IACS Institutional Repository

Advisory Board and Editors

Kumar, Krishna

Paul, Supriyo

Jain, Dharmesh

Rhombic Patterns Near a Bicritical Point in Periodically Forced Surface Waves

Kimothi, Aradhana

Tiwari, V K

Saxena, V K

Saini, J S

Bhatnagar, D

Theoretical Analysis on the Performance of a Circular Sector Microstrip Antenna Under Re-Entry Conditions

Barman, Jayanta

Sarma, K C

Indian I Phys. 82(7) 855-862(2008) 
Structural and optical properties of CdS nanoparticles 
Jayanta Barman and K C Sarma* 
Department of Instmmentation & USIC, Gauhati University, Guwahati-781 014, Assam, India 
E-mail: kanak_sarma50@rediffmail.com 
Received 16 August 2007, accepted 4 February 2008 
Abstract : 11-VI semiconductor nanoparticles have attracted a lot of attention due to the possibility of their 
applications in various devices. Among the materials used for fabrication of quantum dots (Q0) and nanccrystals, 
cadmium sulphide (CdS) has drawn wide attention because of its potential for applications and availability of a 
variety of methods for its preparation. In the present study, chemical method has been used for synthesis of CdS 
nanoparticles in polyvinyl alcohol (PVA) as a matrix. X-Ray diffraction and electron microscopy (JEM) were used 
to determine the phase, shape and size of quantum dots. UV-VIS spectrophotometer and PL-study shows the 
blue shift and defect level formation within the band gap. Particle size was found to be between 4 and 7 nm. 
Keyword : CdS quantum dot, bandgap, nanocrystallites. 
PACS Nos.: 73.63.Kv, 68.37.Lp, 68.37.H 
1. Introduction 
Nanostructured materials, especially ll-VI semiconductors have become a subject of 
intensive research for their extraordinary properties compared to their bulk counterpart 
[1—5]. The properties of semiconductor nanoparticles depend mainly on their shape 
and size due to high surface to volume ratio [6,7]. The structural and optical properties 
of CdS are of increasing interest due to the size dependent optical properties in the 
regime where the nanocrystal is smaller than bulk exciton radius. Blue shift of optical 
absorption spectrum, size dependent luminescence are some examples of the interesting 
properties exhibited by the nanoparticles [8-12]. All these properties are various 
manifestations of the so called quantum size effect which arises due to the increasing 
quantum confinement of the electrons and holes. However, when the dimensions of 
the crystalKtes become comparable to or less than the Bohr radius of exciton wave 
function, there is a significant change in properties. In CdS such quantum size effect 
is observed for crystallites below 5 nm diameter [13-15]. 
^ra 
'Corresponding Author © 2008 IACS 
856 Jayanta Barman and K C Sarma 
Nanometer-scale electronics has opened new areas of application in device 
technology. One-dimensional nanoscale materials may be utilized in various nanodevices 
including nanologic circuits [16], nanolasers [17], nanosensors [18], nanothermometers 
[19], etc. Quantum dot of semiconductors and metallic alloys have been found to 
exhibit interesting optical properties. Several techniques such as thermal evaporation 
[20], sol gel process [21], metal-organic vapor phase epitaxy [22], wet chemistry 
deposition [23], spray pyrolysis [24], and chemical bath deposition (CBD) [25] have 
been used to produce CdS nanoparticles. Among them, chemical deposition is well 
known as a prevalent low-temperature technique for depositing large area of 
semiconductor nanoparticles, the simplest and the most economical one. A solution 
growth technique (SGT) is a relatively inexpensive and convenient technique for large 
area preparation of semiconductor nanoparticles at low temperature. The physical 
properties of the film are reproducible by this method. The growth process involves 
the application of a controllable chemical reaction which proceeds at a low rate in an 
aqueous solution containing the various reactants. The reaction rate in this technique 
is controlled by adjustment of pH, temperature, and the relative concentrations of the 
various reactants in the solution. 
CdS nanoparticles are attractive candidate for optoelectronic applications, as the 
band gap of the semiconductors can be tuned to suit to any specific application. Bulk 
CdS has a hexagonal wurtize type crystal structure with lattice constant a = 0.4160 
nm and c = 0.6756 nm. The single cubic phase has been observed in synthesised 
nanoparticles. In the present work, we report the chemical deposition of CdS nanoparticles 
and their characterization by XRD, TEM, SEM. UV-VIS spectro-photometry and PL 
spectrometry. The effect of pH on structural, morphological and optical properties of 
these films are investigated with the objective to optimize the conditions of the 
deposition process. 
2. Experimental 
CdS nanoparticles were synthesised using polyvinyl alcohol (PVA) as a matrix. PVA 
being a good solute to multiple phase systems capable of provoding uniform gap in the 
form of array. 5 wt% solution of PVA, CdCI2 was added with various concentration (2, 
3, 4 wt%) under a high stirring rate (200 rpm) condition using magnetic stirrer. The 
constant temperature 70°C for 3 hours was maintained. The sample under preparation 
was kept for 12 hours for complete dissolution to get a transparent solution. To this 
solution 2 wt% Na2S was added till the whole solution turns into yellow colour. The 
CdS nanoparticle containing PVA were caste over precleaned glass slides to produce 
thin film form. The size of the particle is controlled by changing the pH value of the 
whole solution. The chemical reaction occurs as follows 
CdCI2 + Na2S -» CdS + 2NaCI 
pH controlled the rate of reaction due to the common ion effect At higher pH the 
solubility product increases and as a result formation of CdS particle stops. 
Structural and optical properties of CdS nanoparticles 857 
3. Results and discussion 
3.f. XRD observation : 
The XRD patterns of prepared sample were taken by Seifert XRD (3003 TT) operating 
at 40 KV-30 mA. The radiation source used was CuKa (A = 1.542 A) and a nickel 
filter was used to block K0 radiations. Figure 1 represent four different spectra of CdS 
nanoparticles with variation of pH with bulk sample (99.999% pure). 
s ; 
20 30 40 50 60 70 
Angle of diffraction (two theta) 
Figure 1. XRD traces of CdS for bulk, CdS, at pH 1.6, CdS2 at pH 1.8 and CdS3 at phi 2.0. 
The XRD pattern of CdS! contain broad peaks at 20 = 26.6 indicating the 
formation of nanostructure at pH = 1.6 and in CdS2 shows broad peaks at 20 = 26.8 
and CdS3 at 2d - 27.1 respectively at pH = 1.8 and pH = 2 
The XRD spectrum of both CdS^ CdS2 and CdS3 apparently exhibit weak and 
broad peaks, suggesting small crystallite sizes of cubic phase (JCPDS 89-0440). 
Again it is seen that when the pH increases peaks are more broadened and shifted 
to higher diffraction angle with decreasing crystal size. The increase in diffraction angle 
is clearly a result of lattice contraction which is expected to occur because of higher 
surface to volume ratio [26]. Some separate peaks are also observed which may be 
due to CdO because XRD patterns were recorded in ambient condition. The change 
of phase from hexagonal wurtzite to cubic is observed in nanocrystallite as compared 
to their bulk counterpart. The grain size of the crystallite are estimated by Sherer 
formula [27] 
Dhki « Kx/w Cos0 (1) 
w being FWHM, 0 is the Braggs angle and K m 0.89 for spherical shape (from TEM). 
For calculation w is observed by zooming the peak position using origin graphics 
CdS, 
i$ninm ii IIIMMIII MI ii < i i ^ n i 
• * • • • • i i • • 
860 
Jayanta Barman and K C Sarma 
2.5 2.6 3.0 3.1 2.7 2.8 2.9 
Band gap energy (eV) 
Figure 4. Band gap energy vs. particle size. 
and at 610 nm for CdS3. These spectra were recorded using (Thermospectronic 
AMINCO) luminance spectrometer. Emission bands redshifted from the absorption 
edge are observed in almost all of the semiconductor nanoparticles studied so far and 
are generally assigned to defect states. These redshifted emissions are usually 
associated with trapped states such as vacancies, interstitials, impurities, and surface 
defects [29]. It is seen that when the size of the nanocrystallite decreases the 
luminance is dominated by the band impurity. Also, surface passivation by sulpher has 
resulted in reduced emission intensity of this band indicating involvement of surface 
defects. 
n — • — i — » — i — i — i — i — i — i — p 
300 350 400 450 500 550 600 650 
wave length (nm) 
r 
700 750 
Figure S. PL spectrum of CdS quantum dot at same excited wavelength 300 nm. 
The nanostructure observed from TEM (model JEOL JEM-100cx) shows the 
Structural and optical properties of CdS nanoparticles 861 
nanocrystallites to be spherical in shape. The size of the crystallite decreases with 
increase of pH which agrees with the result obtained from XRD observation. 
The TEM photograph patterns Figures 6 and 7 were recorded using selected area 
observations and the particle sizes were determined in higher magnification (~2 x 105). 
The size of the particles obtained under various conditions are shown in Table 1. It is 
seen that optimum pH value for getting smaller nanoparticles is approximately 2. The 
PVA : CdCI2 ratio also affects the size of the nanoparticles. 
Figure 6. TEM image of CdS quantum dot at pH « 1.8 Figure 7. TEM photograph (zoomed view) of CdS 
quantum dot at pH = 2. 
4. Conclusions 
CdS nanoparticles of different crystallite sizes have been synthesized by the chemical 
route. The particle size is controlled by stirring rate, temperature of the solution, pH 
and time of stirring. The structural and optical characterization of the samples by XRD, 
TEM, UV-VIS spectrophotometer and PL show formation of CdS quantum dots having 
sizes between 4 and 7 nm. The optimum pH value for minimum size is found to be 
2 and PVA : CdCI2 ratio is found to 5 : 2 in our observations. 
Acknowledgment 
The authors would like to acknowledge Physics Departmen* and CIF, IITG for 
providing XRD and PL facilities. We would also like to acknowledge RSIC, NEHU, 
Shillong for TEM observations and Department of Chemistry, Gauhati University for 
spectrophotometer observation. 
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862 
Jayanta Barman and KC Sarma 
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[25] K C Sarma, R K Bordoloi, M Sarma and J N Ganguli J. Instrum. Soc. India 31 216 (2001) 
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[27] H P Klug and L E Alexander X-Ray Diffraction Procedures for Polycrystaliine and Amorphous Materials 
(2nd ed) (New York: John Wily and Sons) p491 (1954) 
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Structural and Optical Properties of CdS Nanoparticles

