Chapter 5
Ultrasonic study of elastic properties and phase transitions in Potassium Sulphamate crystal
The siiidies currietl out on the elastic properties of or/horhor~tbic Potcrssiuin
.sulph(iirrcite cg,srti/ tire described in this chapter. The synthesis of tile iiiaferiul,
~1urificliiir1)7. oys fo i grow~rh, identification ofthe r~zorpho1og)i of the gro,l,ri cr~~s to l
cind /~re/~c~ratiorr ( i f the sj~ecimen are described All ihe nine indeperidei~t second
order elcistic .iiill,ies.s constants, compliance constant, Poissoi?'.~ i.ti/io.s [ire
estir~ici~ed A /ho~.oiigh st~idy ofthe temperalure vnriu/ioii of elasfic constnr~ts over
ti rui~go qf iOli h-400 K has heen undertuken. /;'or tile sub.s~cii~/iiitio,i of /lie
ii/ti.(i.~oiii(~ .t/ii(!). Oi/fireriliul cunning calorin7etric .stirdy a/ cr very .~loli' iiecitirig
id? i . ~ (i/.so coii~ii~c~eii iirid presented.
Ultrasonic study of elastic properties and phase transitions
in Potassium Sulphamate crystal
5.1. Introduction
I'otassium sillpharnate is a very interesting piezo electric crystal. The
structure of the ct-!'stal has been studied using x-ray diffractio~l technique by
Brown and Cox [ j . l ] and Jeffrey and Stadler [5.2]. In both cases authors
suggested that the nitrogen valency directions are co-planar and that hydrogen
atoms lie in the smne mirror plane of symmetry of the molecule. 01dy Little
infor~naiion about crystallographic and physical properties of Potassium
sulphamatc crystals can be found in the literature. Hence the elastic properties of
these saniples using ultrasonic Pulse Echo Overlap (PEO) technique are carried
out. From earlier reports [5.1,5.2] it. is obvious that it crystallizes in the
orthorhombic symmetry with space group Pbcm [ ~ 2 h l ' ] with lattice parameters
a = 5.907b., b = 8.333A and c = 8.302A, Z = 4 molecules/unit cell. Density =
2.2078 gmlcc. Molccular volume = 102.16 x 10." m3 The illtermolecular 0-H
distance of Potassiuln Sulphamate is found to be 2.15 which is sig~iificantly
less than the van der waais contact distance of 2.8 A and constitutes a weak
hydrogen bond. 'I'hc prominent hkl planes of the crystal are [-loo], [OlO], (1201,
[I lo], [ I 001, [I -201, [O-1 01 [-]-lo], [OO-11 and have perpendicular distances
iioni thc origin as I . 14.1.49,1.04,0.97,1.14,1.68,1.75,0.09,1.7~ respectively. The
investigation shows conclusively that the bond angles around the nitrogen atom
are S-N-lI 1 1 0 . 2 and H-N-H llO.lO. These are close to the ideal tetrahedral
angle of 109.5" ant1 it must be assumed that the nitrogen atom has a S P ~
configuration with a lone pair electron in the fourth tetrahedral position. The
conclusion is in disagreement with the expected relationship between bond length
and bond order for the S-N bond. In potassiu~n sulphamate the S-N bond is 1101
significantly longer but thc configuration is SF".
Cox et al. [5.4] carried out neutron diffraction study on the crystal
structure of Potassium Sulphamate. In tneir study it was observed that the
hydrogen bonds were so disposed throughout the structure that no well-defined
cleavage was found in any direction of .he crystals. They also observed that
sulphamate ion in potassium sulphamate was not switterionic. It has pinacoid in
the directions [loo], [OlO], [OOl], prisms iri the directions [120], [ l o l l , dipyramid
in the direction [ l 111.
The thermo elastic constant T, and anisotropy of the thermal expansion a,,
and Gruneisen tensor G were studied 3y Haussiihl el ul. 15.31. Potassium
Sulphamate possesses large values of Ti] since it has strong Hydrogen bond. Thc
ui, of potassium sulphamate is smaller than in perchlorates or tetrafluroborates
and G is having almost the same valuc: as that of perchlorates. The elastic
constants of this crystal were measu~ed by using Ultrasonic Resonance
Technique [5.3].
DSC studies by Rapp [5.6] suggested that the crystal exhibits a first order
phase transition at about 450 K.
