Correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters
In the paper, a correlation between acoustic velocities V, elastic moduli M, and densities ρ, with surface tension σm, and work of adhesion Wad of different liquid metals on a given ceramic is studied and demonstrated. Simulation program is developed and used for scanning acoustic microscopy (SAM) u...
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| Опубліковано в: : | Успехи физики металлов |
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2018
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| Цитувати: | Correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters / Z. Hadef, A. Doghmane, K. Kamli, Z. Hadjoub // Progress in Physics of Metals. — 2018. — Vol. 19, No 2. — P. 168-184. — Bibliog.: 43 titles. — eng. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859838510517714944 |
|---|---|
| author | Hadef, Z. Doghmane, A. Kamli, K. Hadjoub, Z. |
| author_facet | Hadef, Z. Doghmane, A. Kamli, K. Hadjoub, Z. |
| citation_txt | Correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters / Z. Hadef, A. Doghmane, K. Kamli, Z. Hadjoub // Progress in Physics of Metals. — 2018. — Vol. 19, No 2. — P. 168-184. — Bibliog.: 43 titles. — eng. |
| collection | DSpace DC |
| container_title | Успехи физики металлов |
| description | In the paper, a correlation between acoustic velocities V, elastic moduli M, and densities ρ, with surface tension σm, and work of adhesion Wad of different liquid metals on a given ceramic is studied and demonstrated. Simulation program is developed and used for scanning acoustic microscopy (SAM) under operating conditions, which favour the generation of acoustic waves. As found, for the given systems, all investigated acoustic parameters exhibit good dependences with both σm and Wad. Analysis and quantification of the results lead to the determination of semi-empirical formulas.
В статті вперше демонструється кореляція між акустичними швидкостями V, пружніми модулями M, густинами ρ та поверхневим натягом σm, а також роботою адгезії Wad різних рідких металів на фіксованій кераміці. Використовується обчислювальна програма для сканувального акустичного мікроскопу (САМ) за умов, що є сприятливими задля ґенерування акустичних хвиль. Виявлено, що для конкретних систем усі досліджені акустичні параметри демонструють хорошу залежність від поверхневого натягу та роботи адгезії. Аналіза та кількісне визначення результатів привели до встановлення напівемпіричних формул.
В статье исследуется и впервые демонстрируется корреляция между акустическими скоростями V, упругими модулями M, плотностями ρ и поверхностным натяжением σm, а также работой адгезии Wad различных жидких металлов на фиксированной керамике. Используется вычислительная программа для сканирующего акустического микроскопа (САМ) в условиях, благоприятных для генерирования акустических волн. Обнаружено, что для конкретных систем все исследованные акустические параметры демонстрируют хорошую зависимость от поверхностного натяжения и работы адгезии. Анализ и количественное определение результатов привели к установлению полуэмпирических формул.
|
| first_indexed | 2025-12-07T15:36:26Z |
| format | Article |
| fulltext |
168 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 2
https://doi.org/10.15407/ufm.19.02.168
PACS numbers: 43.20.+g, 62.20.De, 68.03.Cd, 68.08.Bc, 68.08.De, 68.35.Np, 68.60.Bs, 81.05.Mh,
81.40.Jj, 81.70.Bt
Z. HAdEf, A. dOgHmANE, K. KAmLI, and Z. HAdjOub
Laboratoire des Semi-Conducteurs, Département de Physique,
Faculté des Sciences, Université Badji-Mokhtar,
Annaba, BP 12, DZ-23000, Alegria
correlation Between sUrface
tension, work of aDhesion in liqUiD
Metals/ceraMic systeMs,
anD acoUstic ParaMeters
In the paper, a correlation between acoustic velocities V, elastic moduli M, and den-
si ties ρ, with surface tension σm, and work of adhesion Wad of different liquid met-
als on a given ceramic is studied and demonstrated. Simulation program is devel-
oped and used for scanning acoustic microscopy (SAM) under operating conditions,
which favour the generation of acoustic waves. As found, for the given systems, all
inves tigated acoustic parameters exhibit good dependences with both σm and Wad.
Analysis and quantification of the results lead to the determination of semi-empir-
ical formulas. The expressions are as follow: log (Vi) = 0.49log (σm/ρsm) + Bi, M = Amσm,
Wad = CiVi, and Wad = ξ (M/ρsm)1/2 + Dc, where Ai, Bi, Ci, Am, ξ, and Dm are characteris-
tic constants for velocities and elastic moduli, the subscripts m relate to the elastic
moduli (Young’s or shear ones), and i = L, T, R — to the propagating longitudinal,
transverse, and Rayleigh waves’ modes. The importance of the deduced formulas
lies in the possibility of prediction of surface tension and work of adhesion of such
metal/ceramic interfaces depending on the elastic and acoustic characteristics.
