
position and the piezo tube’s maximum scanning area, respectively. We show
three different topographies on a HOPG graphite taken at room temperature
($I_{T}=100pA$, $V_{bias}=5mV$) where are evident consecutive atomic layers on
the HOPG (i, ii and iii). (b) Control window for topography mode. (c) Control
window for spectroscopy mode. (d) Trace and (e) retrace images on atomically
resolved HOPG surface, taken with the parameters of (a). (f) Height vs.
distance profile and IV Curves in HOPG graphite (g) while the scan in (d) and
(e) is measured.

###  IV.1 Description of the STM electronic Hardware

High-quality STM electronic hardware designs are essential for reducing the
noise level in the tunnel currentKamlapure  _et al._ (2013). To achieve this,
we designed and built a low-noise electronic system in-house for our STM
setupBermudez-Perez, Salgado, and Galvis (2023a). The electronic hardware
block diagram of the system is shown in Fig. 3. We use a low-noise current
trans-impedance amplifier by Femto®("I/V converter") to measure the tunnel
current, which is linearized and filtered through a "signal processing" module
(light blue box in Fig. 3). The current and the analog signals are acquired
using an 18-bit Analog to Digital (ADC) chip (ADC8698) with eight separate
input channels. The system uses two 16-bit Digital to Analog (DAC) DAC8734
chips, each with four output lines, which provide control voltage signals from
$-$5 V to $+$5 V. Six output lines are amplified to a factor of 25 to drive
the STM piezoelectric tube, using an ultra-low noise amplifier (TD 250)
provided by PiezoDrive®. For each piezo axis, we use two DACs which are added
together, allowing a resolution in the position well below Å. Furthermore,
these output lines have RC filters as close as possible to the piezo tube
electrodes to decrease the noise coming from the system power supply. Another
output line is amplified to a factor of 12 by a high-voltage amplifier circuit
(PA343DF) and is optocoupled into the board to avoid high-frequency
interferences. This output line is used to slip the prism (stick-slip motion)
applying a sawtooth signal up to +60 V at the piezo stacks (O in Fig. 1(d))
(see section II). The final output is used on the sample voltage to establish
the tunneling current between the tip and the sample. This output is mounted
in a voltage gain configuration to apply Bias voltage signals from $-$10 V to
$+$10 V, which directly relates to the energy scale. This voltage range allows
us to measure semiconductors with a high band gap such as Boron Nitride (6.08
eV) or low band gaps such as $Ag_{2}Te$ (0.31 eV)Cassabois, Valvin, and Gil
(2016); Zhu and Pandey (2019).

To operate the STM in constant current mode we can switch between analog or
digital feedback. The first takes the control signal after the signal
processing module (light blue box in Fig. 3). The second operates continuously
in the user interface background ("AtomicView") (see section IV.3). Both
systems operate on reading the current value and applying a voltage on the Z
electrode of the piezoelectric tube. To control the electronics hardware, we
have written a C++ code in an embedded micro-controller (STM32F407VGT6) by ST
Electronics®. We use a standard USB 2.0 to communicate the hardware with the
control user interface. The USB is isolated to the main board to avoid noise
from the main computer, using a Digital Isolator (ADUM3160BRWZ) chip.
Furthermore, we reduce the whole system noise coming from the 60 Hz line,
grounding all the instruments to a separate master ground exclusive for the
STM set-up (The whole STM/SthEM system hardware is shown in detail in
Supplementary Fig. S2).

###  IV.2 Description of the SThEM electronic Hardware

The equipment that we incorporate into the STM hardware to operate it as SThEM
is shown in the pink box in Fig. 3. We use an infrared pyrometer thermometer
(CT model SF22-C3V) with a temperature range from 223 K to 1248 K and 1 K
accuracy to measure the tip temperature. To measure the temperature of the tip
in the 1 mm range of the tip holder (L in Fig. 1(d)), we add to the pyrometer
a close-focus optics lens (TM-APLCF-CT). The specifications of this lens
determine the optical path of the sensor, achieving a 0.5 mm focus spot size
at 10 mm distance. The sensor transmits data by TTL/USB standard output, used
to acquire the tip temperature.

The sample temperature control operates through a commercial Controller (Lake
Shore®CRYOTRONICS 336). One output provides up to 100 W and connects to a 16
$\Omega$ Nichrome heater (G in Fig. 1(c)). To measure the sample temperature
we use a silicon diode (DT-670-SD-1.4H) anchored to the cooper piece of the
sample holder (F in Fig. 1(c)) which is connected to one sensor input of the
temperature controller. To obtain the thermovoltage when the tip makes a
nanoscale contact with the heated sample, we use a commercial low-noise
preamplifier (SR560) by Stanford Research. The tip and the sample are
connected through a BNC wire at the DC-coupled low-noise differential input
signals of 1 to 50000 gain. We acquire the output through a filtered line and
using a NI USB-6341 National Instruments data acquisition system
(Supplementary Fig. S2).

###  IV.3 Description of the user control interface

The STM/SThEM electronic hardware described in the previous section is handy
and can be easily connected to different USB ports on the same computer,
allowing remote operation through a user application. We have developed a
Windows desktop application (AV) in VisualStudioBermudez-Perez, Salgado, and
Galvis (2023b). The AV user software includes all the routines needed to
operate the STM/SThEM system, using C as the programming language on a Windows
Presentation Foundation (WPF) platform. We communicate with all the electronic
hardware via USB standard using AV.

Fig. 3 shows a block diagram of the user software modules (dark blue box) and
how it is linked to the electronic hardware (light blue, pink, and white box).
The software interacts with the hardware through the main page ("AVMain"),
which includes four additional menus. The "General" menu (Fig. 4(a)) is
accessed from the main controls and provides options for manual piezo tube
movements, operations related to the bias voltage, closed-loop feedback
settings, and the operation of the coarse approach system (Fig. 4(b)). The
"Topography STM" menu (shown in Fig. 4(c)) enables the operation of the STM in
topography mode. For feedback of this mode, AV shows the voltage applied on
each piezo axis as curves as a function of time (Fig. 4(d)). In the following
sections, we will provide a detailed explanation of this menu along with an
example of STM operation. The "Thermopower SThEM" menu (Fig. 4(e)) is used to
operate the STM as SThEM. The "SThEM/Graphs" display shows the tip and sample
temperature measurements as a function of time, communicated through GPIB and
TTL/USB communication (see the section IV.1). The "LakeShore/Heater
Properties" menu is used to control the sample temperature setpoint, limit the
current of the silicon sensor, and the PID operations. The tip-sample
nanocontact can be made through piezo tube movement in the
"LocalNanocontac/Tools" option or using the coarse approach system in the
"General" menu. The last menu "Properties" is used to control the tunnel’s
current acquisition and processing settings.

In Fig. 4(b), we show the "Feedback Controller" window in AV, where we can set
the control parameters settings, switch between digital and analog controller,
and slip the slider prism (A in Fig. 1(b)). Another AV routine captures the
tunnel current and the voltage in the piezoelectric electrodes through an ADC
(see section IV.1) chip. These values are stored and plotted on an "Electrode
voltage" window on the page "General" (Fig. 4(d)) as a time function. AV also
uses several DAC outputs through USB standard to the electronic hardware (see
section IV.1). One output is to control the bias voltage between the sample
and tip. AV also employs other outputs to apply voltage to the piezo tube
electrodes. During the scanning routine, it is possible to manually or
automatically stop the closed-loop controller at each pixel and perform
spectroscopy measurements.