In this chapter, measurement of elastic stiffness constant by PEO
technique and investigation of the phas: transition of Potassium Sulphamate
crystal above room temperature of the cr),stal are presented, since no such study
using ultrasonic PEO technique is repor:ed in the literature. Measurements of
elastic constants have proved to be an excc:llent probe for phase transition studies.
Elastic constants are measured using PEO technique. The anisotropy in the elastic
properties of Potassium Sulphamate is well studied by measuring ultrasonic
velocity in the crystal in certain spe;ified crystallographic directions and
evaluating all the nine Elastic stiffness constants, Compliance constants and
Poisson's ratios and the results are presen.ed in Section 5.2.3. In Section 5.2.1 the
details of sample preparation are given and in Section 5.3.4 the surface plots in a-
h. a-c :incl I>-c pl;inL,, 01 ' phase velocity, slowness, Yo~ing's n~odulus ailti lincar
compl-cssihility art gilten.
5 .2 Experimental Technique
I a g c single crystals of size ( 35x30~12 ) mn13 have been gr01v11 from
supersat~~rated ailueous solution of the salt by slow evaporation technique ovei- a
period ~1 '60-65 days. The solution was prepared by dissolving equimolar fraction
of I(2CO; and Sulpl?;linic acid in double distilled water
The tcinl,erature of ~ h c bath was maintained constant at 305K and col~t~.olIed to
an accuracy 01' i 0 ' ' ' K . The details of the apparatus used for thc crystal growth
were ilcscribcd in scction 2.2. The photograph of the grown Potassiuln
Sulpha~ncitc crystal i s as shown in Figure 5.1. A drawing of tlie morpliology of'
the cr)stal is sliown in Figure 5.2. At 305 K the solubility is 70.4
g~~~l lOO~i i l . l - l~C) .~f l ie solubility curve is a s shown in Figure 5.4. The solubility
reaches a saturation value near 305 K. The positive temperature coefficient
i~~dica tcs that thc crystal can be grown by slow evaporation technique. The
arrangement of inolccules in the unit cell o f the crystal viewed along a- axis is
depicted in Figure 5 . 3
Figure 5.1 Photograph of grown Potessium Sulphamate Ctystal
Figure 5.3 Arrnngernent of molecule.^ in the Unil cell oJPota.vsium Sulyhanmte crvstnl ubozr~ a4mis .
Figure 5. B Morphology of Potassium Sulphamate Crystal
4 -
Figure 5.4 Soiubii~ty curve of Potassium sulphamate crystal
Tabfc 5 1 Comparison of the computed Mferfacial angles of the crystal with the
nieasured value.
I
( Crystal faces
1 100- 110
Interfacial angle between the faces
1 iO0-i l i
I- I r i00- i i i I -
102- i 02 t I 110-i10
021-02i t I 101-111
Computed
144.7
Measured
144 I-.
135
ioo-ilop 125 2
I35
70 4
70 4
104 5 --- 135
135
124
135
70 1 70
104
135
135 100-1 1 1 135
F i q u ~ 5.5 (a) Stereographic proaction of Potassium Subhamate crystal about c-axis
4
F@um 5 3 ( b ) S b ~ m p h i c pmjgcffun of Potassium Sulphamate crystal about h x i s
I'hc iirtcl-facial angles of the crystal were measured using an accurate
co~itact golriu~~lc~cr. Uy k110~~illg the lattice parameters, crystal systc111 and spacc
group one can construct a stereographic plot (Figures 5.5(a)-5.5(c)) by using thc
computer programme '.[crystal'. The natural faces of the sample have been
identified by tlie technique discussed in Section 2.2.3. The samples have been
cut using a slow speed diamond wheel saw. The samples have been cut so as to
have propagation directions along [loo], [OlO], [OOl], [ l o l l , [I 101 and 101 I].
The sample lengths along the measured directions are in the range of 0.8 to 1.2
cni. Since all c~lttings are niade very accurately, the error due to misorientation is
below -I 0.5". Thc samples are polished cautiously using ceriuni oxide powder to
optical reflection icvel so as to ensure good bonding of the transducer to the
sample surface.