Bayal, Nirmalya

Chanda, Arunabha

Mazumdar, Asis

Indian J. Phys. 81(7) 673-681 (2007) 
UP' 
Numerical investigation on hydrodynamic developement 
length of gaseous flow in parallel plate micro channel 
Nirmalya Bayal* 
Department of Mechanical Engineering, Haldia Institute of Technology, Haldia West Bengal, India 
and 
Arunabha Chanda and Asis Mazumdar 
Department of Mechanical Engineering, Jadavpur University, Kolkata-700 032 India 
E-mail n_bayal<Wrediffmail com 
Received 6 June 2007 accepted 28 June 2007 
Abstract : Fluid flows in the entrance region of micro channels have been investigated using fourth order 
finite difference implicit scheme Considerations have been given to investigate the compressibility and rarefaction 
effects assuming slip and no slip on developing laminar flow at very close to the entry It has been seen that the 
effect of compressibility is important for higher Reynolds number (Re) and rarefaction is significant for lower 
reynolds number Re. The result suggests that rarefaction has a significant effect on hydrodynamic development 
length A new relation has been proposed for entry length for Reynolds number in between 1 to 400 The 
proposed relation provides an efficient, practical and accurate toot in predicting the hydrodynamic development 
length It has also been observed that due to slip flow mass flow rate increases 
Keywords : Hydrodynamic developement length, Reynolds number, Knudsen number 
PACS NOS. : 47 10 -g, 47 11-j, 47 40 -x 
1. Introduction 
Small channels (Dh < 1 mm) have been used in a large variety of novel applications, 
such as heat exchangers, micro sensors, biological cell reactors and selective 
membranes. The study of fluid transport phenomena in small channels on the scale of 
micro-electro-mechanical systems (MEMS) have been an interesting research topic not 
for a very long time. At these length scales the dimensions of the system itself 
become comparable to the mean free path of the particles flowing. For liquid, this in 
turn tends to make the flow granular in nature, whereas for gases the flow becomes 
•Corresponding Author ® 2 0 0 7 , A C S 
674 Nirmalya Bayal, Arunabha Chanda andAsis Mazumdar 
rarefied. Studies reflect that the continuum approach with no-slip wall conditions can no 
longer predict the fluid flow characteristics in micro devices. As the flow becomes 
molecular, a number of parameters such as : surface force, roughness, rarefaction, 
viscous force, thermal creep start playing significant role which are not dominant in 
macroscopic flows. 
The degree of rarefaction and applicability of continuum model is determined by 
local Knudsen number and is defined as, the ratio of molecular mean free path A to 
the characteristic length L This means that rarefaction becomes an important 
phenomenon and accordingly, micro scale flow is no longer in the continuum flow 
regime. Therefore, conventional continuum computational fluid dynamics (CFD) method 
with no-slip wall boundary conditions are not suitable. First order slip boundary 
conditions should be applied in the Navier-Stokes equations in the range of 0 001 < Kn 
< 0.1. The transition region spans the range of 0 1 < Kn < 10 and second order slip 
boundary conditions are applicable. 
Nomenclature 
H Channel height 
u,v Velocity components in axial and transverse directions 
u0 Free stream velocity 
Ma Mach number 
R Gas constant 
Dh Hydraulic diameter 
L0 Hydrodynamic development length 
Pr Pressure ratio = Inlet pressure/Outlet pressure 
7"g a 8 Temperature of gas adjacent to wall 
rwa„ Wall temperature 
T Temperature 
Re Reynolds number 
ma Mach number 
Kn Knudsen number 
Over the past two decades researches are going on to determine the fluid flow 
characteristics in micro devices. Pfahler et al [1], Harley et al [2] reported that 
conventional analyses are unable to predict the transport of gas in silicon micro-machined 
channels. Harley et al [2], focused on nonlinear effects of gaseous slip flows. Peng 
et al [3] found experimentally that transition to turbulent flow begins at Re = 200 to 
700 and that fully turbulent flow occurs when Re is in between 400 to 1500. Chen et al 
[4] studied the flow of nitrogen and helium in micro channel. They concluded that due 
to larger wall shear stress, pressure gradient is high even at lower velocities. The 
capability of Navier-Stokes equations to adequately represent the flow and heat 
transfer behavior in the micro channels has been called into question in some of these 
studies. 
n Number of time steps 
a Aspect ratio {W/H) 
p Density 
// Viscosity 
A Molecular mean free path 
rw Wall shear stress 
y Ratio of specific heats 
crv Tangential momentum 
accommodation coefficient 
Numerical investigation on hydrodynamic developement length etc 675 
The focus of the analytical and computational models cited above were mainly 
limited to flow characteristics in the hydro dynamically developed zone with the 
exception of the work of Atkinson et al [5] and Chen [6] who however assumed no-
shp flow (i.e Kn - 0.0) There are a number of applications where the entrance region 
fluid flow characteristics become important There is a clear need for additional 
investigations Most notably, flow through micro geometry has become a topic area of 
the utmost research activity and increased interest with the dawn of the new century 
This has motivated the present work where a study of the fluid flow characteristics in 
the entrance region of micro channels has been carried out The present study 
examines the role of Reynolds number and Knudsen number on the hydrodynamic 
development length at the entrance of the parallel plate micro channel 
2. Governing equations 
The channel is assumed to be long enough such that fully developed flow may safely 
occur The channel is wide with an aspect ratio (a = W/H) of approximately 40 Since 
the three dimensional effects are proportional to the inverse of the a2 [7], The flow can 
be safely assumed to be two dimensional 
The governing equations for this study are 
The continuity equation . 
dt dx dy 
The unsteady two-dimensional compressible Navier-Stokes equations 
(2) ( du du du ) dp 
'[dt dx Dy dy 
(4d2u d2u 1 d2v 
, dv dv dv\ dp 
1
 dt dx dy) 3y 
3 dx2 dy2 3 dxdy 
r4d2v d2v 1 d2u ) 
(3) 3dy2 dx2 3 dxdy J 
The equation of state . 
P = pRT (4) 
The above equations are non dimensional zed by introducing 
H H u0 u0 p0 pi4 T0 v ' 
