Int J Thermophys (2017) 38:108
DOI 10.1007/s10765-017-2235-5
TEMPMEKO 2016

A Thermodynamic Consideration on the Mechanismof Ultrasensitive Moisture Sensing by Amorphous Silica
Y. Tsukahara1,2 · T. Tsuji1 · T. Oizumi1 ·S. Akao1,2 · N. Takeda1,2 · K. Yamanaka1,2Received: 23 July 2016 / Accepted: 27 April 2017
© The Author(s) 2017. This article is an open access publication

Abstract We detected the change in the phase velocity of surface acoustic waves
propagating along the amorphous silica layer on a spherical single-crystal quartz due
to the very small amount of moisture at less than 1 µmol·mol−1 in the ambient N2gas. This was made possible for the first time because this system, called a ball SAW
moisture sensor, was extremely sensitive. The measured phase velocity changed asa function of moisture density and temperature and was fitted strikingly well witha thermodynamic model assuming that dissociated H+ and OH− dissolved into theamorphous silica layer at room temperature.
Keywords Ball SAW sensor · Humidity · Hygrometer · Trace moisture

1 Introduction
The diffraction-free propagation of surface acoustic waves around a spherical object,
namely a steel ball bearing, was first generated and observed by the acousto-optical system [1] and analyzed [2] in the year 2000. Immediately after that, the ball SAW(surfaceacoustic wave) device by means of fabricating the interdigital transducers on a spherical solid was conceptualized and demonstrated [3,4]. In the following years, a variety
of gas sensors were developed by depositing various sensitive layers on the ball SAW

Selected Papers of the 13th International Symposium on Temperature, Humidity, Moisture and Thermal
Measurements in Industry and Science.

B Y. Tsukahara

tsukahara@ballwave.jp

1Tohoku University, Sendai, Miyagi, Japan

2

Ball Wave Inc., Sendai, Miyagi, Japan

123

108

Page 2 of 8

Int J Thermophys (2017) 38:108

device: a hydrogen gas sensor [5], an odor sensor [6], and a portable gas chromatograph [7]. Then it was accidentally found [8] that a very small amount of moisture in
the ambient gas affected the acoustic properties of a SAW device made of a singlecrystal quartz ball without any sensitive layers. It had been a mystery for a while
how the surface of a single-crystal quartz was affected by the ambient moisture atroom temperature, until a mechanism was eventually proposed that the surface of theparticular quartz ball was damaged during the production process and a thin amorphous layer of silica was produced which captured water molecules resulting into thechange of the acoustic properties. According to this hypothesis, a ball SAW device
was fabricated by depositing a thin layer of amorphous silica using sol-gel method
on a single-crystal quartz ball without a surface damage, and the detection of watermolecules in the ambient N2 gas at densities far less that 1 µmol·mol−1 was achieved[9]. Now it remains an unanswered question how the water molecules enter into theamorphous silica layer at room temperature.

2 Ball SAW Moisture Sensors
The design and fabrication of the ball SAW device for the moisture sensor is foundin [9]. The water molecules in the ambient gas surrounding the ball SAW device
somehow react with a thin layer of amorphous silica deposited on the spherical surfaceof the device. When the moisture density is low as 1 µmol·mol−1 , the phase velocityof the surface acoustic wave increases with the moisture density, which means thatthe amorphous silica film stiffens with moisture. When the moisture density furtherincreases the phase velocity starts to decrease which can be explained by the standard
model of mass loading effect on the surface acoustic waves [10]. In the following, weconcentrate on the new phenomena, stiffening of the amorphous silica layer due to theambient moisture in the low-density limit.

3 Thermodynamic Model
It is known [11] that the water molecules dissociate and the resultant H+ and OH−
chemically react with the network in silica glass at an elevated temperature as(bulk) ≡ Si–O–Si ≡ (bulk) + H2 O < − > 2 · (bulk) ≡ Si-OH .

(1)

As an elementary process in the reaction described by Eq. 1, we start with a simple
model of dissolving the “solutes” of H+ and OH− into a “solvent” of amorphous silica
and analyze the thermodynamics of the system. Let H2 O in the ambient gas and thedissociated H+ and OH− dissolving in the matrix of silica are in the equilibrium as
H2 O < − > H+ + OH−(2)

where the chemical potentials, µ, for the constituents satisfy the equationµ(H2 O) = µ(H+ ) + µ(OH− ) .