5.3. Results and discussions
5.3.1 Velocity measurements
X-cut and )'-cut quartz transducers of fundamental frequency 10 MHz are
rnountcd on the sa~iiple using suitable material like silicone grease. Silicone
grease is a good bonding material in the range 300 K-420 K. Absolute velocities
at room tempcraturc (303K) have been measured for the selected direction and
modes. Details of measurement of the elastic constants of orthorhonibic crystals
are reported in the Section 1.2.1. The ultrasonic velocities are measured using thc
PEO technique [5.71. MATEC model 7700 pulse niodulator and receiver systeni
with its associated subunits have been used for the velocity measurements and tlie
details are described in the literature [5.8]. The basic experimental set up is
described in Scctioli 2.1
'fhe McSkimin At criterion [5.9,5.10] for bond correction has been
applied ~ising conil~utcr programme [5.16] to correct for the phase lag introduced
by the bonding rnidiunr o n the RF echoes. The details of the McSkirnin At
criterion and the dc~ails of the lnethodology of tlie computer programme are
discussed in Scctioir 2.1. Taking into account the uncertainties in measuring the
length and various other experimental limitz.tions, one has obtained an absolute
accuracy better than 0.3% in velocity nleasurr:ments.
KNH2S03, being an orthorhombic clystal, has the following nine second
order Elastic stiffness constants C1j, C22, C3), C44, C55, C66, C12, C13 and C23. The
diagonal elastic constants C, ,, C 22, C]), C44, C55 and C66 have direct relationship
with the suitable ultrasonic mode velocity given by C,; = p ~ 2 . It is found by PEO
technique that no cleavage is present in an!, direction of the crystal as suggested
by Cox ei al. [5.4].
Table5.2 Measured ultrasonic velocities and elastic constants of Potassium Sulphamate crystal at 300K L, T and QL represent longitudinal,
trarlsverse and quasi longitudinal modes respectively
$1. No
Direction of propagation
Direction Velocity of measured
polarisation 1 (mls)
Elastic constalrt
( G P ~ ) V-C, ~relatiori
Table. 5.3 Elastic stiffness constants, Elastic compliance constants, and PoissonS
ratios of Potassium Sulphamate at 300K
Elastic siilliiess Elastic colnpliance 1 constallt (GPa) constant( l~~ '~~n~N") Poisson's ratio
I'hc relationship between elastic constants for relevant ultrasonic wave
velocity for the orthorhombic system is explained in Section 1.2.1. The off
diagonal constants have the following relationship.
1 2 c,, =t.. =j7le2cl, + c 2 c S 5 -PV,, )e2c5. +c'c,~ - p v I 2 )I -'55 . ( 5 . 3 ) C S r
where s = sin 0 c = cos 8 where v is the velocity of propagation of respective modes. The
angle 0 = 3 5 l 3 ' for. a-b plalie, 0 = 45' for b-c plane and I3 = 35' for a-c plane. 'The
density of potassiu~ii siilphaiiratc is p = 2.207gmlcc
Of thc 18 propagation modes, 12 .ire sufficient to evaluate all the ninc-
second order elastic constants and rest of t l ~ e modes can be applied to cross check
the measured constants. Considering all experimental uncertainties the absolute
accuracy of elastic value is estimated to bt: better than 0.2% for diagonal elastic
constants and 1% for off diagonal elastic cc~nstants.
Starting with the well-known Christoffel equation [5.16], one can deduce
the relationship between the elastic cor~stants. Velocities of propagation of
various ultrasonic modes measured along, selected directions in the crystal arc
listed in Table 5.2. By measuring ultrasoni,: velocity in the Potassium Sulphamate
crystal in certain specilied crystallographic directions, the anisotropy in the
elastic properties of the c~ystal is studkd and the elastic stiffness constant.
compliance constants, Poisson's ratios [5.14,5.15] are evaluated and tabulated in
Table 5. 3. There are nine values of compliance constant which are the
components obtained from the matrix invt:rse of elastic constants. Also there are
six values for Poisson's ratio in orthorhombic crystal. The equations for
Poisson's ratio are discussed in Section 1.4.5
Some elastic stiffness constants c~f this crystal measured by this study
(PEO method) show appreciable deviation from those measured using Resonant
Ultrasound Technique [5.3] (RUT) Out of the diagonal constants, the constants
C 1 1 (0.91%), C 33 (4.8%) and Cd4 (1.9Y0) show deviation below 5%, wliercas
constants C2? (7.7%), Cj5 (41.6%) andC(6 (60.7%)exhibit deviation above 5%.
While off diagonal constants C 1 2 (21.5%), C13 (375%) and Cz3 (268%) havc
exhibited large deviation fro111 the RUT values.