Figure 6: Screenshots of AV in thermopower mode. (a) Thermovoltage graph
window. We show time profiles of the measured voltage in $Bi_{2}Te_{3}$ when
at room temperature tip make a nanoscale contact with the sample at 343 K. The
tip and sample temperature profiles are shown in (b) as a function of the
time. In Both graphs we can identify three regions; (i) the tip is away from
the sample, (ii) the tip makes a nanoscale contact, and (iii) the sample
heater is turned off with the tip in nanoscale contact with the sample. (c)
Sample heater setting window. Figure 7: (a) STM/SThEM set-up. (b) Inert-
controlled environmental Stainless steel glove box. (c) Deterministic layer-
transfer exfoliation stage.

In Fig. 5, we present screenshots to illustrate a typical STM topography
operation mode. The measurements were taken at room temperature using a gold
tip and a HOPG sample. In Fig. 5(a), we show the X-Y positioning window used
for topography operation mode. We can select the size and position of the
scanning window, covering areas from 1 nm2 to $2\mu$m2. The images (i), (ii),
and (iii) show topographies in different places of the same scanning window,
where some HOPG steps are easily identifiable as atomic layer steps between
consecutive graphite planes (more details in Fig. 5). The small red circle in
(iii) indicates the actual tip position in real time. The image settings, such
as resolution and time per pixel, can be configured in the "Topography (STM)"
menu (Fig. 5(b)). Furthermore, the "Spectroscopy (STS)" menu controls the
parameters for the STM spectroscopy mode (Fig. 5(c)). Fig. 5(d) and (e)
present a topography in a field of view (FOV) of some tens of nm, where the
HOPG atomic lattice is visible (red and blue lines in Fig. 5(f)) and its
metallic behavior in the I-V curves (red and blue lines in Fig. 5(g)). The
data and image analysis were done using WSXM as processing softwareHorcas  _et
al._ (2007).

In Fig. 6, we show AV examples screenshots for the SThEM configuration mode.
In these measurements, we use a commercial platinum-iridium (805.ASY.STM) tip
and a $Bi_{2}Te_{3}$ sample. We measure the temperature and the tip-sample
voltage as a function of time in "Thermovoltage Graph" (Fig. 6(b)) and
"Temperature Graph" (Fig. 6(c)) windows. We start with the tip away from the
heated sample ((i) in Fig. 6(b)(c)). Then the tip, at room temperature, makes
nanoscale contact with the heated sample, and a temperature gradient is
generated, which produces a thermoelectric voltage ((ii) in Fig. 6(b)). During
the tip-sample nanocontact, the tip is slightly heated up by thermal
conduction, causing a decrease in the thermovoltage after a few seconds. While
the tip remains in nanoscale contact with the sample in (iii), the heater is
turned off. As the temperature gradient begins to disappear, the
thermoelectric voltage also decreases, demonstrating that the voltage is
caused by the temperature difference between the tip and the sample. Note that
the thermovoltage decreases at a faster rate than the temperature gradient.
This is because the thermovoltage is measured at the nanocontact, where the
tip and sample reach thermal equilibrium faster than in other regions.

Figure 8: (a) Schematic representation of a typical STM image of bulk HOPG
graphite using an Au tip. The distances between neighboring atoms are
indicated, with 1.42 Å between adjacent atoms and 2.84 Å between opposite
vertex atoms. The large solid circles in the image represent the size of an Au
atom (2.9 Å in diameter), which coincides with the size of one hexagon in the
graphite structure. Due to its comparable size, When the tip is positioned
directly over the center of a hexagon, all six carbon atoms within that
hexagon are simultaneously detected by the tip, resulting in a bright spot in
the image at that location. This corresponds to the hollow-site lattice points
in the HOPG structure. (b) STM image of HOPG graphite (taken at room
temperature, $V_{bias}=200mV$, $I=1nA$), The bright spots in the image
correspond to the locations where the tip is positioned over a hollow site
(center of the hexagon), resulting in a strong tunneling current contribution
from the six carbon atoms within that hexagon. The small image in the bottom
corner shows the Fourier transform of the image, revealing the hexagonal
lattice of the hollow sites. (c) Inverted Contrast Image Revealing the True
Position of Carbon Atoms in a Honeycomb Lattice. The inset provides a
magnified view of the region enclosed by the black square, highlighting a
single atomic position. The profiles shown at the bottom of the image
illustrate the height difference between the A and B atoms in the HOPG
structure.

##  V Electrical and Mechanical Noise Isolation

In conventional STM operation, the resolution, scanning speed, and quality of
topography/spectroscopy are often limited due to the interplay between the
tunneling process and mechanical and electrical noiseGe, Ovadia, and Hoffman
(2019). In Fig. 7, we show a picture of the STM/SThEM set-up. To minimize
mechanical noise (vibration and acoustic) from the building, we have designed
a home-built 1.5m x 1m soundproof box, made of brass and covered with
soundproofing foam. The soundproof box (SB) is located on an optical table
(VERE) with integrated passive vibration isolation. The STM/SThEM is mounted
on a platform inside a home-built stainless steel high-vacuum (HV) chamber
(Supplementary Fig. S1(a)). The vacuum chamber is placed inside the SB and
suspended from the laboratory roof using vibration isolation springs and
ropes. To reduce mechanical vibrations transmitted through the wires, we have
carefully routed all of them through the soundproofing foam. Additionally, we
have placed the most noise-sensitive instruments, including the STM electronic
control, the I-V converter, and the voltage amplifier (DLPCA-200 by Femto®) to
measure the tunnel current inside the soundproof box. The remaining
instruments are positioned on an adjacent rack, as illustrated in Fig. 7(a).

The home-built vacuum chamber is capable of achieving a base pressure of $\sim
1\times 10^{-5}$ mbar through a turbo molecular pump (TS85W1002) and measured
with a Pirani gauge (RS-EV-D14642000). Additionally, the system incorporates a
stainless steel glovebox (2800-2-B by Cleatech®) (Fig. 7(b)), which provides a
controlled atmosphere and clean environment that prevents humidity and
oxidation during sample preparation when backfilled with nitrogen or argon.
The STM/SThEM vacuum chamber is located inside the main chamber using a
transfer chamber. We include a deterministic layer-transfer exfoliation stage
(Fig. 7(c)) through a 300x optical microscope and an XYZ manual stage, as
described in Refs (Castellanos-Gomez  _et al._ , 2014; Zhao  _et al._ , 2020;
Valencia-Ibáñez  _et al._ , 2022). Inside the glovebox, we can exfoliate the
sample or transfer layers to the STM sample holder in a clean atmosphere. The
iris ports on the glovebox allow us to manipulate the STM/SThEM and access the
vacuum chamber with bare hands while maintaining an inert atmosphere inside
the glovebox.