3. Boundary conditions 
At the inlet a uniform velocity profile has been assumed Therefore du/dx « 0 and 
v = 0. The pressure at the inlet is specified by pressure ratio. The temperature at the 
inlet has been assumed to be same as the wall temperature. At the outlet, the axial 
gradients of the flow variables has been set to zero. 
676 Nirmalya Bayal, Arunabha Chanda andAsis Mazumdar 
4. Model descriptions 
The cross section of the micro channel is shown in Figure 1 Several experimental and 
numerical results have been documented for this popular geometry Table 1 lists the 
dimensions of the micro channel and relevant physical properties of the working fluid 
Figure 1 Geometry of micro channel analysis 
Table 1 Model dimensions 
Flow parmeters Range or mean value 
Length L 
Width W 
Height H 
Pressure ratio P, = PJPout 
Gas temperature at inlet 7, 
Wall temperature Tw 
Outlet mach number 
Outlet Kundsen number 
Reynolds number 
3000 urn 
40|im 
1 2 urn 
1 35 1 50 1 95 2 20 2 35, 2 51 2 7 
300 K 
300 K 
(2 25-25) x 10* 
0 001 < Kn < 0 1 
1-400 
5. Hydrodynamic entrance length 
When a viscous fluid enters in a duct, a velocity boundary layer develops along the 
inside surfaces of the duct The boundary layer fills the entire duct gradually The 
region where the velocity profile is developing is called the hydrodynamic entrance 
region and its extent is the hydrodynamic entrance length For practical engineering 
calculations, the hydrodynamic development length (Le) is arbitrarily defined as the axial 
distance required for centre line velocity to reach 99% of the fully developed value The 
region beyond the entrance region is referred to as hydro dynamically fully developed 
region In this region, the boundary layer completely fills the duct and velocity profile 
becomes invariant with the axial co-ordinate 
The important features of gas flow in micro ducts are mainly due to rarefaction 
and compressibility effects The inertia force is negligible compared with the viscous 
force for small Reynolds number If MaKn « 1, all stream wise derivatives can be 
ignored except the pressure gradient, which is the only driving force. Hence, the fully 
developed assumptions imply that the velocity profile can be determined by solving • 
Numerical investigation on hydrodynamic developement length etc. 677 
d2u _ dP 
1
 dyz " dx (6) 
Integrating eq. (6) twice with respect to y assuming P = P(x), Arkilic et al [9] the 
derived following fully developed velocity profile : 
u(y) = H
2
 dP 
8A* dx 
2y 
H 
+ 4Kn(x) (7) 
In addition the maximum velocity along the centerline of the duct (y = H/2) can be 
derived as 
1.5aM = 8-~- - Kn 
1 + 1 2 2 — Kn 
(8) 
As Kn -> 0.0 the above equation reverts to familiar Poiseuille flow. L/max = l.5ty. 
6. Slip flow 
Slip flow occurs when gases are at low pressure or for flow in micro passages. In 
presence of wall-normal and tangential temperature gradients, for an ideal gas the 
complete slip-flow (first order) will be in no dimensional form [2] : 
"gas ^wall 
2-ou Kn du* 3 y-\Kn
2Re 
{dy*)w In Y Ec 
DJ2 
dx* 
(9) 
The second term of the above equation is known as 'thermal creep' which generates 
slip velocity in the direction opposite to the increasing temp. For isothermal wall the 
above equation becomes : 
2-crv 
"gas Kn 
du * \ 
ay J (10) 
For continuum flow, Kn 0, above equation reduces to the familiar no-slip conditions. 
°
v =
 V~S7~ Tangential momentum accommodation coefficient (TMAC). Subscripts /, r, 
w stands for incident, reflected and solid wall conditions respectively; r is a tangential 
momentum flux. For real walls, a portion of the momentum of the incident molecules 
is lost to the wall and a portion is retained by the reflected molecules. TMAC is 
defined as the fraction of molecules reflected diffusively. The value of this coefficient 
depends on the fluid, the solid, temperature, local pressure and surface finish and has 
been determined experimentally to be in between 0.2 and 0.8. The lower limit is for 
exceptionally smooth surface while the upper limit is typically of most practical surface. 
678 Nirmalya Bayal, Arunabha Chanda andAsis Mazumdar 
Recently Arkilic et al [9] experimentally concluded that TMAC varies in between 0.75 
to 0.85. 
7. Methodology 
The governing equations are nonlinear in nature. The numerical computations were as 
carried out by solving the governing equations along with boundary conditions by forth 
order accurate finite difference implicit scheme with uniform grids. Forward difference in 
time and central difference in space has been used. Convergence and accuracy were 
obtained by MacCormack multi step (intermediate step between n and n+1) scheme. 
Stability has been checked for every grid points for each iteration by the following 
empirical formulas proposed by Tannehill et al [10]. 
At< G{At)^i where cr = 0.7-0.9, (11) 
1 + 2/Re, 
(*)r- * /CFL \u\ \v\ 1 1 Ax Ay V**2 Ay2 (12) 
ReA=m\n(ReAX,ReAy)zQ, tf^x = — — , ReAy = -±j— (13) 
The flow is assumed to be two dimensional and flow variation in z-direction is assumed 
negligible. The governing equations for this study are the continuity equation, unsteady 
two-dimensional compressible Navies-Stokes equation and the equation of state. Two 
types of boundary conditions on the channel wall, a slip wall and a no-slip wall has 
been employed in this simulation to study their effects on the micro channel flow. The 
governing equations along with the boundary conditions are solved using a numerical 
code written in FORTRAN-90 language to get velocity and pressure at every internal 
point. The solution procedure starts by applying the initial guesses values and obtained 
a converged solution by iteration. The iterations were continued until the sum of 
residuals for all computational cells became negligible i.e. max|((/,+1 -un}jun*A\< 10~5 is 
satisfied, where u is the velocity in the x direction and n denotes the number of the 
time steps. The distribution of cells in the computational domain was determined from 
a series of tests with a different number of cells in x and y directions. A uniformly 
distributed 5000 x 30 grid is found to be adequate and has been used throughout the 
study. 
8. Results and discussions 
A thorough investigation at very close to the entry has been performed by varying the 
different mass flow rates for nitrogen flow through the channel under isothermal 