123

(3)

Int J Thermophys (2017) 38:108

Page 3 of 8108

Since we assume that the reaction is a simple dissociation, we propose that for eachconstituent(4)µ = µ0 (T ) + kT log([x]),where T and [x] are the temperature and the density, respectively, of the constituent
x, and k is the Boltzmann constant. T has the same value in all constituents in theequilibrium. By substituting Eq. 4 into Eq. 3, we haveµ0 (H2 O)+kT log([H2 O]) = µ0 (H+ )+kT log([H+ ])+µ0 (OH− )+kT log([OH− ]).
(5)Then[H+ ]·[OH− ]/[H2 O] = exp{(µ0 (H2 O) − µ0 (H+ ) − µ0 (OH− ))/kT }.
Assuming(6)[H+ ] = [OH− ](7)[H+ ] = [H2 O]1/2 · exp{!µ0 /2kT },where !µ0 = µ0 (H2 O) − µ0 (H+ ) − µ0 (OH− ).

(8)
(9)

Eq. 6 leads to following Eq. 8,

4 Dependence on Moisture Density and Temperature
Let us look at the dependence on the moisture density. Maintaining the temperature
at a constant value and changing the density of H2 O, [H2 O], in Eq. 8, the dependenceof [H+ ] on [H2 O] islog[H+ ] ∼ (1/2) log[H2 O] + Const.

(10)

It seems reasonable to assume that the variation in the phase velocity of the surface
acoustic wave is proportional to [H+ ] in its low-density limit; then the dependenceof the phase velocity on the moisture [H2 O] in the N2 carrier gas was measured tovalidate the model.
4.1 Method of Measurements and its Uncertainty
The setup of experiment, the method of generating the trace moisture and its accuracy,
and the procedure of the measurement were detailed in [12]. We generated the tracemoisture in the N2 carrier gas by means of a diffusion tube method and measured
the SAW delay time change which was negatively proportional to the phase velocitychange of SAW. First, we consider the amount of uncertainty in the set values of tracemoisture in the experiment. The amount of generated moisture was inversely proportional to the temperature of the diffusion tube [12]. The temperature of the diffusion
tube was affected by the room temperature which fluctuated as large as ±5 ◦ C. This led

123

108

Page 4 of 8Int J Thermophys (2017) 38:108

to an uncertainty of about ±2 %. The generated humid N2 gas was mixed with a dry N2
gas by means of a mass flow controller to obtain a target humidity. The uncertainty due
to the allowance of the mass flow rate was ±2 % according to the specification of theinstrument. Hence, the accumulated uncertainty in the amount of generated moisture
was estimated to be less than ±4 %. Next, we consider the amount of uncertainty inthe delay time measurement of SAW. The elasticity of single-crystal quartz depends
on temperature. Thus, the amount of success in the temperature compensation was
the most important factor affecting the uncertainty in the measurement of delay timechange. Its procedure is described in depth in [12], and the uncertainty was estimated aslow as ±0.05 ppm. Another factor affecting the delay time change was the presence of ajitter or a phase noise in the electronic circuitry. By observing the phase noise in the output signal from the instrument when the dry nitrogen gas was introduced into the sensor
head, the standard deviation of the phase noise was estimated to be 0.016 ppm [12].
4.2 Result of the Measurement
Figure 1 shows the experimental dependence of delay time change on the trace moisture[H2 O] in the N2 carrier gas. A detection limit is indicated in the plot which is threetimes the standard deviation of the phase noise: 0.016 ppm. Error bars in the plots arebarely seen because the error due to the phase noise(0.016 ppm) was too small to be
seen in the plot except for a few points where deviations due to the fluctuation in room
temperature were not fully compensated. The experimental dependence was in goodaccordance with Eq. 10 at small values of [H2 O]. When [H2 O] becomes larger, theexperimental data deviate from the relationship in Eq. 10 and start to behave according