Figure 9:  SThEM measurements: (a) Thermovoltage measurement on Au sample to
demonstrate the behavior when the tip makes a nanoscale contact with a heated
non-thermoelectric sample. In (i), the platinum-iridium tip is away from the
heated sample. The tip made nanoscale contact with the heated sample at room
temperature in (ii), and then retracted in (iii). (b) Induced thermoelectric
voltage in $Bi_{2}Te_{3}$ due to different temperature gradients from 14.2K
(black) to 72.87K (turquoise). (c) Seebeck coefficient of bulk $Bi_{2}Te_{3}$
sample versus temperature obtained from the curves in (b). (d) Thermoelectric
voltage measured on four different zones of the sample surface as shown in the
inset STM topographic image (size = 67 nm, Vbias = 350mV, I = 100pA).

##  VI results

Studies on the atomic resolution of graphite using STM can reveal important
issues related to the microscopy technique itself. It is well-known that
atomic resolution images on graphite only show the position of three out of
six atoms in the same hexagon of the lattice (A and B atoms in Fig. 8(a))
Atamny, Spillecke, and Schlo (1999). Several theoretical and experimental
studies attempt to explain this electronic asymmetry between neighboring
carbon atomsAbraham, Williams, and Wickramasinghe (1988); Soler  _et al._
(1986); Mate  _et al._ (1989); Whangbo  _et al._ (1994); Soler  _et al._
(1986); Tománek  _et al._ (1987); Binnig  _et al._ (1986); Atamny, Spillecke,
and Schlo (1999); Ouseph, Poothackanal, and Mathew (1995); Mizes, Park, and
Harrison (1987); Tersoff (1986). Achieving atomic resolution in graphite (and
in most materials) depends not only on the electronic properties of the sample
but also on the coupling between the tip and the sample.

Fig. 8(b) shows atomic-resolution images of HOPG graphite in positive
contrast. In this case, the bright spots do not correspond to individual
atoms, but rather to the hollow centers of each hexagon of the lattice,
creating a hexagonal lattice with a distance of 4.26Å between spots (see Fig.
8(a)). This effect occurs because the radius of the atomically sharp tip
(which is approximately 1.46Å for Au atoms) which is comparable to the radius
of a hexagonal ring (1.42Å). When the tip is at the center of a hexagon, it
receives tunneling current contributions from the six neighboring atoms,
producing a bright spot at the center. However, by inverting the image
contrast, it is possible to distinguish individual atoms, and the position of
atoms A and B, as shown in Fig. 8(c). Atoms A and B exhibit an asymmetry in
their charge density. This is due to the fact that A atoms have neighbors
directly below them in the second layer, whereas atoms B are located at the
center of a hexagon in the second layer, with no atom immediately below them
Atamny, Spillecke, and Schlo (1999); Tománek and Louie (1988). This effect,
resulting from electronic interactions between atoms of different layers, can
be observed in the z-profile of figures 8 (bottom panels of b and c), where A
atoms appear higher than B atoms. The observation of this small effect proves
the high topographic resolution of our STM.

To evaluate the SThEM mode, we conducted thermovoltage measurements using a
Platinum-iridium tip, in both, a non-thermoelectric sample (Au) and a
thermoelectric sample ($Bi_{2}Te_{3}$). In Fig. 9(a), we present the
measurements on the Au sample. The sample was heated to 348K while the tip
remained at room temperature. In (i), the tip was not in contact with the
sample. We observed electrical noise around zero voltage induced by the sample
heater. When the tip made contact with the sample in (ii), a noticeable
reduction in electrical noise was observed. Although the noise level
decreased, the thermovoltage response remained at zero. In (iii), the tip was
moved away from the sample, similar to the configuration in (i). As expected,
the noise level returned to its initial state. Consequently, in the absence of
a thermoelectric response, the thermovoltage remains zero.

In Fig. 9(b), a similar process was performed using a $Bi_{2}Te_{3}$ sample, a
well-known thermoelectric compound with a thermopower ($S$) ranging from
$-61\mu V/K$ to $-170\mu V/K$ near room temperature. The specific thermopower
value depends on factors such as crystal structure, impurities, defects, or
carrier concentration levels Radek  _et al._ (2016); Wu  _et al._ (2013);
Goldsmid (2014); Situmorang and Goldsmid (1986). The $Bi_{2}Te_{3}$ sample was
heated to different temperatures, ranging from 308K to 348K. Seven nanoscale
contacts were made with the tip at room temperature at the same sample point
for each temperature. The resulting thermovoltage curves are depicted in Fig.
9(b) (from black to turquoise curves). To calculate the thermopower ($S$)
using the formula $S=-\Delta(V)/\Delta(T)$, the maximum value of the
thermovoltage at each nanoscale contact was utilized for each temperature (as
shown in Fig. 9(c)). For temperature differences below 70K, the thermovoltage
response steadily increases until it reaches a maximum value, which then
remains constant for several minutes. However, for temperature differences
above 70K, a reduction in the thermovoltage is observed after a few seconds,
which can be attributed to heat transfer to the tip. This behavior is evident
in the turquoise curve in Fig. 9(b), and it demonstrates that the
thermovoltage response is a sensitive indicator of heat transfer processes,
particularly for high-temperature differences ($\Delta T$).

Fig. 9(c) shows the calculated Seebeck coefficient as a function of the sample
temperature for the measurements presented in Fig. 9(b). As expected for
$Bi_{2}Te_{3}$, the absolute value of the Seebeck coefficient increases with
the sample temperature. To further validate the homogeneity and isotropy of S
at the sample surface, we conducted four S measurements at different locations
on a flat surface area, as shown in the inset STM image in Fig. 9(d). The
tests were performed consecutively at a sample temperature of 348K and $\Delta
T=54.4K$, and all four measurements yielded a value close to $\bar{S}=98\pm
0.03\mu V/K$. Notably, we did not observe any evidence of a reduction in
thermovoltage due to heat transfer between measurements.

##  VII Conclusion AND OUTLOOK

In summary, we have presented the design, construction, and performance of a
custom-built high-resolution scanning tunneling microscope operating at room
temperature and pressure up to $1\times 10^{-5}$ mbar. This setup enables
atomic resolution imaging and achieves $\mu eV$ energy resolution. We have
also detailed the modifications made to the STM to enable its operation as a
Scanning Thermoelectric Microscopy. This allows for the simultaneous
characterization of structural, electronic, and thermodynamic properties
within a single technique. Furthermore, we discussed the integration of
temperature control capabilities into the STM, enabling precise thermal
manipulation of the sample. A user-friendly control interface was developed,
allowing the entire system to be operated via USB standard communication
through a computer. To ensure accurate measurements, we implemented various
mechanisms to minimize noise stemming from mechanical vibrations and acoustic
interference caused by the STM and SThEM operations. Lastly, we showcased the
performance of the STM/SThEM system by demonstrating its ability to achieve
atomic-resolution topographic imaging on monoatomic layers of Highly Oriented
Pyrolytic Graphite (HOPG), as well as thermoelectric measurements on
$Bi_{2}Te_{3}$ samples. Overall, the described STM/SThEM system provides a
powerful platform for investigating nanoscale phenomena, offering a
comprehensive range of capabilities for studying various materials and their
properties.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

###### Acknowledgements.

The authors thank the support of the Ministerio de Ciencia, Tecnología e
Innovación de Colombia (Grants No.122585271058), and the Cluster NBIC of
Universidad Central (Colombia). E.H., and J.A.G. acknowledge the support of
Departamento Administrativo de Ciencia, Tecnología e Innovación, COLCIENCIAS
(Colombia) Convocatoria 784-2017. J.A.G. acknowledge the support of School of
Engineering, Science, and Technology at Universidad del Rosario. P. G-G., K.
V-B., and G. C-C., acknowledge the support of the School of Sciences and the
Vice Presidency of Research Creation at Universidad de los Andes, and the
financial support of FPIT-Fundación para la promoción de la investigación y la
tecnología, of Banco de la Republica de Colombia, project number 4.687.