Keywords: surface tension, work of adhesion, acoustic velocities, elastic constants,
ceramics, liquid metals, interfaces.
introduction
The interfacial phenomena between metals and ceramics are one of
interest subject in science and engineering. The performance of sev-
eral technological applications such as ceramic metal bonding, metal–
ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 2 169
Tension, Adhesion in Liquid Metals/Ceramic Systems, and Acoustic Parameters
ceramic joining, ceramic–metal
matrix composites [1], thermal-
barrier coatings (TBC) [2], hard
TiN-coating [3], photovoltaic mate-
rials [4], and thin metal films on
ceramic substrates [5] is directly
linked to the nature of the metal/
ceramic interfaces. The behaviour
of this interfacial phenomenon is related directly to the nature of in-
terfacial bonding between metal and ceramic [6]. The adhesion of the
metal/ceramic system is the most important factor of all metal bonds.
It is defined by the change in the free energies of two materials when
they come into contact [7].
The work of adhesion Wad , between liquid metal and ceramic sub-
strate is given by Young–Dupré equation relating surface tension of
molten metal above melting temperature σm and measured equilibrium
contact angle θ formed between deposited liquid metal and its ceramic
substrate (see Fig. 1) [6]:
Wad = σm(1 + cosθ). (1)
Various non-destructive techniques are established to characterize
the metals-ceramic interfaces [8]. Scanning acoustic microscope (SAM)
is one of the important tools for non-destructive determination of adhe-
sion [9]. It can be used quantitatively (microanalysis) and qualitatively
(imaging). The microanalysis mode is employed to characterize not only
the elastic properties of materials but also the interfacial adhesion via
propagation of acoustic wave’s measurement. It is possible when the in-
ter face lies in a plane xy and disturbs the propagation of surface acous-
tic waves (SAWs) [9].
In this paper, a new acoustical approach has been proposed to in-
terpret and estimate the surface tension and work of adhesion in the
molten metal/ceramic systems; it shows that these parameters are de-
termined simultaneously via acoustic velocities and elastic moduli ac-
cording to the semi-empirical formulas.
Methodology and Materials
saM technique
Scanning acoustic microscope can be applied for a quantitative charac-
terization of the interfacial adhesion via the investigation of acoustic
material signature, V(z). This analog signal received by transducer and
focused by the position of the acoustic lens at the sample against the
distance z, under an incidence angle with the reflected ones [10–12]. Its
determination based on the calculation of the reflection coefficient R(θ).
The V(z) is the result of the several interferences of all the leaky wave
Fig. 1. Contact angle θ in equilibrium
liquid/solid system [7]
170 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 2
Z. Hadef, A. Doghmane, K. Kamli, and Z. Hadjoub
modes, such as leaky SAW, leaky pseudo-SAW, leaky surface-skimming
compressional wave, leaky Lamb wave, and harmonic waves. However,
only the velocity of leaky SAWs has been extracted from the V(z) curves
in microanalysis mode [13]. In effect of the operating conditions of the
SAM, only one significant mode dominates all other leaky SAWs modes.
Hence, the introduction of fast Fourier transformation (FFT) analysis
of V(z) is adopted for the directly determining of the acoustic velocities
of materials [13].
V(z) Calculation. The most important quantitative method for elas-
tic parameters determination, in particular, SAW velocities in scanning
acoustic microscopy are acoustic material signatures, also known as
V(z), which are obtained by recording the output signal, V, as the dis-
tance, z, between the sample and the acoustic lens is varied. Such curves,
that can be measured experimentally, can also be calculated theoreti-
cally, via the angular spectrum model [14], from the following expres-
sion:
max
2
0
0
( ) ( ) ( )[exp(2 cos )] sin cos .V z P R jk z d
θ
= θ θ θ θ θ θ∫ (2)
Here, P(θ) is the distribution function, k0 = 2π/λ is the wave number
in the coupling liquid, j2 = −1, θ is the angle between the wave vector
k and the lens axis, and R(θ) is the reflectance function of the speci-
men. The latter function, for acoustic waves, can be found by solving
the acoustic Fresnel equation. The reflection coefficient [15, 16] from
a layer reads as
( ) ,
in liq
in liq
Z Z
R
Z Z
−
θ =
+
(3)
where Zliq is the impedance of plane wave in the liquid, Zin is the input
impedance of the layer that is the impedance at the layer–liquid bound-
ary, which is expressed by the formula:
tg
tg
sub ch
in ch
ch sub
Z iZ
Z Z
Z iZ
− ϕ
=
− ϕ
(4)
with ϕ = kLhLcosθL being the phase advance of the plane wave passing
through the layer of an h thickness, and Zsub and Zch are the acoustic
impedances of substrate and layer given by
Zi = ρi Vi/cos θi, (5)
where subscript i = liq, ch or sub stands for liquid, layer, or substrate,
respectively. It is clear that, at normal incidence, the acoustic imped-
ance becomes simply the product of density and velocity. Hence, the
intensity reflection coefficient of a layer on a substrate is as follows:
2 2 2 2 2 2
2 2 2 2 2 2
( ) cos ( ) sin
.
( ) cos ( ) sin
ch sub liq ch sub liq ch ch
ch sub liq ch sub liq ch ch
Z Z Z k h Z Z Z k h
R
Z Z Z k h Z Z Z k h
− + −
=
+ + +
ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 2 171
Tension, Adhesion in Liquid Metals/Ceramic Systems, and Acoustic Parameters
Note that the reflection coefficient is a complex-valued function
with an amplitude and a phase and the total reflections obtained for
|R(θ)| = 1. Therefore, the V(z) calculation from relation (1) can readily be
carried out by just knowing the SAW velocities and material densities.