conditions. The results are first compared with the available experimental and numerical 
results of prominent researchers. Then they were applied to the rarefied flow in the 
hydrodynamic flow region has been investigated. 
Numerical investigation on hydrodynamic developement length etc 679 
In channel flows the velocity of the molecules near the wall decreases and the 
molecules near the centerlme accelerated to converse the mass flow rate Figure 2 
shows the axial velocity at three different cross sections with a pressure ratio of 2 65 
y/H 
- 0 4 
-O2J 
00 j 
0 2-J 
0 4-j 
V 
I • UnMlpx/U25% 
-o-U.apX/L^S0 /*, 
• Uno.-(px/L=50% 
—0 -U«pX/L=50% 
• Uno.rtpx/L=100% 
__o_ U i ( i p X /U i00% 
• Chenetal [4] 
m. 
• q ^ V c 
* \ 
• 
• 
p 
m£ 
u %xr 
«-tf^icr^^-i: 
mv4&£^-y r- 1 — 
1 1 » ! 1 j 
k^ P ^"^"O. | 
P N J P ^"^P \ 
b p \ J p \ \ 
p » VJ • P -| 
i m p m p \ 
/ / • 
r ^ I 
x 1CT 
00 05 10 15 
u/ao 
Figure 2. Velocity profile along the transverse direction of nitrogen flow at three locations subjected to slip and 
no slip boundary conditions, Pr = 2 65 
The axial velocity has been non-dimensional zed with the velocity of sound To check 
the accuracy of the present work, the fully developed velocity profile has been 
compared with the numerical results of Chen et al [4] It is important to note that Chen 
et al [4] utilized an explicit finite difference method on a 6000 x 23 grids, while 5000 
x 30 grids have been used for present numerical simulations Nitrogen was used as 
the working fluid in the micro channel The geometrical dimensions and other 
parameters used for nitrogen flow have been shown in Figure 1 and listed in Table 1 . 
It is also visible from this figure that at downstream velocity increases with a 
corresponding increase in wall velocity due to slip effect In order to satisfy the 
conservation of mass along the channel direction, the flow is accelerated to make up 
the density drop that result from the decrease of the pressure In case of compressible 
flow the velocity is obviously accelerated because of local bulk density decreases due 
to low local pressure there and the average velocity increases. This is due to the 
volume expansion of the gas caused by the pressure drop. Consequently fully 
developed flow never occurs due to this compressibility effect even if the micro channel 
is very long or the length is very large. In this present model it has been assumed 
that if the axial velocity gradient is less that 10"6, the flow is fully developed At inlet, 
uniform velocity distribution has been assumed. 
Hydrodynamic development lengths were computed for a wide range of Knudsen 
numbers (0 001 < Kn < 0 1) and a range of Reynolds number in between 1 to 400 
Some of them are presented here. Figure 3 shows the non-dimensional development 
length as a function of Re. The results of the present investigation have been super 
imposed on the results of Atkinson et al [8]. Atkinson et al [8] found that hydrodynamic 
680 Ntrmalya Bayal, Arunabha Chanda andAsis Mazumdar 
1 5H 
L/Dh 
—O— Atkinson 
O Chen 
A Present Numerical 
03 H 
20 40 60 
Re 
80 100 
Figure 3. Comparison variation of development length Kn = 0 0 
development length for parallel-plate duct flow at lower Reynolds number can be related 
by the following linear relations * 
Dw 
= 0.3125 + 0.011 Re (14) 
However their calculations were based on Kn = 0.0, that means there results are 
based on continuum flow. 
However Chen [6] proposed that the above relation could be related to : 
* • _ 0315 
DH 0.0175 Re+ 1 
+ 0.011 Re (15) 
Figure 3 indicates that the results obtained from the present numerical investigation at 
Kn = 0 are in good agreement with the results of Atkinson et al the deviation is only 
-5%. More importantly the present results show that as the Reynolds number 
increases the hydrodynamic development length also increases. From Figure 4 it is 
L/Dh 
100 200 300 400 
Figure 4. Vanation of development length for rarefied slip flow in parallel plate microchannel 
Numerical investigation on hydrodynamic developement length etc 681 
also clear that Kn has a significant role on development length. As Kn increases 
development length also increases. It has also been observed that, for Reynolds 
number 400, the entrance length when Kn = 0.1 is 21.7% longer that corresponding 
no-slip solutions (i.e. Kn = 0 0). The formulae developed by Atkinson et al and Chen 
are not valid in the slip flow regime Therefore consequently a new development length 
equation must be evaluated. Based on numerical calculations over a wide range of 
Reynolds number (1 < Re < 400) and Knudsen number with in the slip flow regime 
using curve fitting methods we propose a new relation for hydrodynamic length has 
been suggested as follows : 
- 2 - ^ 0.350 +0.011 \.Re /1R \ 
It can be seen that the proposed equation provides a good representation of the 
numerical development length data. The proposed entrance length equation should be 
used in the laminar (Re = 1 to 400) flow regime only 
9. Conclusions 
A finite difference based micro scale formulation has been developed and compared 
with the available experimental and numerical results for two dimensional compressible 
flow. The investigation is mainly concentrated on the low Reynolds number behavior at 
the entrance of parallel plate micro channel. A new relation for hydrodynamic entry 
length has been proposed. The numerical model has indicated that the Knudsen 
number has a significant effect on the hydrodynamic development length. As Knudsen 
number increases hydrodynamic development length increases and nonlinear pressure 
distribution becomes more pronounced. 
References 
[1 ] J Pfahler, J Harley, H Bau and J N Zemel Micromechanical Sesors, Actuators and Systems, ASME 32 49 
(1991) 
[2] J Pfahler, J Harley, H Bau and J N Zemel J Fluid Mech 284 257 (1995) 
[3] X F Peng, G P Peterson and B X Wang Experimental Heat Transfer? 265 (1994) 
[4] C S Chen, S M Lee and J D Sheu Numerical Heat Transfer, Part A, 33 749 (1998) 
[5] B Atkinson, M P Brocklebank, C C H Card and J M Smith A I Ch.E Journans 548 (1969) 
[6] R Y Chen Trans ASME, J Fluids Engineering, 95 153 (1973) 
[7] F M White Viscous fluid flow, {2nd Edn) (McGaw-Hill, Inc ) (1991) 
[8] A Beskok and Ge Kamiadakis J Thermo Physics Heat Transfer 8 647 (1994) 
[9] E B Arkilhc, M Schmidt and K Breuer J Fluid Mechamics 437 29 (2001) 
[10] J C Tannehill, T L Hoist and J V Rakich AIAA 75 (1975) 