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# Analytical Solution for Turbulent Flow in Channel

(Date: July 10, 2023)

###### Abstract

In this work the exact and approximate analytical solution of the GHE for
turbulent flow in channel are presented. It was discovered first by numerical
simulations, Fedoseyev and Alexeev (2010), and now the explicit formula are
obtained. The solution is a superposition of the laminar (parabolic) and
turbulent (superexponential) solutions. The analytical solution compares well
with the experimental data by Van Doorne (2007) for axial velocity and data by
Nikuradse (1933) for axial velocity, for flows in pipes.

It is proposed to explain the nature of turbulence as oscillations between the
laminar (parabolic) and turbulent (superexponential) solutions. Good
comparison of the analytical formula, a difference of the parabolic and
superexponential solutions, for turbulent velocity fluctuations with the
experiment by Van Doorne (2007) confirmed this suggestion. The Navier-Stokes
equations do not have the superexponential solution.

The obtained analytical solution provides a complete structure of the
turbulent boundary layer that compares well with the experiments by Wei and
Willmarth (1989). It also presents an explicit verifiable proof that Alexeev’s
generalized hydrodynamic theory (GHE) is in close agreement with experiments
for turbulent flows.

Alex Fedoseyev

Ultra Quantum Inc., Huntsville, Alabama, USA

Email<EMAIL_ADDRESS>
##  1 Introduction

Generalized Hydrodynamic Equations (GHE) have been proposed by Boris Alexeev
(1994)[1]. They were used for the simulations of incompressible viscous flows
for a wide range of problems and flow parameters, including high Reynolds
number turbulent flows with thin boundary layers in 3D driven cavity flow at
Re = 3200 and 10,000, 2D backward facing step flow at Re = 132,000, flow in
channels for Reynolds number up to $Re=10^{6}$, magnetohydrodyamic flows [7,
8, 9, 10, 12], resulting in good agreement with experiments [20], [19] and
other works.

GHE model has been applied to compressible hypersonic flows that exhibit both
continuum and non-continuum flow regimes. The shock wave (bow shock) can be
detached from the vehicle at high altitude, and near boundary slip-flow is
typical for such regimes. Results for hypersonic GHE model have been reported
in [13, 14, 15] are in close agreement with the experiments by [3, 17, 18] and
other works even for highly rarefied flows.

In this work an approximate analytical solution of GHE for turbulent flow in
channel is presented and compared with the experimental data.

Generalized Hydrodynamic Equations (GHE) are based on a new set of
conservation equations obtained from a generalized version of the Boltzmann
equation (GBE) by Alexeev that includes more details of the molecular
collision processes [1], [2]. The model combines continuum-to-free molecular
flow physics in one consistent formulation by accounting for the kinetic
effects (intermediate Knudsen number, fluctuations and turbulence) in the
continuum approximation. A brief outline of the basic ideas of GBE and
derivation of governing Generalized Hydrodynamic Equations is presented in
Section 1.1.

###  1.1 Generalized Boltzmann Transport Equation (GBE)

Physical derivation of the standard Navier-Stokes equations (NS) can be
obtained from the kinetic theory of gases, which is based on the solution to
the Boltzmann transport equation for space-time evolution of particle velocity
distribution function, $f$, written in the form

$\displaystyle\frac{Df}{Dt}=J,$ (1)

where $D/Dt$ represents material derivative in space, velocity space and time
and $J$ is the collision integral. The standard Boltzmann transport equation
takes into account the changes in distribution function $f$ on hydrodynamic
and mean time between collision time scales of infinitesimal particles.
Accounting for a third time scale associated with finite dimensions of
interacting particles gives rise to an additional term in the Boltzmann
transport equation resulting in a generalization of the form as

$\displaystyle\frac{Df}{Dt}-\frac{D}{Dt}(\tau^{*}\frac{Df}{Dt})=J,$ (2)

where $\tau^{*}$ is the mean time between particle collisions. The new term is
thermodynamically consistent and is proportional to the Knudsen number, $Kn$,
and therefore in the hydrodynamic limit, to viscosity. More details on the GBE
are provided in the Alexeev’s book [2].

##  2 Generalized Hydrodynamic Equations

Hydrodynamic equations can be obtained from Eq. (2) by multiplying the latter
by the standard collision invariants (mass, momentum, energy) and integrating
the result in the velocity space. These equation are for incompressible
viscous flow, presented originally in [12], are the following:

$\it\frac{\partial\bf V}{\partial\rm t}+({\bf V}\nabla){\bf
V}-Re^{-1}\nabla^{2}\bf V+\nabla\rm p-{\bf
F}=\tau\left\\{2\frac{\partial}{\partial t}(\nabla\rm p)+\nabla^{2}(\rm p\bf
V)+\nabla(\nabla\cdot(\rm p\bf V))\right\\}$ (3)

while continuity equation is

$\it\nabla\cdot\bf V=\tau\left\\{2\frac{\partial}{\partial t}(\nabla\cdot{\bf
V})+\nabla\cdot({\bf V}\nabla){\bf V}+\nabla^{2}\rm p-\nabla\cdot{\bf
F}\right\\}$ (4)

where ${\bf V}$ and $p$ are nondimensional velocity and pressure,
${Re=V_{0}L/\nu}$ \- the Reynolds number, $V_{0}$ \- velocity scale, $L$ \-
hydrodynamic length scale, $\nu$ \- kinematic viscosity, ${\bf F}$ is a body
force and a nondimensional ${\tau=\tau^{*}L^{-1}V_{0}}$. Note that the right-
hand side of (3) is the divergence of the fluctuation part of flow velocity,
expressed explicitly through original primitive variables according to [1].

We made the following assumptions deriving Eq. (3, 4):

  * •

$\tau$ is assumed to be constant,

  * •

Neglected the nonlinear terms of the third order in the fluctuations, and
terms of the order $\tau$/Re and smaller,

  * •

Assumed slow flow variation, so neglect second derivatives in time.

Additional boundary conditions on walls are for fluctuations to be zero. The
boundary condition for pressure on walls is

$({\nabla\rm p-{\bf F})\cdot{\bf n}=0},$ (5)

where ${\bf n}$ is a wall normal.

Note, that the dimension of a product $\tau^{*}\nu$ is a square of length. We
introduce fluctuation length scale $l$, $l^{2}=\tau^{*}\nu$ and rewrite
nondimensional ${\tau}$ as ${\tau=l^{2}L^{-2}Re=K\cdot Re}$, where
${K=l^{2}/L^{2}}$. The value of $\tau^{*}$ is a material property and not
known in advance, but we provide speculation on a choice of $\tau^{*}$ value
in Section 5 and in the discussion of results, Section 7.

The obtained GHE model is not a turbulence model, and no additional equations
are introduced. Kinetic effects (small flow scales) have been successfully
captured with the GHE, and the obtained small scale of turbulence compared
well with observations in experiments by Koseff and Street (1984) [20], and 2D
and 3D Naver-Stokes solutions and k-$\varepsilon$ turbulent model solutions
have been outperformed by GHE results, Fedoseyev and Alexeev (2012)[12].