Acoustic Velocity Determination. The schematic representation of
V(z) curves, given by Eq. (1), is shown in Fig. 2, a; it consists of many
peaks and valleys due to constructive and destructive interference be-
tween different propagating modes, with a main peak at the focal dis-
tance (z = 0) representing the lens response. However, successive peaks
decay exponentially when z increases, because of the influence of the
acoustic lens signal, Vlens (Fig. 2, b). Thus, the real signal of the speci-
men, Vs(z), would be
Vs(z) = V(z) – Vlens(z). (7)
Thus, the obtained signal (Fig. 2, c) is a periodic curve character-
ized by a spatial period ∆z. Hence, its treatment can be carried out via
fast Fourier transform (FFT), which exhibits a large spectrum consist-
ing of one or several peaks (Fig. 2, d).
The dominant mode (usually Rayleigh one) appears as a very sharp
and pronounced peak, from which the Rayleigh velocity can be deter-
mined [8] according to the relation:
( )2
,
1 (2 )
liq
R
liq
V
V
V f z
=
− ∆
(8)
where Vliq is the sound velocity in the coupling liquid and f the operat-
ing frequency.
Elastic Constants Determination. It is well known that the Ray-
leigh velocity is generally determined experimentally from SAM, satis-
Fig. 2. Schematic diagram showing different calculation steps: a — V (z) signature,
b — lens response, c — sample signature, and d — spectrum of the fast Fourier
transormation (FFT)
172 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 2
Z. Hadef, A. Doghmane, K. Kamli, and Z. Hadjoub
fying the standard equation. In order to determine the elastic constants
E and G, Viktorov’s formula was used [10]:
2
2
0.718 ( )
.
0.75 ( )
T L
R T
T L
V V
V V
V V
−
=
−
Elastic constants can be expressed in term of density ρ and velo-
cities of the longitudinal VL and transverse VT modes of acoustic waves
[11].
2 2 2
1 2 2
[3 4 ]
,
T L T
L T
V V V
E
V V
ρ −
=
−
(10)
2
1 .TG V= ρ (11)
On the other hand, another approach has been proposed [17] to
find the relationships between the velocities of the different modes
(Rayleigh, longitudinal, and transverse ones) of acoustic waves in order
to determine the Young’s module E and the shear modulus G by an ex-
pression that contains only one of these terms:
2
2 2.99 ,RE V= ρ (12)
2
2 0.757 ,LE V= ρ (13)
2
2 2.586 ,TE V= ρ (14)
2
2 1.156 ,RG V= ρ (15)
2
2 0.293 .LG V= ρ (16)
The application of these equations removes several limitations re-
lated to SAM operational conditions.
Materials and Simulation Conditions. It is important to note that
the determination of Rayleigh velocities of several deposited liquid met-
als is impossible in using the SAM technique. For this reason, the simu-
lation of deposited metals was taken in a bulk state for determining
these Rayleigh velocities, and comparison between obtained results and
experimental sound velocities of several deposited liquid metals was
made to enrich this study.
The calculations were approved out in case of a reflexion scanning
acoustic microscope; Rayleigh mode dominate and appears under normal
operating conditions (half-opening angle of lens 50°, working frequency
is 142 MHz and water as a coupling liquid whose wave velocity Vliq =
= 1500 m/s and density ρ = 1000 kg/m3) or with annular lenses.
The final step consists in determination of the Rayleigh velocities
from reflection coefficient and the acoustic signature; for example sim-
ulation, it will be taken two metals tin (Sn) and silicon (Si).
Reflection Coefficient. The reflection function, R (θ), was first cal-
culated for two deposited metals (Sn and Si) to show their effects in the
experimental calculation of Rayleigh velocity. The curves obtained are
ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 2 173
Tension, Adhesion in Liquid Metals/Ceramic Systems, and Acoustic Parameters
shown in Fig. 3. For a better representation of the curve and since R (θ)
is a complex function, we have separated the amplitude curves (Fig. 3,
a) from those of the phase (Fig. 3, b). Then, for the deposited metals
mentioned above, the real parts and the imaginary parts were super-
posed as a function of the angles of incidence θi.
Following Fig. 3, a, representing the amplitude of R (θ) as a func-
tion of the angle of incidence θi, we can clearly observe a first amplitude
fluctuation when the angle of incidence reaches the values of the criti-
cal longitudinal angles θL. A change from θL to higher values. Then, a
second fluctuation when the angle of incidence reaches the values of
the critical transverse angles, θT. Between θL and θT, amplitude of R (θ)
remains constant. Finally, beyond θT, the amplitude of R (θ) increases to
reach the unit corresponding to total reflection.
Following Fig. 3, b, representing the phase of R (θ) as a function of
θI, it can be easily noticed that almost a 2π transition is obtained for
Si. This transition occurs at the critical angle, θR, which corresponds to
the Rayleigh mode, which is the most important in the current simu-
lation conditions. Thus, the Rayleigh mode dominates all other modes
leading to the fact that the longitudinal critical angle, θL, is not very
noticeable.