English

Numerical Investigation on Hydrodynamic Developement Length of Gaseous Flow in Parallel Plate Micro Channel

From Editor's Desk

Chammasheril, C J

Vinodkumar, P C

Indian J. Phys. 81 (8), 793-801 (2007) 
Sensitivity of nuclear matter EOS parameters 
on neutron star properties 
C J Chammasheril1* and P C Vinodkumar2 
'St. Xavier's College, Navaragpura, Ahmedabad-380 009, Gujarat, India 
department of Physics, Sardar Patel University, Vallabh Vidyanagar-388 120, Gujarat, India 
E-mail : clementcj@rediffmail.com 
Received 6 November 2006 
Abstract : Neutron stars are composed of matter at extremely high densities. There is considerable 
uncertainty regarding the exact nature of neutron star matter. This uncertainty is reflected in the types of EsOS 
employed in theoretical study of neutron star. In this paper, a comparative study of three different types of 
parametric EsOS is presented. Sensitivity of parameters like compression constant at saturation (K0) and the 
symmetry parameter (y ) on EOS and neutron star properties are studied and results are presented graphically. 
Keywords : Neutron star, nuclear matter EOS. 
PACS Nos. : 97.60.Jd, 21.65.+f 
1. Introduction 
Study of neutron stars is one of the important areas of modern astrophysics. Neutron 
stars are the smallest and densest stars known. The density in the neutron star can be 
up to ~7 times the equilibrium density p0 -0.16 fnrr3 (which is equivalent to matter density 
e0 ~ 4x1014 gm cm*"3) of charged nuclear matter in nuclei. Matter at such densities has 
not yet been produced in the laboratory; its properties must be theoretically deduced from 
the available terrestrial data with guidance from observed neutron star properties. Due to 
this fact, study of neutron star has its importance in nuclear physics also. Neutron stars 
are made up of relatively cold, neutral matter that has been fully processed by nuclear 
combustion, so that all available energy has been exhausted at each density. Such matter 
in its ground state is often referred as cold catalyzed matter. The quantities of interest, in 
study of neutron stars, are the phase and composition of cold catalyzed neutral dense 
* Corresponding Author 
©2007IACS 
794 C J Chammasheril and P C Vinodkumar 
matter, its energy density e(p) and pressure P(p), where P denotes the baryon number 
density. The baryon number is conserved in all known interactions. Therefore, it is 
convenient to find the composition by minimizing the total energy e(p) per baryon, including 
the rest mass contribution. This gives 
e(p) = pe{p), 
The equation of state (EOS) is found by eliminating p from eq. (1). 
A major part of the densities inside neutron star can be well represented by nucleonic 
degrees of freedom only. But the EOS for neutron matter, particularly at high densities is 
still not very clearly known. The large uncertainties are reflected in considerable variety of 
significantly different parameterizations which are applied for the description of structure of 
neutron star [1]. As for as neutron star structure is concerned the only relevant quantity 
is the energy per baryon e(p). From e(p), neutron star properties such as its radius, 
gravitational red shift, moment of inertia etc. can be determined as a function of mass [1]. 
If we consider only the nucleonic degree of freedom for such a study of neutron star, one 
can adopt a simple form for binding energy per nucieon in nuclear matter consisting of a 
compressional term and a symmetry term. In the present work we have made a comparative 
study of three different forms for compressional term £comp» fcomp» and e""mp • 
The compressional term ef0^p, as given by Sierk and Nix (SN) [2-4] has the following 
form 
4mP(P) = f M V ^ 2 , (2) 
whereas compressional term £° m p known as quadratic (Q) form is given by [5-7] 
^ ( P ^ Q M " - ! ) 2 . (3) 
and £ ^ p as given by Heiselberg and Hjorth-Jensen (HH) has the form [7] 
,HH u-2-S 
1 + 5u * c o m P - * o " — ~ - (4) 
In all the three cases u = p/p0 and Po = 0.16 fnrr3 is saturated nuclear matter density. 
The compressional term at P0 (ie. KJ is related to S by K0 = 18£0/(1 + <5) where e0 = 
-15.8 MeV is binding energy for symmetric nuclear matter at Po. 
Sensitivity of nuclear matter EOS parameters on neutron star properties 795 
The other important term in the binding energy per nucleon in nuclear matter is the 
symmetry term S(p). S(p) is defined as the difference in energy for symmetric nuclear 
matter (SNM) and pure neutron matter (PNM) and can be given as 
S(p) = £(p,x p =0)-<-(p ,x p =1/2) , (5) 
where xp is the proton fraction, which corresponds to ratio of protons as compared to 
total nucleon number (ZM), defined as xp = Pp/p. If we expand the energy per baryon 
in case of nucleonic degree of freedom only in the proton concentration xp about the value 
of the energy for SNM (xp =1/2), we obtain 
e(p,xp) = e(p,xp=V2)+ld2^{U2xpf+ (6) 
where d2e(p)/dx2p is to be associated with the symmetry energy S(p) in empirical 
mass formula. Various studies [8-10] have revealed that the energy per baryon of a neutron-
rich matter can be nicely approximated as 
*(P, xp) = e(p% xp = 1/2)+ S(p) ( l - 2 x * ) . (7) 
The symmetry energy term can be parameterized as [7] 
S(p) = S 0 u y ( l -2x p ) 2 . (8) 
One can show that the symmetry parameter xp can be given by 
(4^(12 , , , ) °
 <9) 
One can have an analytical solution of eq. (9) and get xp as function of p, but the 
soluvation as given by [7] seems to be incorrect. In the present work, we solve eq. (9) 
numerically which allows the inclusion of muon's contribution. For this calculation we take 
an empirical value of S0 = 32 MeV [11]. 
Here we calculate the bulk energy using a simple form of energy density of nuclear 
matter consist of a compressional term, a symmetry term, and an electron energy density 
as depicted below. 
e = p[m + £comp(p) + S(p)] + £e , (10) 
796 C J Chammasheril and P C Vinodkumar 
where eoomp(P) can be obtained using any of the eqs. (2, 3 or 4), S(p) can be obtained 
using eqs. (8, 9) and ee = n*/(\2n2) where Me, the chemical potential for electron is 
obtained using condition for charge neutrality and chemical equilibrium given as 
A*i>ssA*p+/'». (12) 
Me = / V (13) 
In all the three form of compressional term discussed before, K0 is the parameter. We 
have investigated the sensitivity of gross properties of neutron star of KQ and Y appearing 