In this paper to obtain the analytical solution we further simplify the GHE
equations: (a) temporal derivatives are neglected in the fluctuations, (b) the
nonlinear terms are neglected in the fluctuations.

###  2.1 Governing Equations for 2D Incompressible Flow

The case of 2D incompressible fluid flow is considered. The equations are
taken from [12] where they are presented explicitly, and further
simplification is done by dropping all the terms (with coefficient $\tau$) in
momentum equations, and keeping only Laplacian term in the continuity
equation.

The resulting continuity equation of GHE model is the following:

$\displaystyle u_{x}+v_{y}$ $\displaystyle=$ $\displaystyle\tau\nabla^{2}p$
(6)

while the momentum equations are :

$\displaystyle u_{t}+u\,u_{x}+v\,u_{y}+p_{x}$ $\displaystyle=$ $\displaystyle
Re^{-1}\nabla^{2}u$ (7) $\displaystyle v_{t}+u\,v_{x}+v\,v_{y}+p_{y}$
$\displaystyle=$ $\displaystyle Re^{-1}\nabla^{2}v$ (8)

where $Re=LV_{0}/\nu$ is Reynolds number, $L$ is the length scale, $V_{0}$ is
velocity scale, $\nu$ is kinematic viscosity, $\tau=\tau^{*}V_{0}/L$ is the
nondimensional time scale from GHE ($\tau^{*}$ is the dimensional time).

One may point out that the continuity Eq. (6) is not correct, and the mass is
not conserved for incompressible fluid (may have sources and sinks). Let us
note, that

  * •

Eq.(6), as well as Eq.(7, 8) are obtained by a the standard procedure, by
multiplying Eq. (2) by standard collision invariants (mass, momentum, energy)
and integrating the result in the velocity space. Then the simplification was
done, and only the highest derivative was left in Eq. (6).

  * •

Recall that the particles now are not the material points, but finitely-sized
particles, that may be partially in and partially out of any control volume,
and that is the origin of GBE, and the pressure Laplacian term in Eq.(6).

  * •

When solving Eq.(6, 7, 8) numerically [11], the residuals of solution of
Eq.(6) with Laplacian term and without that term have been verified. The
residuals have been small and of the same order of magnitude in both cases, so
no sources and sinks have been observed for wide range of flows. oth cases, so
no sources and sinks have been observed for wide range of flows. Still the
results using the Navier-Stokes equation with div(V)=0 were far from the
experimental data while the GHE results with Eq.(6) fit the experimental data
well [11].

##  3 Laminar Flow Solution

The 2D channel flow problem: a stationary flow in $x$ direction in a 2D
(horizontal) channel of width $L$ with a center line velocity $V_{0}$ is
considered. So the time derivatives are dropped in Eq.(7, 8). We consider the
flow where everything is the same in any $x$ cross section, so all the partial
derivatives in x are zero, except for the applied pressure gradient $p_{x}$.
The boundary conditions are:

$u=0$, $v=0$ and the normal derivative of the pressure $p_{n}=0$ at the walls
$y=0$, and $y=L/2$.

A flow with $p_{x}=$ const has a parabolic velocity $u$ profile:

$\displaystyle p_{x}$ $\displaystyle=$ $\displaystyle Re^{-1}u_{yy},$
$\displaystyle u_{y}$ $\displaystyle=$ $\displaystyle Re\ p_{x}y+C_{1}$
$\displaystyle u$ $\displaystyle=$ $\displaystyle\frac{1}{2}Re\
p_{x}y^{2}+C_{1}y+C_{2}.$

Taking into account the boundary conditions, the laminar flow solution is

$\displaystyle U_{L}=u(y)=4V_{o}y(L-y)/L^{2},$ (9)

where $V_{0}=-\frac{1}{8}Re\ p_{x}L^{2}$, and $v=0$ everywhere.

This solution of Eq.(6),(7),(8) is also a solution of the Navier-Stokes
equations, as $\nabla^{2}p=0$.

The equations (6),(7),(8) also have a second solution, that was initially
discovered numerically for a flow in 2D channel by Fedoseyev and Alexeev
(2012) [11], Figure 1(a): both turbulent and laminar numerical solutions are
shown for $Re=5000$.

In this paper the second solution is obtained analytically, it is called as a
turbulent solution, and it is a super exponential function:

$\displaystyle U_{T}=V_{0}\left(1-e^{1-e^{y/\delta}}\right).$ (10)

Figure 1(b) shows example of both $U_{L}$ (Eq.(9), parabolic, green line) and
$U_{T}$ (Eq.(10), super exponential, blue line) solutions for laminar and
turbulent flows respectively.

(a) (b)

Figure 1: (a) Simulation of 2D channel flow where the turbulent solution has
been discovered, Fedoseyev and Alexeev (2012) [11]: horizontal velocity U(x),
for $Re=5\cdot 10^{3}$ both laminar and turbulent solutions are shown (the
brown and red lines), $10^{4},10^{5}$ and $10^{6}$ (each solution has the same
total flux). (b) Example of solutions for laminar and turbulent flows,
$U_{L}$, Eq.(9) (parabolic solution, green line), and $U_{T}$, Eq.(10) (super
exponential solution, blue line).

In Section 4 it is explained how to obtain this approximate analytical
solution (second solution). The construction of general solution of GHE is
proposed in Section 5, and a comparison of this analytical solution with the
experimental data is provided in Section 6.

##  4 Analytical Solution of GHE

###  4.1 Exact Solution of GHE

A general solution can be constructed from a particular solution of a problem
with non-zero right hand side (pressure gradient) and general solution with
zero pressure gradient. Below, it is described how to find a second solution
of GHE and then obtain a general solution.

We consider stationary flow regime

$\displaystyle u\,u_{x}+v\,u_{y}+p_{x}$ $\displaystyle=$ $\displaystyle
Re^{-1}(u_{xx}+u_{yy})$ (11) $\displaystyle u\,v_{x}+v\,v_{y}+p_{y}$
$\displaystyle=$ $\displaystyle Re^{-1}(v_{xx}+v_{yy})$ (12) $\displaystyle
u_{x}+v_{y}$ $\displaystyle=$ $\displaystyle\tau(p_{xx}+p_{yy})$ (13)

Let us designate $\alpha$ =$Re^{-1}$ and assume $u_{x}=v_{x}=0,p_{x}=$cont.