It can also be seen that the amplitude of the transition in the
Rayleigh mode phase becomes lower than the usual 2π value for Sn.
While the position, i.e. the value of θR moves to lower values (similar
behaviour to that observed with θL in Fig. 3, a. In addition, it is clear
that all modes are generated with angles less than 20°. These critical
angles strongly depend on the simulation conditions, in particular the
coupling liquid densities.
Acoustic Signature. The acoustic signature can be calculated from
the spectral angular model. The curves obtained for the two deposited
metals (Sn and Si), are shown in Fig. 4.
It is clear that the two curves of V (z) exhibit an oscillatory be-
haviour, with a spatial period ∆z, due to constructive and destructive
Fig. 3. Reflection coefficients: amplitude (a) and phase (b) as a function of incidence
angles at deposited Sn (dash line) and Si (solid line) metals
174 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 2
Z. Hadef, A. Doghmane, K. Kamli, and Z. Hadjoub
interferences between the propagation modes. It should be noted that
the two curves are distinct in amplitude as well as in the periods, ∆z. In
amplitudes, the curves attenuate faster for Si corresponding to a large
period. Such behaviour is the result of previously observed changes in
the module and phase curves of the reflectance functions.
The FFT spectral analysis of these periodic curves V(z) is shown in
Fig. 2, b. These spectra are characterized by a principal peak represent-
ing the most dominant mode, which is that of Rayleigh, under the pres-
ent conditions. However, the efficiency of this mode represented by its
height is more important for higher VR. Moreover, a small shift is ob-
served for the main ray highlighting the spatial differences ∆z obtained
in the curves V(z).
Several deposited metals parameters used in this investigation are
listed in Table 1; sound velocities at melting temperatures are tabulated
by Blairs [18], surface tension values are proposed by Keene [19], liquid
densities are taken by Crawley [20] and by Baykara et al. [21].While
the elastic constants and solid densities are obtained from Briggs [10],
Rayleigh velocity determined by SAM and Rayleigh velocity calculated
from one-parameter approach are given in Table 1.
results and quantification
The objective of this quantification is to find correlations applicable for
the estimation of surface tension and the work of adhesion for different
liquid metals in contact with ceramic as function of acoustic velocities
and elastic constants of these metals.
Fig. 4. Acoustic signatures (a) and FFT spectra (b) for deposited Sn (dash line) and
Si (solid line) metals
ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 2 175
Tension, Adhesion in Liquid Metals/Ceramic Systems, and Acoustic Parameters
Table 1. Experimental sound velocities, c, surface tensions, σm, densities, ρlm,
of different liquid metals at the melting temperature, elastic moduli, densities,
ρsm, and calculated Rayleigh velocities, VR , of these metals at solid state
Metal
c
(m/s)
σm
(mj/m2)
ρlm
(kg/m3)
E
(GPa)
G
(GPa)
ρsm
(kg/m3)
VR (m/s)
SAM O. P. approach
Na 2526 203 951 10 3.9 968 1720 1875
Mg 4065 577 1589 45 17.4 1738 2879 3078
Al 4561 1075 2390 70 27.1 2700 2929 3130
Si 6920 859 2524 169 18.2 2330 4598 4863
Ca 2978 362 1378 20 7.7 1550 2036 2203
Fe 4200 1909 7042 211 81.6 7874 2806 3003
Co 4031 1928 7740 209 80.8 8900 2808 3005
Ni 4047 1834 7889 207 80.0 8908 2607 2796
Cu 3440 1374 8089 130 50.3 8920 1993 2159
Zn 2850 817 6552 108 41.8 7140 2175 2348
Ge 2693 631 5487 89,6 34.6 5323 2165 2337
Ag 2790 955 9329 83 32.1 10490 1414 1558
Cd 2256 637 7997 50 19.3 8650 1306 1446
Sn 2464 586 6973 50 19.3 7310 1262 1400
Sb 1900 382 6077 55 21.3 6697 1589 1740
Ba 1331 273 3343 13 5.0 3510 1011 1140
La 2030 728 5940 37 14.3 6146 1303 1443
Ce 1694 750 6550 34 13.1 6689 1183 1318
Pr 1926 716 6500 37 14.3 6640 1243 1380
Yb 1274 320 6720 24 9.3 6570 966 1093
Ta 3303 2083 14353 186 71.9 16650 1919 2082
Pt 3053 1746 18909 168 65.0 21090 1574 1724
Au 2568 1162 17346 78 30.2 19300 1008 1136
Sc 4272 939 2680 74 28.6 2985 2841 3039
Ti 4309 1475 4141 116 44.8 4507 2862 3061
V 4255 1856 5340 128 49.5 6110 2641 2831
Y 3258 872 4180 64 24.7 4472 2093 2263
Zr 3648 1463 5650 68 26.3 6511 1846 2006
Nb 3385 1757 7830 105 40.6 8570 1954 2118
Pd 2657 1482 10495 117 45.2 12023 1637 1789
Hf 2559 1517 11550 78 30.2 13310 1361 1503
Nd 2212 685 6890 41 15.9 6800 1272 1411
Sm 1670 431 7420 50 19.3 7353 1359 1501
Eu 1568 264 5130 18 7.0 5244 956 1083
Gd 2041 664 7790 55 21.3 7901 1394 1537
Tb 2014 669 8050 56 21.7 8219 1382 1525
Dy 1941 648 8370 61 23.6 8551 1417 1561
Ho 1919 650 8580 65 25.1 8795 1447 1592
Er 1867 637 8860 70 27.1 9066 1480 1626
Lu 2176 940 9750 69 26.7 9841 1395 1538
176 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 2
Z. Hadef, A. Doghmane, K. Kamli, and Z. Hadjoub
estimation of c and ρlm in terms of Vi
and ρsm for Different solid Metals
In this article, analytical study has been proposed to express the rela-
tion between experimental sound velocities of liquid metals at the melt-
ing temperature and determinate acoustic velocities of these metals at
solid state by SAM program.