in symmetry term. 
Neutron star structure can be calculated by applying Einstein's general theory of relativity. 
The space time geometry of a non rotating, spherically symmetric star can be described 
by a metric function having following form in Schwarzchild coordinate as 
ds? = -ef^dt2 + [1 - 2M(r) / r]"1 dr2 + r2 [d©2 + sin2 0o> 2 1 . (14) 
This equation can be reduced to well known Tolman, Oppenheimer and Volkoff (TOV) 
equation [7,12] 
dP(r) _ ( f ( 0 + p ( 0 ) ( M ( 0 + ^ r 3 P(r)) 
dr ~
 rzL_mn] ' (15) 
dM(r) .
 2 , , 
"dr ( r )" ( 1 6 ) 
Relativistic changes from Newtonian value are caused by the dragging of the local systems, 
the redshifts and the space-curvature. 
The surface gravitational redshift and Kepler frequency is given as [12] 
- 1 . (17) Z = | 1
- K 
2 M V 1 / 2 
K
 {(R/kmf) x104 a"1. (18) 
Sensitivity of nuclear matter EOS parameters on neutron star properties 797 
In the present work, we use eqs. (15-18) to calculate properties of neutron star sequence, 
like mass (M) radius (fl), surface gravitational redshift (Zs), and Kepler frequency (QK) 
2. Results and discussion 
There exist, in literature, study of neutron star properties at different values of the parameters 
of the EsOS like /^ and y [1,3,4,6,7,10]. Since, there are allowed range for the numerical 
values for the numerical values for K0 and y, it is interesting to study effects of variation 
(sensitivity) of these parameters on EOS as well as on neutron star properties obtained 
using them. In this paper, we have studied the sensitivity of the parameters like compression 
constant at saturation (K0) and the symmetry parameter (y) of three different types of 
parametric EsOS generally employed in study of neutron stars. We define relative sensitivity 
of pressure on parameter X(X = K0,y) as 1/P(4P/4X). In Figure 1(a) we show this 
sensitivity as function of KQ for a fix value of energy density t = 1000 MeV 
firr3 and y = 0.6 Similarly the relative sensitivity of pressure on y is presented in 
Figure 1(b) with same energy density and K0 = 250 MeV. 
0 OOO-i . 1 . 1 . r-
150 200 250 300 350 400 450 
Kft (MeV) 
I 
i r . *v -
0 1 5 -
0.10-
0 05 -
OOO-
-0 0 5 -
— r - ' t 
i ' ' '• » 
i - •• 
— HH 
~ o 
SN 
"lb>l 
J 
\ 
\ 
J 
— . , , . i I 
0 6 0 7 08 
Y 
0 9 1 0 
Figure 1. Relative sensitivity of pressure, (a) on K0 at e = 1000 MeV fm 3 and y a 0 6 and (b) on y at i = 1000 
MeV fm-3 and K0 = 250 MeV 
All the parametric EsOS discussed in this paper are being used to solve the TOV 
equations to get mass, radius and other properties of neutron star sequence. In 
Figure 2 (a and b) we present relative sensitivity of mass of neutron star on K0 and y 
respectively. 
From Figure 1(a) we observe that the sensitivity of pressure on K0 is always positive 
for all the three EsOS. Of the three EsOS discussed HH-EOS is the most sensitive to KQ 
and the way this sensitivity varies, is also very different form SN and Q-EOS. While 
sensitivity of SN and Q-EOS show similar behavior (Figure 1(a) and 2(a)). Sensitivity of y 
798 C J Chammasheril and P C Vinodkumar 
on pressure found to be positive or negative depending on value of y and e (Figure 1(b)). 
SIM-EOS is the most sensitive to variation in y followed by HH and Q-EOS. Neutron star 
mass obtained using these EsOS shows similar variation with y. We observe that, though 
pressure sensitivity on y can be negative, mass sensitivity on y is positive (Figure 2(b)) 
I.e. mass increases with increasing y. 
U . W f 1 
0.006-
^ 
O.OOSi 
> 
I 0.004-
2 
< 
H 0.003-
0.002-
0 . 0 0 1 -
0 0 0 0 -
• 1 ' 1 
•^  
"\ 
x
» ^  
— i r—• 1 — • — i — 
• i 
l.. . 
.,—r 
• " • « i 
— H H 
— Q 
•••• S N 
("»]j 
H 
J 
X. J 
0.25 
0.20-1 
0.15 
0.10-L 
0 05 
150 200 250 300 350 400 450 
K 0(MeV) 
06 
Figure 2. Relative sensitivity of masses of different neutron stars sequences at central energy density ec = 
1000 MeV fnrr3, (a) for y = 0.6 and (b) for K0 = 250 MeV on KQ and y, at e = 1000 MeV fm"3. 
In Figure 3, we present limiting mass properties of neutron star obtained using all the 
three EsOS studied in this work. In Figure 3(a) we plot maximum stable mass /W|imit> in 
units of MQ, as function of KQ. In Figure 3 (b, c and d), we plot, respectively, neutron 
star radius (fl„m,t), surface gravitational redshift (Z^um)) and Kepler frequency (0K{Um)), 
all corresponding to maximum stable mass. From Figure 3(a) we found that, Q-EOS 
being most stiff among the three that gives highest value of maximum mass, but the HH-
EOS is the most sensitive to variation in KQ for mass, radius and gravitational redshift. In 
the case of Kepler frequency all the three parametric EsOS shows similar variation with 
Ko-
to Figure 4, we present limiting mass properties of neutron star as function of y. In 
Figure 4 (a-d), we plot, respectively, neutron star mass (Mlimrt) radius (A?limlt), surface 
gravitational redshift (Z^J and Kepler frequency (oK{Um)), all corresponding to 
maximum stable mass. From Figure 4(a) it can be seen that, maximum mass varies very 
little with Y, while Figure 4(b, c, & d) shows radius (fl||mit), gravitational red shift ( Z ^ ^ ) 
and Kepler frequency ®K(um\t) relatively more sensitive to variation in y. 
Sensitivity of nuclear matter EOS parameters on neutron star properties 799 
0.70 
0 65 
060 
| 0 55 
050 
045H 
040 
200 
K.fMcV) 
220 240 
Ko(McV) 
VMeV) 
i.uo-
106-
• 
~ 104-
CO 
O 
>
—' 
f 1.02-
sr ^M 
CT 
100-
0.98-
' i • ; i 
V 
HH 
SN 
M| 
\ 
H 
i 
J 
k^  > ^ H 
^ ^ \ 1 
** ^^^N*. J 
" " ^ ^ 1 
^ 1 
• • • "• • — — 1 
220 240 
K.CMcV) 
260 
Figure 3. Variation of limiting mass properties of neutron star with /C0 
3. Conclusion 
From the present work we conclude that HH-EOS is the most sensitive to variation in K0, 
particularly in the experimentally predicted range of KQ (200-300 MeV). Here we find that 
though the expression for symmetry term is the same in all the parametric EsOS, it 
manifests differently. For example here we find stiff EOS is less sensitive to variation in 
800 C J Chammasheril and P C Vinodkumar 
N' 
0.65 
0.60 H 
0.55 
0.50 
0.45 
0.40 
Wj 
-Q 
•HH 
SN 
— r | i | > | " • ' 
0.6 0.7 0.8 0.9 
11.0 
10.5 
* lo.o-r 
9.5 
9.0 
, , 1 ^ — | r 
1 ~~*~" 
1 -*'*' 
J ..••'" 
» 1 ' 1 '" 
1 r 1 
.A 
Q 
HH 
SN 
" " i '— 
J 
0.6 
J 
1.04 
1.02 
1.00 
0.98 
0.96 
0.94 
0.92 
0.7 0.8 
Y 
0.9 1.0 
1 ' 1 ' 1 •-
1 
1 ^ " * * " * > f c ^ * » 
J > 
1 **^ 
1 ' 1 ' 1 ' 
} ' — 
V 
HH 
SN 
i * 
m 
\ 
/•-
.
 