Then Eq.(12, 13) become

$\displaystyle p_{x}$ $\displaystyle=$ $\displaystyle\alpha v_{yy}-vu_{y}$
(14) $\displaystyle\tau p_{yy}$ $\displaystyle=$ $\displaystyle v_{y}$ (15)

Integrating Eq.(15) one obtain

$\displaystyle\tau p_{y}$ $\displaystyle=$ $\displaystyle v+C_{1}$ (16)

As $p_{y}=0,v=0$ at the wall, then $C_{1}=0$. Substitute that in Eq.(14) we
get

$\displaystyle\alpha v_{yy}-vv_{y}-v/\tau=0$ (17)

The analytical solution of Eq.(17) is available (e.g. [23], Sec.2.2.3-2, Eq.2,
and Sec.1.3.1-2 Eq.1, or WolframAlpha.com) in implicit form as Eq. (18),

$y+C_{2}=-\int_{1}^{v(y)}\tau/(W(-e^{-\tau(\xi^{2}+2\alpha\tau
C_{1})/(2\alpha)-1})+1)d\xi$ (18)

where $W(z)=W[k,z]$ is Lambert W function (or the Product Log function), the
solution for $w$ in $z=w\exp(w)$. Constants $C_{1}$ and $C_{2}$ are chosen to
satisfy boundary conditions. There is the $k-th$ solution of the Eq.(17)
corresponding to $W[k,z]$ in Eq.(18). There are multiple solutions of $v(y)$,
and then multiple solutions $u(y)$ can be found by substituting $v(y)$ into
Eq.(11). The pressure can be found using Eq.(16). This can not be done
analytically to obtain the explicit formulas for $u(y)$ and $p(y)$. The
example of solutions for $v(y)$ and $u(y)$ are shown in Figure 2.

In the Section 4.2 we obtain explicit analytical expressions for the
approximate solution.

Figure 2: Example of the exact solution $v(y)$ for some $C_{1}$, $C_{2}$ of
Eq.(18), and $u(y)$ from Eq.(11): (a) $u(y),v(y)$; (b) $v(y)$ details are
shown.

###  4.2 Approximate Solution of GHE

Let us designate $\Psi=v_{y}$, $\alpha$ =$Re^{-1}$ and assume
$u_{x}=v_{x}=0,p_{x}=$cont, and neglect small $v\,v_{y}$ term. Then

$\displaystyle p_{x}$ $\displaystyle=$ $\displaystyle\alpha u_{yy}-vu_{y}$
(19) $\displaystyle p_{y}$ $\displaystyle=$ $\displaystyle\alpha\Psi_{y}$ (20)
$\displaystyle\tau p_{yy}$ $\displaystyle=$ $\displaystyle\Psi$ (21)

Substituting Eq.(20) into Eq.(21) obtain

$\alpha\tau\Psi{}_{yy}=\Psi$, or

$\alpha\tau\Psi_{yy}-\Psi=0$.

The solution is $\Psi=Ae^{\pm\frac{y}{\sqrt{\alpha\tau}}}$, and recalling what
is $\Psi$:

$v_{y}=Ae^{\pm\frac{y}{\sqrt{\alpha\tau}}}$, integrating that one obtains:

$v=A_{1}e^{\pm\frac{y}{\sqrt{\alpha\tau}}}+B_{1}$, then applying boundary
condition:

$v=0$ at y=0, obtain the solution $v$:

$\displaystyle v=A(1-e^{\pm\frac{y}{\sqrt{\alpha\tau}}}),$ (22)

Here

$\displaystyle\alpha\tau=\tau
Re^{-1}=\frac{\tau^{*}V}{L}\cdot\frac{\nu}{VL}=\frac{\tau^{*}\nu}{L^{2}}=\delta^{2},$

or

$\displaystyle\delta=\frac{\sqrt{\tau^{*}\nu}}{L}$ (23)

###  4.3 Case of the Positive Sign in the Exponent

In this case the solution for $v=A(1-e^{{y}/{\delta}})$ grows exponentially,
still for $y\leq L/2$ it is limited. So we try to get the solution $u$ using
obtained solution for $v$. Let us recall Eq.(19) and put the solution for $v$
there, we have:

$\alpha u_{yy}-A(1-e^{{y}/{\delta}})u_{y}-p_{x}=0,$ where $p_{x}=0$, as we
search for a solution of a homogeneous equation

$\alpha u_{yy}-A(1-e^{{y}/{\delta}})u_{y}=0,$ (24)

To integrate once in $y$ we rewrite Eq.(24) as:

$\frac{u_{yy}}{u_{y}}=\frac{A}{\alpha}(1-e^{{y}/{\delta}}),$

and obtain integral as

$\ln u_{y}=\frac{A}{\alpha}(y-\delta e^{{y}/{\delta}})+C_{1}$

or

$\ln u_{y}=\frac{A\delta}{\alpha}({y}/{\delta}-e^{{y}/{\delta}})+C_{1}$

or

$u_{y}=\exp(\frac{A\delta}{\alpha}({y}/{\delta}-e^{{y}/{\delta}})+C_{1})$ (25)

The explicit expression for the solution of $u(y)$ (integral of $u_{y}$,
Eq.(25)) is available if

$A=\frac{\alpha}{\delta}.$ (26)

Since $A$ is an arbitrary constant, we may choose $A$ as in (26). With this
choice, the general solution is

$u(y)=-\delta e^{C_{1}}e^{-e^{{y}/{\delta}}}+C_{2}.$

If $u(0)=0$, and $u(\infty)=V_{0}$, then the constants $C_{1}$ and $C_{2}$ are

$\displaystyle C_{2}=V_{0},$

and

$\displaystyle\delta e^{C_{1}}=V_{0}e^{1}.$

We get the solution for $u(y)$ that we call a turbulent solution $U_{T}$ as:

$\displaystyle U_{T}=V_{0}\left(1-e^{1-e^{y/\delta}}\right),$

that is limited, $u(0)=0$, $u(L/2)\approxeq V_{0}$, and is acceptable.

###  4.4 Case of the Negative Sign in the Exponent

Again recall Eq.(19) and putting the solution for $v$ there, we have:

$\alpha u_{yy}+A(1-e^{-\frac{y}{\delta}})u_{y}-p_{x}=0,$ where $p_{x}=0$, as
we seek a solution of a homogeneous equation.

$\frac{\alpha u_{yy}}{u_{y}}=A(1-e^{-\frac{y}{\delta}}),$

Solution for $u_{y}$ :

$\ln u_{y}=Ay+A\delta e^{-\frac{y}{\delta}}+C$

or

$\ln u_{y}=A\delta(\frac{y}{\delta}+e^{-\frac{y}{\delta}})+C$, and

$u_{y}=\exp(A\delta(\frac{y}{\delta}+e^{-\frac{y}{\delta}}))=D\exp(\frac{y}{\delta}-e^{\frac{y}{\delta}})$.

$u=\int\exp(A\delta(\frac{y}{\delta}+\exp(-\frac{y}{\delta}))dx=A_{1}\left[\exp(\frac{y}{\delta}+\exp(\frac{y}{\delta}))-Ei(\exp(-\frac{y}{\delta}))\right]+A_{2}.$

As $u(0)=0$, then $A_{2}=0$ and solution $u$ is the following:

$\displaystyle
u=A_{1}\left[e^{\frac{y}{\delta}+e^{\frac{y}{\delta}}}-Ei(e^{-\frac{y}{\delta}})\right],$
(27)

where Ei(x) is the integral exponential function. This solution for $u$ grows
indefinitely, so it is not physical, and will be disregarded.

##  5 General Solution for the Turbulent Flow

The general solution is proposed as a linear superposition of two solutions,
laminar and turbulent:

$\displaystyle U_{GHE}=\gamma U_{T}+(1-\gamma)U_{L},$ (28)

where the coefficients $\gamma$ and $(1-\gamma)$ are introduced, to get
$V_{0}$ at the center line. The first term containing $U_{T}$ grows
superexponentially in the boundary layer, while the second term containing
$U_{L}$ is nearly zero. Outside the boundary layer, the first term is
constant, and the second term starts to grow (see Figure 7).