It is noted that Rayleigh velocities of bulk metals determined by
SAM have almost the same values that these calculated by one param-
eter approach using Eq. (11).
The variation of VR values as function of c was made; it shows a
linear increase of VR with c increasing. Simple fitting was made and
resulted in a well-defined linear correlation between the quantities, as
can be seen in Fig. 5.
The quantified correlation between VR and c can be written as
VR = 0.674 c. (17)
Longitudinal and transverse velocities follow similar behaviours
that take the following form:
VL = 1.342c, (18)
VT = 0.724c. (19)
Relation between acoustic velocities of solid metals and experimen-
tal sound velocities of liquid metals can be generalized with following
analytical form:
Vi = Aic (20)
where Ai is a characteristic constant for velocities; the subscripts i = L,
T, R represent the propagating longitudinal, transverse, and Rayleigh
waves modes.
Fig. 6. Correlation between densities for liquid (ρlm) and solid (ρsm) metals
Fig. 5. Correlation between experimental sound velocities c for liquid metals and
calculated Rayleigh velocities VR of these metals in solid state
ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 2 177
Tension, Adhesion in Liquid Metals/Ceramic Systems, and Acoustic Parameters
One can see also a clear tendency between the liquid-metals’ densi-
ties, ρlm, with that of these metals at solid state, ρsm, as can be seen in
Fig. 6. The relationship that expresses this tendency can take the fol-
lowing form:
ρsm = 1.088 ρlm. (21)
The importance of Eqs. (20) and (21) lies in the prediction of acous-
tic parameters from liquid to solid states of metals and vice versa.
Determination of σm in terms
of the acoustic Velocities for Different Metals
Many statistical theories established to associate the surface tension
and sound velocities [22–25]. In this context, Auerbach proposed semi-
empirical relation to express the sound velocity of such liquid metal at
the melting temperature in term of σm and ρlm [26]:
c = σm /6.33 ⋅ 10−10ρlm. (22)
According to Mayer [27], the previous equation can be written as:
c = A (γ1/2 Vm
1/6) (σm /ρlm)1/2, (23)
where A is a constant, Vm is the molar volume, and γ is the ratio of the
isobaric, CP, and isochoric, CV, heat capacities. The plot of log (c) versus
log(σm/ρlm) may be linear with a slope equal to 0.67 for Auerbach rela-
tion and equal to 0.50 for Mayer relation. This point was study by Blairs
[18]; slope equal to 0.552 is found.
In this context, to analyse the functional dependence of σm and VR
of solid metals, a linear correlation between the behaviours of quantities
is found, where the deduced VR values increase with increasing of the
quantity (σm/ρsm), as can be seen in Fig. 7.
To quantify the relation between VR and (σm/ρsm), a logarithmic plot
was made in the present work; a linear correlation between the quanti-
ties is defined by:
log(VR) = 0.49 log (σm/ρsm) + 8.53. (24)
It would be noted that similar behaviours were deduced for longi-
tudinal and transverse velocities; this is evident in the following rela-
tions:
log(VL) = 0.49 log (σm/ρsm) + 9.22, (25)
log(VT) = 0.49 log (σm/ρsm) + 8.60. (26)
The above relations of acoustic velocities take the following general
form:
log(Vi) = 0.49 log (σm/ρsm) + Bi, (27)
where Bi is characteristic constants for velocities.
Equation (27) shows that the slope of the plot of log (V) versus
log (σm/ρsm) is closer to the Mayer proposition than that of the Auerbach
178 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 2
Z. Hadef, A. Doghmane, K. Kamli, and Z. Hadjoub
relation. It is finds application for the estimation of unknown surface
tension of metallic liquids using available acoustic velocities and den-
sity values in solid state.
Determination of σm in terms of elastic constants
for Various Metals
A close comparative of Eqs. (12), (17), and (23) shows a linier depen-
dence between E and σm, as can be seen in Fig. 8, where the results’
analysis shows increase comportment for surface tension with Young’s
modulus increasing according to linear relationship; similar behaviour
was also obtained for shear modulus.
To quantify the relation between elastic moduli and σm, a simple
plot was made; a linear correlation is defined that it can be written as
E = 0.083 σm, (28)
G = 0.032 σm. (29)
A close comparative analysis of the above equations derived expres-
sions shows that they can be taking the following form:
M = Am σm, (30)
where Am is characteristic constant for elastic moduli.