.
 
/_
L
..
..
 
, 
0.6 0.7 0.8 0.9 1.0 
Y 
Figure 4. Variation of limiting mass properties of neutron star with y. 
Y and soft EOS is more sensitive. We also found that, variation in y does not affect the 
limiting mass Mlimlt much but does have influence on f?ljmjt, Z^m and 0Kmn). 
References 
[1] W Arnett and R Bowers Astrophys. J. Suppl. Series 33 415 (1977) 
[2] A Sierk and J Nix Phys. Rev. C22 1920 (1980) 
[3] S Khadkikar, A Mishra and H Mishra Mod. Phys. Lett. A10 2651 (1995) 
Sensitivity of nuclear matter EOS parameters on neutron star properties 801 
[4] A Rosenhauer, E Staubo, Csernai, T OvergSrd and E Ostagaard Nucl. Phys. A540 630 (1992) 
[5] H St&cker and W Greiner Phys. Rep. 137 277 (1986) 
[6] H Heiselberg, C J Pethick and E Staubo Phys. Rev. Lett. 70 1355 (1993) 
[7] H Heiselberg and Hjorth-Jensen Phys. Rep. 328 237 (2000) 
[8] R Wimga, V Fiks and Fabrocini Phys. Rev. C38 1010 (1981) 
[9] M Prakash and T Ainsworth Phys. Rev. C36 346 (1987) 
[10] H Muther, M Prakash and T Ainsworth Phys. Lett. B199 469 (1988) 
[11] P Haustein At. Data Nucl. Data Tables 39 185 (1988) 
[12] N Glendenning Compact Stars (Springer : Verlag) (2000) 