We get

$\displaystyle
U_{GHE}=V_{0}\left[\gamma\left(1-e^{1-e^{y/\delta}}\right)+(1-\gamma)4y(L-y)/L^{2}\right].$
(29)

The value of $\gamma$ can be calculated from the Eq.(19) at some point, better
for small $y$, for example, $y=\delta$.

Substituting the solution $U_{GHE}$ into Eq.(19) we get

$\displaystyle\alpha{U_{GHE},}_{yy}(\delta)-v(\delta){U_{GHE},}_{y}(\delta)=p_{x}$
(30)

or, dropping the argument $\delta$ , and substituting the expression of
$U_{GHE1}$ from Eq.(28):

$\displaystyle\alpha(\gamma{U_{T},}_{yy}+(1-\gamma){U_{L},}_{yy})-v(\gamma{U_{T},}_{y}+(1-\gamma){U_{L},}_{y})=p_{x},$

or

$\displaystyle\gamma(\alpha({U_{T},}_{yy}-{U_{L},}_{yy})-v({U_{T},}_{y}-{U_{L},}_{y}))+\alpha{U_{L},}_{yy}-v{U_{L},}_{y}=p_{x},$

and

$\displaystyle\gamma=(p_{x}-\alpha{U_{L},}_{yy}+v{U_{L},}_{y})/(\alpha({U_{T},}_{yy}-{U_{L},}_{yy})-v({U_{T},}_{y}-{U_{L},}_{y}))$
(31)

evaluated at $y=\delta$. Here ${U_{T},}_{yy}$ and ${U_{L},}_{yy}$ are the 2nd
partial derivatives of ${U_{T}}$ and $U_{L}$ in $y$, ${U_{T},}_{y}$ and
${U_{L},}_{y}$ are the first partial derivatives of ${U_{T}}$ and $U_{L}$ in
$y$, $v(y)$ is calculated using the Eq.(22) with $A$ from Eq.(26),
$p_{x}=-8V_{0}/(ReL^{2})$.

The parameter $\delta$ and the value of $\gamma$ in the comparisons with the
experiments below will be chosen to fit the experimental data, as we do not
have the material property $\tau^{*}$ for particular liquids used in the
experiments, as $\delta={\sqrt{\tau^{*}\nu}}/{L}$. The value of $\gamma$ can
be calculated from the parameters of experiments using Eq.(31) if the values
of $V_{0}=\frac{1}{2}Re\ p_{x}$ and $L$ are known.

Each of the two solutions has different similarity parameters. The laminar
parabolic solution depends on the Re number, while the turbulent solution
depends on the parameter $\delta={\sqrt{\tau^{*}\nu}}/{L}.$

The parameter $\tau^{*}$ (dimension of time) is a mean time between particle
collisions. For gases this mean time between collisions can be calculated
using hard-sphere model of particles at pressure p, and viscosity $\mu$ as
$\tau^{*}=\Pi\mu/p$, where $\Pi$ is a constant, $\Pi=0.786$ [4]. For liquids
this is more complicated (see [2], p.324 discussion about Frenkel work [16]),
there is no explicit formula, and the experimental measurements are needed. It
can be found by fitting the experimental velocity data (as will be shown in
the next Sections) finding $\delta$. Then $\tau^{*}$ can be found as
$\tau^{*}=\delta^{2}L^{2}/\nu$.

##  6 Comparison of Analytical Solution with Experiments

###  6.1 Pipe Flow Experiment by Van Doorne at Re=720

Here a comparison will be provided for the approximate GHE and other
analytical solutions with the pipe flow data Van Doorne 2004, 2007 [5, 6]
(Prof. Bruno Eckhardt group). The analytical solution was obtained for 2D flow
in Cartesian coordinates, but still we try it for pipes (cylindrical
coordinates) assuming that in the vicinity of the wall it will work. Figure 3
shows the experimental data digitized from [5], and a number of different
plots from (a) laminar (parabolic) flow profile (green line); (b) turbulent
solution (blue line) and and (c) GHE solution. The figure demonstrates that
neither the laminar nor turbulent solution fit the data, but the superposition
$U_{GHE}$ provides an excellent fit to the experimental mean velocity profile
for $\gamma$=0.66 and $\delta$=0.047, and $\delta$ is assumed to be a
Kolmogorov length scale. In [11] the GHE solution compared well with the
experiment for the driven cavity flow [20], and the dimensional value of
$\delta=0.58mm$ was a good approximation to the experimentally observed
”Kolmogorov microscale” $\delta_{exp}=0.5mm$ ( [20], p. 398).

Figure 3:  A comparison of experimental data for horizontal velocity by Van
Doorne(2007) [5], water at Re=720, with the GHE model. The GHE solution which
is the linear superposition of the turbulent and lamina profiles ($U_{GHE}$ on
the plot) fits the experimental velocity profile.

The next Figure 4 is a comparison of the values of experimental turbulent
intensity for horizontal velocity rms(u’) with the corresponding analytical
turbulent intensity. As the solution of the unsteady equations is not provided
(not possible analytically), it is suggested that a turbulence is the
oscillation between different stationary solutions, two solutions in our case,
the laminar (parabolic) velocity profile $U_{L}$ and the turbulent one given
by $U_{T}$. Therefore, the rms(u’) at every point is

$rms(u^{{}^{\prime}})=\sqrt{\frac{\sum{(U_{m}-u(t_{n}))^{2}}}{N}}$

where $U_{m}(y)$ is a mean velocity, $u(y,t_{n})$ is a velocity at time
$t_{n}$, and $N$ is the number of measurements.

The analytical rms(u’) was presented as a difference between two solutions
$U_{T}$ and $U_{L}$:

$\displaystyle rms(u\text{\textquoteright})={\mid\theta\cdot U_{T}-\beta\cdot
U_{L}\mid}$ (32)

and shown in Figure 4. As one can see, the data and analytical curve are in
concordance. The parameters of the analytical curve that define the difference
of $U_{T}$ and $U_{L}$ are the following:
$\theta=3.552,\beta=2.736,\delta=0.047$.

Figure 4: Comparison of experimental rms(u’) data [5] (blue circles) turbulent
intensities and analytical solution from GHE (red curve). The comparison is
quite satisfactory.

###  6.2 Experiments by Wei and Willmarth

For the experiments with a turbulent boundary layer, the velocity $U^{+}$
plots versus $y^{+}$ does not depend on Re number, see e.g. data from Wei and
Willmarth (1989) [24], Figure 5. The parameter $y^{+}=\frac{yu_{\tau}}{\nu}$
where $u_{\tau}$ is so called friction velocity, y is the absolute distance
from the wall, and $\nu$ is a kinematic viscosity. One can interpret $y^{+}$
as a local Reynolds number. The velocity scale $u_{\tau}$ is defined as
$u_{\tau}=\sqrt{\frac{\tau_{w}}{\rho}}$, where wall shear stress $\tau_{w}$,
$\tau_{w}=\rho\nu\frac{dU}{dy}$ at y=0, and the dimensionless velocity is
given by $u^{+}=\frac{u}{u_{\tau}}$.

The analytical solution $U_{GHE}$ is presented in Figure 5 as red line, and is
in a good concordance with data (parameters are $\delta=0.052,\gamma=0.65$).