An important point that can be interpreted from Eq. (25) is the pos-
sibility of determining the unknown surface tension for liquid metals as
function of elastic moduli.
Fig. 8. Correlation between surface tension σm of different liquid metals and Young’s
modulus E of these metals
Fig. 7. Correlation between deduced Rayleigh velocities (VR) and ration of surface
tension to density in solid state (σm /ρsm)
ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 2 179
Tension, Adhesion in Liquid Metals/Ceramic Systems, and Acoustic Parameters
Table 2. Experimental values of the work of adhesion Wad
for different metal/ceramic systems
Ceramics Metal Atmosphere
Wad
(mj/m2)
Refs. Ceramics Metal Atmosphere
Wad
(mj/m2)
Refs.
AlN Au Vacuum 550 [28] BeO Cu Ar 600 [31]
Co Vacuum 1270 [28] Fe He 717 [32]
Cu Vacuum 1060 [29] Ni Vacuum 680 [32]
Fe Vacuum 1320 [28] Pb Vacuum 130 [33]
Ga Vacuum 750 [28] BN Au Vacuum 205 [34]
Ge Vacuum 911 [28] Cu Vacuum 345 [34]
In Vacuum 448 [28] Si Vacuum 664 [34]
Ni Vacuum 1305 [28] Sn Vacuum 128 [34]
Pb Vacuum 203 [28] MgO Ag Ar 421 [34]
Pd Vacuum 858 [28] Fe Vacuum 820 [35]
Sn Vacuum 461 [29] In Vacuum 172 [36]
Al2O3 Al Vacuum 948 [30] Ni He 585 [37]
Au Vacuum 577 [30] Sn Vacuum 278 [36]
Co Vacuum 1141 [30] NiO Ag Ar 1267 [35]
Fe Vacuum 1202 [30] Cu Ar 1738 [35]
Ga Vacuum 537 [30] Ni Ar 2652 [32]
In Vacuum 335 [30] Sn Vacuum 921 [35]
Ni Vacuum 1191 [30] SiO2 Au Vacuum 165 [1]
Pb Vacuum 218 [30] Cu Vacuum 390 [34]
Pd Vacuum 704 [30]
Si Ar 708 [38]
Sn Vacuum 305 [30]
relationship between wad in Metals/ceramic
systems and acoustic Velocities
The principle of the present approach is to find a relation between the
acoustic velocities of different metals and the work of adhesion in this
several metals on a given ceramic. Experimental results of the work of ad-
hesion and the contact angle for various metal-ceramic systems are sum-
marized in Table 2. It should be taken that the criterion of metal−ceramic
systems selected in Tables 2 and 3 is the existence in the literature.
From the plotting of Wad values given in Table 2 against the Rayleigh
velocities of the various metals, a linear correlation between these quan-
tities is observed, where Wad increase with VR increasing as can be seen
in Fig. 9 for different metal–aluminium nitride systems.
It is reasonably to express the quantified correlation between Wad
and VR as can be written as
Wad = 0.434 VR . (31)
180 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 2
Similar behaviours of other aco-
us tic velocities would also be dedu-
ced; this is evident in the form:
Wad = 0.864 VL , (32)
Wad = 0.467 VT . (33)
The above relations take the fol-
lowing general form:
Wad = Ci Vi, (34)
where Ci is slope parameter, which
gives the interfacial adhesion as
function of acoustic velocities.
The points presented in Fig. 9
yields a slope parameter for Al−N. These results, together with the CR va-
lues obtained for other metals–ceramics systems, are given in Table 3.
Moreover, it should be noted that Eq. (34) and CR values presented
in Table 3 determine directly the work of adhesion of different metals−
ceramics systems depending on the acoustic proprieties of these
metals.
relation between wad in Metals/ceramics
systems and elastic constants
To enrich this estimation, it would be useful to quantify the influence
of elastic constants on the work of adhesion. The plot of log (Wad) against
log(E/ρsm) is evaluated. Some typical results are summarized in Fig. 10;
it is clear that the general tendency is for an increase in Wad as (E/ρlm)
increases.
Using a simple logarithmic plot, it is possible to find linear depen-
dence between logarithms of (E/ρsm) and Wad as follows:
log (Wad) = 0.504 log (E/ρsm) − 0.648, (35)
log (Wad) = 0.502 log (G/ρsm) + 2.138. (36)
The above elastic moduli (Young’s and shear ones) follow similar
behaviours, which take the following form:
log (Wad) = 0.502 log (M/ρsm) + Cm, (37)
where Cm is characteristic coefficient for the elastic moduli depending
on the nature of ceramic.
This behaviour has been remarked for all ceramics in contact with
several liquid metals studied. In this context, a new semi-empirical rela-
tion can be proposed to express the work of adhesion in term of E and
ρsm as can be seen in Fig. 11.