Sensitivity of Nulcear Matter EOS Parameters on Neutron Star Properties

Forthcoming Publications

Shah, A K

Gajjar, P N

Indian J. Phys. 81(1) 131-135(2007) 
Growth and characterization of tungsten substituted 
molybdenum disulfide 
A K Shah and P N Gajjar* 
Sheth R A Bhavan's Science College, Khanpur, Ahmedabad-380 001 Gujarat, India 
* Department of Physics, University School of Sciences, Gujarat University, 
Ahmedabad-380 009, Gujarat, India 
E-mail pngajjar(fJJrediffmail com 
Abstract : Single crystals of Mo0 5W 0 5S2 have been grown by chemical vapour transport technique with iodine as transporting 
agent The identity of the grown crystals has been verified using EDAX technique Electrical resistivity measurements have been 
earned out using Van der Pauw technique The resistivity at room temperature along the basal plane has been found to be 10 62 
ohm cm The temperature variation of electrical resistivity for low temperature (70-300 K) and high temperature (313- 443 K) is 
reported The Hall effect measurements on the grown crystals have also been earned out thereby evaluating the Hall mobility 
and carrier concentration in the grown crystal The obtained value of negative Hall coefficient conform n type nature of the crystal 
having carrier concentration 2 2 x 1014 cm 3 
Keywords : Crystal growth, electrical resistivity, Hall measurement 
PACSNos. : 71 20 Be, 81 10Aj, 81 10 Dn 
1. Introduction 
Transition metal dichalcogenide of Group IVB, VB and VIB elements have received a 
fair amount of attention due to their striking feature and considerable diversity in most 
of their physical properties [1,2]. Tungsten substituted molybdenum disulfide possess 
semi conducting properties which has been used as energy conversion devices as 
photo-voltaic/photo catalytic solar energy converter, Schottaky and liquid junction solar 
cell, catalysts in many industrial applications and secondary batteries [3-10]. Due to 
property of the compound that atomic planes can slide over one another in its 
structure, it found important application as solid-state lubricants [11]. The layered 
structure of tungsten substituted molybdenum disulfide facilitates the process of 
intercalation or insertion of foreign atoms and molecules into host materials allowing 
thereby convenient methods for altering structure and electronic behavior [12,13]. MoS2 
and WS2 of Mo and W is identical and very close as a result due to lanthanide 
contraction which suggests that the amorphous single phase of Mo^WxSa mixed 
crystal could be obtained over entire range of composition [14-20]. All these facts have 
motivated us to undertake the study on growth and characterization of tungsten 
Corresponding Author © 2007 IACS 
132 AKShahandPNGajjar 
substituted molybdenum disulfide. In this paper we report the study of electrical 
resistivity, Hall mobility, Hall coefficient and carrier concentration of single crystals of 
MO0.5W0.5S2. 
2. Crystal growth 
Single crystals of Mo05W05S2 have been grown by chemical vapor transport method 
using iodine as the transporting agent [20,21]. A 10 gm mixture of Mo, W (purity 
99.95%) and S (purity 99.99%) (Koch light laboratory, England make) was filled in 
thoroughly clean quarts ampoules. The quartz ampoules used in the present crystal 
growth have dimension of 22 mm in internal diameter and 180-200 mm in length. A 
small amount of transporting agent iodine was introduced inside, sufficient to give 
concentration of 5 mg/cm3 of ampoule volume. Then ampoule was sealed at the 
pressure 10~5 torr. The sealed ampoule was introduced into two-zone muffle furnace at 
a constant reaction temperature 850°C for five days to obtain the charge. The charge 
so prepared was rigorously shaken to ensure proper mixing of constituents and kept 
in the furnace again for six days with reaction zone at 1200°C and growth zone at 
1130°C temperatures. After this, slow cooling up to room temperature was followed. The 
opaque black shining crystals of the size (4 to 0.55 mm), (4 to 0.6 mm), (0.066 to 
0.002 mm) are obtained. The crystals obtained in the form of flat platelets with straight 
edges and corners having mirror-like surface. Since the crystals grown in the form of 
thin platelets directly over the charge, they are strain-free. The composition of grown 
crystals was studied with the help of energy depressive analysis of X-rays (EDAX). 
LEO 440i with stereo scan 440 computer controlled digital scanning electron microscope 
having 3 x 106 magnification power with 3.5 nanometer resolution was utilized to 
confirm stochiometry of grown crystal. This EDAX analysis has confirmed the stochiometry 
as MOo.5W05S2 From the X-ray diffraction powder method, the evaluated lattice parameters 
for the grown single crystal of Mo0.5Wa5S2 are a = 3.1677± 0.010 A, c = 12.305 ± 
0.0250 A. 
3. Electrical • resistivity 
The room temperature electrical resistance of grown crystals was measured using 
Vander Pauw method. Electrical connections for resistivity measurement were made by 
four copper wires on peripheries of very thin crystals and attached to the crystal 
surface by means of conductive silver paste to ensure the ohmic contacts. The sample 
was flat so that the sample contacts lies in the same plane. The applied electrical field 
was low enough to avoid any kind of breakdown in the sample. Then the resistivity of 
crystals was calculated by 
where d is the thickness of the crystal and R be the measured resistance of the 
crystal parallel to the G-axis. Thus obtained room temperature resistivity of the crystal 
perpendicular to G-axis along the basal plane was 10.62 ohm.cm. 
Growth and characterization of tungsten substituted molybdenum disulfide 133 
The measurements of low temperature electrical resistivity along the basal plane 
were carried out on a four probe set up designed by cryostate LNDP Scientific 
Solutions, Mumbai in the temperature range 77-300 K. The results of low temperature 
electrical resisitivity of MO05W05S2 are shown in the Figure 1. 
2.4-r 
2.3 4 
~ 2.2 \ 
& 2.1 H 
1.9 \ 
1.8 4 
0.002 0.004 0.006 0.008 0.01 
1/T(K) 
Figure 1. Low temperature resistivity of Mo0 5W0 5S2. 
From these measurements, the activation energy for the grown crystal was 
calculated using 
P = P 0 e x p | | 
and toOP-togft+lsJ^jiT. <2> 
where p is the resistivity, po be the constant, Ea the activation energy for electrical 
conduction, kB the Boltzmann constant and T is absolute temperature. Hence from the 
slope of the graph of log p versus 1/7, the obtained value of activation energy is 0.015 
eV. 
The high temperature electrical resistivity parallel to basal plane (pu) was 
measured in the temperature range 303-443 K. A four probe set up manufactured by 
Scientific Equipments, Roorkee was used for the purpose. The measurements of 
electrical resistivity normal to basal plane (p±) were carried out on an experimental set 
up designed and prepared by University Science and Instrumentation Center (USIC), 
Sardar Patel University, Vallabh Vidyanagar. Starting from room temperature (303 K), 
the temperature of sample was increased slowly in steps of 10 K until temperature 
reached to 443 K. At each step the corresponding value of resistivity was calculated. 
The results for high temperature electrical resistivity (p„) and (px) of M005W0.5S2 are 
shown in Figures 2 and 3, respectively. 
The activation energies were also obtained from the graphs and we found that 
in the temperature range 313-383 K it is 0.1818 eV and for temperature range 
y = 78.57x + 1.6301 
R2 = 0.9833 
134 AKShahandPNGanar 
383-443 K it is 0.4232 eV. Anisotropy measurements have been carried out in the 
temperature range 303-443 K and the results are displayed in Figure 4. The anisotropy 
ratio ((p±)l(pu)) increases linearly with increase in temperature. 
38 
36 
* 3 4 
> 
% 3 2 
S 3 
28 
y = 916 45x + 0 7063 
R2 = 0 9869 
35 
3 2 
15H 
1 
y = 2133 4x -2 9554 
R2 = 0 9886 
26 
0 002 
0 0015 0 002 0 0025 0 003 0 0035 
0 0025 0 003 0 0035 
1/T(K) 1/T(K) 
Figure 2. High temperature resistivity (pu) of Figure 3. High temperature resistivity (pL) of 
Mo05W05S? 
Mo0*W05S2 
0 1 
^ 0 08 
Q. 
2 006 
o 
<n 
c 0 04 i 
< 
0 02 
y « 0 0005X - 0 1144 
R2 = 0 9664 
300 350 400 450 
1H"(K) 
Figure 4. The anisotropy ratio ((p±)/(pn)) in Mo05W0SS2 
4. Hall mobility, Hall coefficient and carrier concentration 
The Hall mobility of the crystal was determined by measuring the change in resistance 
Ft upon applying magnetic field perpendicular to basal plane of the sample [20]. The 
Hall mobility, Hail coefficient and net carrier concentration as obtained is -2582 cm2 
voir1sec-\ -27549 cm3 cour1 and 2.2 x 1014 cm"3, respectively. The obtained value of 
negative Hall coefficient conform n type nature of the crystal. 
5. Conclusions 
The single crystals of M005W05S2 have been grown successfully by chemical vapour 
transport technique. The Hall mobility and net carrier concentration as obtained from 
Hall effect measurement is -2582 cm2 voir1 sec"1 and 2.2 x 1014 cm""3. The value of 
activation energy was obtained in the range of 0.1818-0.4232 eV in the temperature 
Growth and characterization of tungsten substituted molybdenum disulfide 135 
range 313-443 K. The negative value of Hall coefficient indicates that the Mo05W05S2 
crystals grown in the present work are n-type. 
References 
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[2] J Pouzet and J C Bernde Metter Chem. Phys. 36 304(1994) 
[3] A A Balchain and F Levy (eds) Crystallography and Crystal Chemistry of Materials with Layered Structure 
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[4] F Hulliger Structural Chemistry of Layered Phases (Dordrecht: D Rfidel) (1976) 
[5] H Tributch J. Electrochem. Soc. 125 1087(1978) 
[6] R Span, M Lux-Steinr, M Oberfell and E Bucher Appl. Phys, Lett. 47871(1985) 
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[8] R R Tenne and A Wold Appl. Phys. Lett. 110 707 (1985) 
[9) Wiesser and S Landa Sulphide Catalysts, their Preparation and Application (Oxford : Pergamon) (1973) 
(10] M S Whittingham and F Levy Intercalated layered material (Dordrecht: Reidel) p307 (1979) 
[11] G V Suba Rao and C S Sunanduna in J M Honing C.N.R.res Preparation and Charaterizatlon of Materials 
(London : Academic) p269 (1981) 
[12] G V Suba Rao and M W Schafer in F Levy (ed) Intercalated Layered Materials (Dordrecht: Reidel) p 99 
(1979) 
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Growth And Characterization of Tungsten Substituted Molybdenum Disulfide

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Rhombic Patterns Near a Bicritical Point in Periodically Forced Surface Waves

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Growth And Characterization of Tungsten Substituted Molybdenum Disulfide

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