5 . 3 . 2 Temperature variation of elastic constants
The temperature variation of the velocity of longitudinal and shear waves
propagating along various directions in t ~e crystal have been deterluined in the
range 300K-400K by keeping the sample mounted on a suitable holder in a
temperature controlled chamber. The r.ite of temperature change i l l all the
measurements is in the range of 0.5 to 1 K per minute. Investigation beyond
400K was not conducted because of bonding problems. The thermal expansion
has been negiected while measuring the variation of ultrasonic wave velocities
with temperature.
Figure 5.6 Temperature variation of C,, and C,, of PS
Elnstic rr~tottinlies urounrl355K
I'he variations in the elastic constants, CI 1 , Czz, C;,, C j j and C66, with
tenlpcrature are investigated and displayed in (Figures 5.6-5.8) The transverse
elastic conslants C41.Cjj and C66 show weak a~~oma i i e s around 355K while the
longitudiilal elastic constants Cli, C22 and Cjj do not show any variations. This
rucans that shearing caused by tangential stress is considerably affected by
thermal energy. Sincc variations of shear elastic constants with thermal energy is
appreciable. From the observed anomalies in the constants a weak phase
transition in the crystal around 355K is suggested. The thermal hysterisis during
cooling and heating cycles through transition point is about 3K.
Present Differential Scanning Calorimetric spectru~u (Figure 5.9)
performed at a very slow heating of 1°/rnil~ shows minor feature centered on
340K (67 '~ ) . The energy associated with this weak feature is 0.8892J/g. The
DSC study do not support a phase transition in potassiu~ii sulpliamatc near 355K.
This may be due to the fact that the weak anomaly around 355K may not be
associated with appreciable thermal change, where as the sensitive ultrasonic
technique can detect such a weak anomaly. Studies of Rapp [5.7] in DSC found
that potassium sulpharnate exhibits a first clrder in the temperature range about
450K. Thermal expansion study of CsNIi2S03 by Haussiihl et,al. [5.3] along
[OIO] direction showed sharp increase i l the value of thermal expansioti
coefficient at about 350K. A similar pheno~nenon was found in betaine NI-L;SO;
where a steep increase in the value of thernal expansion coefficient was found
along [001] direction near 360K. Both cases indicated the occurrence of first
order phase transition. In view of the observations fro111 other family rne~ubers of
Sulphamates one can suspect a phase transision for Potassium Sulphamate crystal
around 355 K.
Figure 5.7 Temperature variation of C,, and C,, of PS
1 1 6
Figure 5 8 Temperature variation of C2, and C33 of PS
I050 300 320 340 360 380 400
Temperature (K)
5.3.3 Investigatio~~ of phase transition using IISC
The thermal changes linked with .his crystal by the inetiiod of Ilift'crential
Scanning Calorimetry (DSC) have been obscrved in the range 30-I0O0C at a slow
heating rate of lo/n~in.
I A - YJ 0 3 TtC , CC
Tmo.rrtun, ( ' E ,
F~gure 5.9 DSC scan of Pot 3ss1urn suipharnate crystal
5.3.4 Surface plots of Phase velocity, Slowness, Young's modulus and Linear compressibility
The anisotropy of elastic wave I,ropagation in KNHlSO, single clystal
can be made clear by drawing the phase velocity surface plots in the a-b, b-c
and a-c planes following a well known ~rocedure [5.13,5.15]. In Section 1.2.1,
the expression for the velocities in the symmetry planes of the orthorhombic
lattice is given. If the velocities are cal2ulated using the corresponding elastic
constants for different values of angle 0 :n the range from 0 to 360 degrees with
some small step like 0.Io, then the resulting velocities can be plotted to get a
phase velocity surface for the correspond ng plane. For plotting the curve on x-y
plane of thc paper, one has to convert the velocity angle H to s- and y- co-
ordinates. These convet-sions are perfo -med using Equations 3.13 and 3.14.
[Figures 5.10(a)- 5.10(c)] show the phase velocity surface in the respective
plants, in which (a) gives the ultrasonic mode corresponding to quasi -
longitudinal [QL] inode with higher velocity of propagation (b) and (c) represent
pure shear [PSI and quasi - shear [QS] modes respectively.
I:rom the above plots, the three velocities in any direction of the
symmctl-) plancs can be easily found out by measurilig tlie length of tlie straight
line drawn Srom llic center to the curve at the required angle from the symmetry
axis. Thi: graphs Iiave bcen plotted with a scale of 1 cm for 1 klii 1s of velocity.