Z. Hadef, A. Doghmane, K. Kamli, and Z. Hadjoub
Table 3. Slope parameter of Rayleigh
velocities, CR, and the coefficient
of the linear regression, R,
determined for various ceramic
Ceramic CR R
AlN 0.434 0.972
Al2O3 0.363 0.931
BeO 0.247 0.986
BN 0.137 0.972
MgO 0.236 0.934
NiO 0.859 0.960
SiO2 0.151 0.976
ISSN 1608-1021. Usp. Fiz. Met., 2018, Vol. 19, No. 2 181
Tension, Adhesion in Liquid Metals/Ceramic Systems, and Acoustic Parameters
Linear correlation between these quantities is defined; it can be
written as:
Wad = 8.08 (E/ρsm)1/2 − 41.5, (38)
Wad = 11.75 (E/ρsm)1/2 + 52.9. (39)
A close comparative analysis of above-mentioned equations derived
from expressions shows that they could be taking the following form:
Wad = ξ (M/ρsm)1/2 + Dc, (40)
where ξ represents the slope parameter of dependence on these quanti-
ties, and DC is characteristic coefficient for the elastic moduli depend-
ing on the nature of ceramic.
An important point that can be taken from Eq. (40) is the possibility
of prediction of work of adhesion in liquid metals/ceramic systems from
elastic moduli and densities of liquid metals.
Fig. 11. Correlation between work of ad-
hesion (Wad) of liquid metals/ceramic
sys tems and elastic constant (E/ρsm) of
these metals
Fig. 10. Correlation between log (Wad) and log (E/ρsm) of the metals
Fig. 9. Correlation between work of adhesion Wad for different metal/AlN systems
and Rayleigh velocities VR of these metals
182 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 2
Z. Hadef, A. Doghmane, K. Kamli, and Z. Hadjoub
conclusion
In this work, surface tension and work of adhesion of different liq-
uid metals on a given ceramic are investigated. Acoustic parameters
(namely, longitudinal, transverse, and Rayleigh velocities) and elastic
constants (namely, Young’s and shear moduli) are calculated for all
solid metals of the system at issue. It is shown that these parameters
change with increasing surface tension as well as with increasing work
of adhesion. Applying different complex-quantitative methods [39–43]
for the analysis and quantification, we found new linear semi-empirical
formulas, which express the variations of velocities and elastic con-
stants. The importance of this estimation consists in the prediction of
acoustic parameters for any surface tension and work of adhesion in
metals/ceramic interfaces and vice versa.
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Received May 1, 2018;
in final version, May 3, 2018
З. Хадеф, А. Догман, К. Камлі, З. Хаджуб
Лабораторія напівпровідників, відділ фізики,
факультет наук, Університет Баджі-Мохтара,
Аннаба, БП 12, DZ-23000, Алжир
КОРЕЛЯЦІЯ МІж ПОВЕРХНЕВИМ НАТЯГОМ,
РОБОТОЮ АДГЕЗІї У СИСТЕМАХ РІДКІ МЕТАЛИ/КЕРАМІКА
ТА АКУСТИЧНИМИ ПАРАМЕТРАМИ
В статті вперше демонструється кореляція між акустичними швидкостями V,
пружніми модулями M, густинами ρ та поверхневим натягом σm, а також робо-
тою адгезії Wad різних рідких металів на фіксованій кераміці. Використовується
обчислювальна програма для сканувального акустичного мікроскопу (САМ) за
умов, що є сприятливими задля ґенерування акустичних хвиль. Виявлено, що
для конкретних систем усі досліджені акустичні параметри демонструють хоро-
шу залежність від поверхневого натягу та роботи адгезії. Аналіза та кількісне
визначення результатів привели до встановлення напівемпіричних формул. Одер-
184 ISSN 1608-1021. Prog. Phys. Met., 2018, Vol. 19, No. 2
Z. Hadef, A. Doghmane, K. Kamli, and Z. Hadjoub
жані вирази мають наступний вигляд: log (Vi) = 0,49 log (σm/ρsm) + Bi, M = Amσm,
Wad = CiVi і Wad = ξ (M/ρsm)1/2 + Dc, де Ai, Bi, Ci, Am, ξ і Dm є характеристичними
значеннями швидкостей і модулів пружности, ніжні індекси m відносяться до
модулів пружности (Юнґа чи зсуву), а індекси i = L, T, R — до поширюваних
поздовжніх, поперечних або Релейових хвиль. Важливість одержаних формул
полягає у можливості передбачення поверхневого натягу та роботи адгезії меж
поділу метал/кераміка, залежно від пружніх і акустичних характеристик.
Ключові слова: поверхневий натяг, робота адгезії, акустичні швидкості, пружні
сталі, кераміка, рідкі метали, межа поділу.