Delay Time Change, ppm1001010.1Detection limit

0.010.010.1

1101001000

10000

Trace Moisture Concentration, μmol∙mol-1Fig. 1 Delay time change versus trace moisture concentration at a constant temperature, 21 ◦ C. The x andy coordinates are in logarithmic scale. The change in the delay time is linearly proportional to the tracemoisture concentration in the logarithmic scale when it is small enough. The long-term fluctuation in thedelay time measurement is ±0.05 ppm, and a standard deviation of phase noise is 0.016 ppm as discussed
in the text. The uncertainty in the trace moisture concentration is ±4 %123

Int J Thermophys (2017) 38:108

Page 5 of 8108

to the familiar mass loading model [10]. This suggests that we are observing a newphenomenon of penetration of H+ and OH− dissociated from water molecules into
the amorphous silica layer at room temperature in the very low level of concentration,
thanks to the very high sensitivity of ball SAW sensors.
Next, we examined the dependence on temperature. Maintaining the trace moisture density of H2 O, [H2 O] at a constant value, and changing the temperature, T, thedependence of [H+ ] on T islog[H+ ] ∼ !µ0 /2kT + Const.

(11)

The absorption of gases by metals is described by Sievert’s law [13,14]. Here wefollow the line of thought and approximate the chemical potential for H2 O gas by thatfor the ideal gas,
µ0 (H2 O) ∼ ε(H2 O) + kT (log P − 5/2 log T + constant),(12)

and the chemical potentials for the H+ and OH− in the amorphous silica layer byµ0 (H+ ) ∼ ε(H+ ),µ0 (OH− ) ∼ ε(OH− ).

(13)
(14)

Substituting Eqs. 12–14 into Eq. 9, Eq. 8 results inlog[H+ ] ∼ !ε/2kT − 5/4 log T + 1/2 log[H2 O] + C,

where !ε = ε(H2 O) − ε(H+ ) − ε(OH+ ).

(15)
(16)

Note that !ε and C are independent of temperature and H2 O density. We obtained
experimental data for this relationship; mean values of the measured delay timechange at each set value of temperature for a constant trace moisture concentration,
1 µmol·mol−1 , are listed in Table 1. The standard deviation in the delay time change
measurement was 0.016 ppm as already discussed above. The uncertainty in temperature was due to the limited accuracy of a Peltier device which controlled thetemperature of the sensor unit [12]. The uncertainty in the temperature controlled by
the Peltier device was ±0.1 ◦ C according to the specification by the manufacturer.
Figure. 2 shows the mean values with a fitted line according to the first term in theright-hand side of Eq. 15. It should be noticed that the experimental data in Fig. 2 areconsistent with a positive value for !µ0 in Eq. 11. This indicates that H+ and OH−
would like to be in the amorphous silica layer rather than in the ambient gas when thetemperature goes down.

5 Discussion
It was shown that a simple model of dissociation of H2 O and dissolution of the resultant
H+ and OH− into the matrix of amorphous silica leads to a strikingly good agreementwith experimental dependences on the moisture density as shown in Fig. 1 and on the123

108

Page 6 of 8

Table 1 Change in the delay
time versus temperature at aconstant trace moistureconcentration, 1 µmol · mol−1

The values of delay time change
are the mean values, and thestandard deviation of thosemeasurements were 0.016 ppm.
The uncertainty in thetemperature was ±0.1 ◦ C

Int J Thermophys (2017) 38:108

Set values of temperature(◦ C)

Mean values of measured delay
time change (ppm)