A greater insight into the elastic anisotropy of a crystal can be obtained
by plotting the in\,crse phase velocity (slowness) surfaces. The surfacc plots of
slowncss for KNH2S03 clystal are plotted in [Figures 5.1 l(a)- 5.11(c)]
Phase velocity-XY Plane 6000 I I I I
L 1 x x
. n o L I I I I I I -60U0 -4000 -2000 0 2000 4300 6000
Phase velocity (r/s)
Figure 5.10 (a) Surface plots of phase velocity ~n the XY plane of PS
Phase velocity-XZ plane
6000 7 I L 1 xx
- -,
600%00u - a 0 - lorn o idm $00 adou
Phase velocity (r/s)
Figure5.l0(b)Surface plots of pha se velocity in the XZ plane
Phase veloc:~ ty-YZ Plane 6000 7 ' L xx 1
- I- 600!600~l -3000 0 3000 6000
Phase velocity (=Is)
Figure 5.10 (c) Suflace plots of phase velocity in the Y Z plane
U11rii.soiiic . , / i i<(~ ~,/Elo.\iic properlies o~ldphosc iratlsiiio~is i t 7 I'oio.s.~iiit~i Szrlplioi~~ure crysrul -- 201
Slovness-XY Plane o .lo-+ I I ' I. xx
QL ++
- 3 10.' - rl \ Vi - L'? Vi IU U CI b 0 -.
Ln . -4 -I 113
Figure 5.17 (a) Surface plots of jnverse phase velocity (slowness) in the XY plane
I I I I -4 .lo-' -3 0 3 10.' 6 10.'
Slovness (sf=)
Figure 5.11 (b) Surface plots of inverse phase velocity (slowness) in the XZ plane
S 1 o w n e s : ; - Y Z P l a n e
F~gure 5.11 (c) Surface plots of inverse phdse velocity (slowness) in the Y Z plane
Young's Hodulus (x 0.1 GPa)
-4 -2 0 2 4 Young's Hodulus
Figure 5.12 Surface plots of Young's moduli in the XY, XZ and YZ planes
~ ~ Linear Compressibility Id I I
I XY xx I
Linear compressibility
Figure 5.13 Surface plots of l~nearcompressibility in the XY, XZ and YZ planes
The velocity surface plots alone cannot completely describe the
anisotropy of tlic elastic properties of a crystal. Young's niodulus s ~ ~ r f a c e plots
are vcry inipor~ant in this regard. The Young's modulus E [5.13] in thc direction
of unit vector i i i for an orthorhombic crystal is given by the equation
Tlie cross scctionj of Young's moduli.. surfaces of KNH2S03
plotted in the a-b, b-c and a-c planes are shown i n (Figure 5.12). The
linear conipressibility of an orthorhombic c~ysta l in matrix form can be
writtct~ as
[i = [SI , + ~ , ~ + s l 3 ] n l ~ + [ ~ 1 2 + ~ 2 2 + ~ 2 3 ] n 2 ~ + [ ~ 1 3 + ~ 2 3 + ~ 3 3 ] n 3 ' (5.5)
The lineal- compressibility of KNH2S03 crystal in the a-b, b-c and a-c
planes Ihave bccn plotted. The plots are as sl?owli in (Figure 5.13). The
I'oissoii's ratios [5.14,5.15] have been evaluated using the equations as csplained
in Section [1.4]. The vol~iii~e cornpressibilit:j Siikk is an invariant parameter for
a crystal. In matrix notation i t is given by
Where S,, 's are the corresponding compliance constants
Hence bulk lnodulus of the crystal is given by K = i/Si,kl (5.7)
Volulne Compressibility of the cryst.11 = 0.45 ~ 1 0 " ~ ~ . ' n r n ~
Bulk modulus of the crystal = 22.22 G Pa.
5.4. Conclusions
PEO technique has been effectivel:! iniplelnented for evaluating all the
nine elastic constants, compliance constants, Poisson's ratios, bulk modulus and
volume compressibility of Potassium sulphamate crystal. The anisotropy in
elastic properties are well studied by the surface plots of phase velocity,
slowness, lincar compressibility and Young s modulus.
PEO technique is successfully implemented in studying the phase
transition in Potassiurn sulphamate single crystal. The variations in the elastic
constants, C I I , Cz2, C33, C44, CSS and C6(,, \vith temperature are investigated. The
transverse elastic coiistalils Ci14 C j j and Ch6 have exhibited iniilor anomalies
around 355 K suggest a possible weak phase transition. But DSC does not
support this possibility. More work is in progress about the nature of the phase
transition occurring in the crystal.
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