З. Хаэф, А. Догман, К. Камли, З. Хаджуб
Лаборатория полупроводников, отдел физики,
факультет наук, Университет Баджи-Мохтара,
Аннаба, БП 12, DZ-23000, Алжир
КОРРЕЛЯЦИЯ МЕжДУ ПОВЕРХНОСТНЫМ НАТЯжЕНИЕМ,
РАБОТОЙ АДГЕЗИИ В СИСТЕМАХ жИДКИЕ МЕТАЛЛЫ/КЕРАМИКА
И АКУСТИЧЕСКИМИ ПАРАМЕТРАМИ
В статье исследуется и впервые демонстрируется корреляция между акустиче-
скими скоростями V, упругими модулями M, плотностями ρ и поверхностным
натяжением σm, а также работой адгезии Wad различных жидких металлов на
фиксированной керамике. Используется вычислительная программа для скани-
рующего акустического микроскопа (САМ) в условиях, благоприятных для гене-
рирования акустических волн. Обнаружено, что для конкретных систем все ис-
следованные акустические параметры демонстрируют хорошую зависимость от
поверхностного натяжения и работы адгезии. Анализ и количественное опреде-
ление результатов привели к установлению полуэмпирических формул. Полу-
ченные при этом выражения имеют следующий вид: log (Vi) = 0,49 log (σm/ρsm) +
+ Bi, M = Amσm, Wad = CiVi и Wad = ξ (M/ρsm)1/2 + Dc, где Ai, Bi, Ci, Am, ξ и Dm явля-
ются характеристическими значениями скоростей и модулей упругости, нижние
индексы m относятся к модулям упругости (Юнга или сдвига), а i = L, T, R — к
распространяющимся продольным, поперечным или рэлеевским волнам. Важ ность
полученных формул заключается в возможности предсказания поверхност ного
натяжения и работы адгезии границ раздела метал/керамика в зависимости от
упругих и акустических характеристик.
Ключевые слова: поверхностное натяжение, работа адгезии, акустические ско-
рости, упругие постоянные, керамика, жидкие металлы, границы раздела.
|
| id | nasplib_isofts_kiev_ua-123456789-167908 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1608-1021 |
| language | English |
| last_indexed | 2025-12-07T15:36:26Z |
| publishDate | 2018 |
| publisher | Інститут металофізики ім. Г.В. Курдюмова НАН України |
| record_format | dspace |
| spelling | Hadef, Z. Doghmane, A. Kamli, K. Hadjoub, Z. 2020-04-15T17:08:25Z 2020-04-15T17:08:25Z 2018 Correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters / Z. Hadef, A. Doghmane, K. Kamli, Z. Hadjoub // Progress in Physics of Metals. — 2018. — Vol. 19, No 2. — P. 168-184. — Bibliog.: 43 titles. — eng. 1608-1021 DOI: https://doi.org/10.15407/ufm.19.02.168 PACS numbers: 43.20.+g, 62.20.De, 68.03.Cd, 68.08.Bc, 68.08.De, 68.35.Np, 68.60.Bs, 81.05.Mh, 81.40.Jj, 81.70.Bt https://nasplib.isofts.kiev.ua/handle/123456789/167908 In the paper, a correlation between acoustic velocities V, elastic moduli M, and densities ρ, with surface tension σm, and work of adhesion Wad of different liquid metals on a given ceramic is studied and demonstrated. Simulation program is developed and used for scanning acoustic microscopy (SAM) under operating conditions, which favour the generation of acoustic waves. As found, for the given systems, all investigated acoustic parameters exhibit good dependences with both σm and Wad. Analysis and quantification of the results lead to the determination of semi-empirical formulas. В статті вперше демонструється кореляція між акустичними швидкостями V, пружніми модулями M, густинами ρ та поверхневим натягом σm, а також роботою адгезії Wad різних рідких металів на фіксованій кераміці. Використовується обчислювальна програма для сканувального акустичного мікроскопу (САМ) за умов, що є сприятливими задля ґенерування акустичних хвиль. Виявлено, що для конкретних систем усі досліджені акустичні параметри демонструють хорошу залежність від поверхневого натягу та роботи адгезії. Аналіза та кількісне визначення результатів привели до встановлення напівемпіричних формул. В статье исследуется и впервые демонстрируется корреляция между акустическими скоростями V, упругими модулями M, плотностями ρ и поверхностным натяжением σm, а также работой адгезии Wad различных жидких металлов на фиксированной керамике. Используется вычислительная программа для сканирующего акустического микроскопа (САМ) в условиях, благоприятных для генерирования акустических волн. Обнаружено, что для конкретных систем все исследованные акустические параметры демонстрируют хорошую зависимость от поверхностного натяжения и работы адгезии. Анализ и количественное определение результатов привели к установлению полуэмпирических формул. en Інститут металофізики ім. Г.В. Курдюмова НАН України Успехи физики металлов Correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters Кореляція між поверхневим натягом, роботою адгезії у сис темах рідкі метали/кераміка та акустичними параметрами Корреляция между поверхностным натяжением, работой адгезии в системах жидкие металлы/керамика и акустическими параметрами Article published earlier |
| spellingShingle | Correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters Hadef, Z. Doghmane, A. Kamli, K. Hadjoub, Z. |
| title | Correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters |
| title_alt | Кореляція між поверхневим натягом, роботою адгезії у сис темах рідкі метали/кераміка та акустичними параметрами Корреляция между поверхностным натяжением, работой адгезии в системах жидкие металлы/керамика и акустическими параметрами |
| title_full | Correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters |
| title_fullStr | Correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters |
| title_full_unstemmed | Correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters |
| title_short | Correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters |
| title_sort | correlation between surface tension, work of adhesion in liquid metals/ceramic systems, and acoustic parameters |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/167908 |
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