Figure 5: Mean velocity profiles in turbulent boundary layer from experiment
[24] (distilled water), non-dimensionalized on inner variables, for the four
Reynolds numbers examined. Re = 2970(circles); Re = 14914 (squares); Re =
22776 (red dots); Re = 39582 (black dots), distilled water. The data for
different Re numbers merge well in coordinates $u^{+},y^{+}$. The red line is
an approximate GHE solution.

Typically, Figure 5 and Figure 7, the inner boundary layer (BL) region
(viscous sublayer) is in the range $0<y^{+}<5$, where $u^{+}=y^{+}$ (blue line
in Figure 7). The near-middle (buffer) BL region is in the range $5<y^{+}<30$,
a strictly nonlinear region. The far-middle (inner) BL region is in the range
$30<y^{+}<200$. Velocity profile in this region can be expressed as the
logarithmic “law-of-the-wall” von Karman law

$\displaystyle U^{+}=1/k\leavevmode\nobreak\ log\leavevmode\nobreak\ y^{+}+B,$
(33)

k = 0.41, B = 5.2 (green line in Figure 7).

The outer (non-linear, essentially inviscid)) region starts at $y^{+}>200$ and
continues to the center line.

The analytical solution $U_{GHE}$ fits well to the experimental data in all
the regions (Figure 7, red line).

(a) (b)

Figure 6:  Experimental data for mean velocity profiles by horizontal velocity
Nikuradse [21] for Re=427,000 (a) and Re=970,000 (b) compared to laminar flow
profile (green), Turbulent flow profile (blue) and analytical solution GHE
flow profile (purple). Approximate analytical solution $U_{GHE}$ (purple
curve) compares well with the experimental data (circles).

###  6.3 Experiments by Nikuradse for High Reynolds Numbers

The solution superposition principle works well for higher values of Reynolds
number, up to nearly $10^{6}$, a range of nearly 3 orders of magnitude.
Nikuradse (1932, 1933) (Prandtl group) experiments have been done for Re from
27,000 to Re=970000 [21, 22]. We present a comparison for Re=427,000 and
Re=970,000 in Figure 6. The approximate analytical solution $U_{GHE}$ (pink
curve) compares well with the experimental data. The parameters for the
analytical solution are the same for all the cases in [21],
$\gamma=0.71,\delta=0.010$.

##  7 Discussion

Let us consider the turbulent boundary layer regions, the linear and the
logarithmic “law-of-the-wall ”von Karman law parts. Let us represent the
turbulent boundary layer profile, Figure 7, where the components of GHE with
the appropriate coefficients from Eq. (28) are plotted: the turbulent solution
$U_{1T}$ and the laminar solution $U_{2L}$, were added. The plot also has the
log law by von Karman (the coefficients have been calculated by Nikuradse
using his data [21]), linear law and empirical 1/7 power law, that was
proposed first by Nikuradse [21].

The linear law is in the range $0<y^{+}<5$, where the parabolic profile
$U_{2L}$ (laminar solution) is very small, and the turbulent solution for
small $y/\delta$ becomes

$\displaystyle U_{T}$ $\displaystyle=$ $\displaystyle
V_{0}\gamma\left(1-e^{1-e^{y/\delta}}\right)=V_{0}\gamma\left(1-e^{1-(1+y/\delta)}\right)=V_{0}\gamma\left(1-e^{-y/\delta)}\right)=V_{0}\gamma\left(1-(1-y/\delta)\right)$

or

$\displaystyle U_{T}$ $\displaystyle=$ $\displaystyle V_{0}\gamma y/\delta,$
(34)

that is a linear law.

Figure 7:  Velocity profiles in turbulent boundary layer: analytical solution
$U_{GHE}$ (red line) and its constituents, the turbulent solution $U_{1T}$ and
the laminar solution $U_{2L}$; log law by von Karman $U^{+}$ = 1/k log $y^{+}$
\+ B, k = 0.41, B = 5.2 (green line), linear law $U^{+}=y^{+}$ (blue line) and
empirical power law $U^{+}={{y^{+}}\leavevmode\nobreak\ ^{1/7}}$ (brown), and
experimental data (dots) by Wei and Willmarth (1989)[24].

In the range of $30<y^{+}<200$, the solution $U_{1T}$ becomes nearly constant
$\approx$ 12.5 (pink line in Figure 7), and the GHE solution changes due to
the growth of the laminar solution $U_{2L}$ (cyan color line in Figure 7).
That $u^{+}$ = (12.5 + log $U_{1L}$), the parabolic solution would fit well
into logarithmic “law-of-the-wall”(von Karman law). The empirical 1/7 power
law, that was first proposed by Nikuradse, also fit well into the “law-of-the-
wall ”.

The parameter $\tau^{*}$, a material property, if not known from the
experiment for a given fluid, can be found by fitting the experimental
velocity data (as we did in the previous Sections). In this case we will
obtain the value of $\delta$ which will be used to calculate $\tau^{*}$:
$\tau^{*}=\delta^{2}L^{2}/\nu$.

## Conclusions

The exact and approximate analytical solution of Generalized Hydrodynamic
Equations (GHE) have been obtained for channel flow (that are valid for the
pipe flow if $\delta\ll 1$), that constructed from two solutions for laminar
and turbulent flows, presented as a linear superposition of these solutions.

The GHE solution for turbulent flows depends on two parameters: the Reynolds
number $Re$, and $Al=\frac{\sqrt{\tau^{*}\nu}}{L}=\frac{\delta}{L}$, the
Alexeev number (author of GHE and GBE). In Figure 7, the data for different
$Re$ numbers fall to a single law in ($U^{+},y^{+}$) coordinates for the same
kind of fluid. This data has the same Alexeev number $Al$ if the same
experimental dimensions hold. Changing the $Al$ number (different fluid or
dimensions) will produce different results.

The time parameter $\tau^{*}$ is a material property. For gases this is a mean
time between collisions, and can be calculated for hard-sphere model of
particles at pressure p, and viscosity $\mu$ as $\tau^{*}=\Pi\mu/p$, where
$\Pi$ is a constant, $\Pi=0.786$. For liquids this is more complicated (see
[2], p.324 discussion about Frenkel work [16]), there is no explicit formula,
and the experimental measurements are needed.

The approximate analytical solution for the GHE model has been compared with
the experimental data for turbulent flows. The linear superposition of two
solutions (turbulent and laminar) provides good agreement with the
experimental axial mean velocity data in a wide range of Reynolds number from
Re=720 to Re = 960,000.

The experimental data for turbulent intensities for axial velocity are in
concordance with the analytical equation (weighted difference of the turbulent
and laminar solutions). Apparently the turbulent oscillations are the
oscillations between two solutions (two in considered case): the turbulent
solution $U_{T}$ and the laminar solution $U_{L}$. GHE intrinsically include a
turbulence model.

Moreover, the obtained analytical solution was able to capture correct
velocity behavior across the whole turbulent boundary layer (BL) and into the
external flow: from the inner viscous sublayer to the outer layer of the
turbulent BL. It also compares well with the experiments by Wei and Willmarth
(1989).

The analytical solution presents the explicit verifiable proof that the
Alexeev generalized hydrodynamic theory (GHE and GBE) is in good agreement
with experiments for turbulent flows.

Availability of data

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Acknowledgments

Author would like to extend the sincere thanks to Dr. Yuri Bozhkov for
discussions of the text during manuscript preparation.

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