24.0

6.401

30.0

4.572

35.0

3.250

40.0

2.234

45.0

1.726

50.0

1.219

55.0

0.914

60.0

0.610

65.0

0.408

70.0

0.305

75.0

0.204

Fig. 2 Delay time change versus temperature at a constant trace moisture concentration, 1 µmol·mol−1 .
The x coordinate is T −1 [K−1 ] and the y coordinate is logarithm of the delay time change in ppm. Each dot
represents a mean value of the measured delay time change. The standard deviation of those measurements
were 0.016 ppm, and the uncertainty in temperature was ±0.1 ◦ Ctemperature as shown in Fig. 2. Based on Eq. 8, we are developing a calibration method
to compensate for the deviations in both temperature and materials characteristics ofthe ball SAW device during the ultrasensitive moisture measurement. The calibration
method and its experimental validation will be reported in a separate paper.
The simple thermodynamic model presented in this paper is a phenomenologicalone where we have neglected the mechanism how the H+ and OH− react with thenetwork of amorphous silica which is an important process for the stiffening of theamorphous silica layer. It was suggested [11] that the water molecules dissociate andthe resultant H+ and OH− chemically react with the network in silica glass at anelevated temperature as in Eq. 1. Recently, it was pointed out by simulation [15] andexperiments [16] that the chemical reaction described by Eq. 1 does occur at roomtemperature. The combination of these studies with our theoretical and experimental

123

Int J Thermophys (2017) 38:108

Page 7 of 8108

work will be an important step toward the understanding of the reaction of watermolecules with the amorphous silica surface at room temperature in the very low
water molecule density.

6 Conclusion
It was shown in this paper that a very small amount of moisture at less than
1 µmol·mol−1 in the ambient N2 gas was measured by a ball SAW moisture sensor. The paper proposed a mechanism to understand how the water molecules in theambient gas affected the surface of the sensor made of single-crystal quartz with athin layer of amorphous silica. Experimental data were found in a good agreementwith a thermodynamic consideration of the physical model of the measurement. But
the model was empirical one, and a further study based on the atomistic model will benecessary for sufficient understanding of the phenomena. The adsorption and absorption of water molecules into silica glass have been a subject of materials research fordecades, but most research focused on a substantial amount of water at an elevatedtemperature. Only recently, a few papers came to appear discovering and simulating
the water molecules chemically bonding to the network of silicon and oxygen at thesurface of silica glass at room temperature. The authors hope that the present paper
paves a way to the research on the behavior of water molecules less than a monolayer
thick on the surface of silica glass at room temperature.
Acknowledgements This work was funded through START Program (program to create start-ups fromadvanced research and technology) by the Ministry of Education, Culture, Science and Technology in Japan.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changes were made.

References
1. K. Yamanaka, H. Cho, Y. Tsukahara, Appl. Phys. Lett. 76, 2797 (2000)
2. K. Yamanaka, H. Cho, Y. Tsukahara, IEICE Tech. Rep. US2000-14, 49 (2000) (in Japanese)
3. Y. Tsukahara, N. Nakaso, H. Cho, K. Yamanaka, Appl. Phys. Lett. 77, 2926 (2000)
4. Y. Tsukahara, N. Nakaso, K. Yamanaka, H. Cho, IEICE Tech. Rep. US2000-55, 31 (2000) (in Japanese)
5. K. Yamanaka, S. Ishikawa, N. Nakaso, N. Takeda, T. Mihara, Y. Tsukahara, IEEE Symp. Ultrason. 1,299 (2003)
6. B. Wyszynski, M. Sekine, T. Nakamoto, N. Nakaso, K. Noguchi, Sens. Actuators B: Chem. 144(1),247 (2010)
7. T. Iwaya, S. Akao, T. Sakamoto, T. Tsuji, N. Nakaso, K. Yamanaka, Jpn. J. Appl. Phys., 51, 07GC24-1-6(2012)
8. N. Takeda, M. Motozawa, Int. J. Thermophys. 33, 1642 (2012)
9. S. Hagihara, T. Tsuji, T. Oizumi, N. Takeda, S. Akao, T. Ohgi, K. Takayanagi, T. Yanagisawa, N.
Nakaso, Y. Tsukahara, K. Yamanaka, Jpn. J. Appl. Phys 53, 07KD08 (2014)
10. H. Seike, J. Acoust. Soc. A. 48, 1098 (1970)
11. M. Tomozawa, New Glass 21(3), 9 (2006). (in Japanese)
12. N. Takeda, T. Oizumi, T. Tsuji, S. Akao, K. Takayanagi, N. Nakaso, K. Yamanaka, in Proceedings ofTEMPMEKO 2013, HMC-1, 173, Oct. 14–18, Funchal, Portugal (2013)
13. A. Sieverts, Z. Metallkunde 21, 37 (1929)

123

108

Page 8 of 8

Int J Thermophys (2017) 38:108

14. T. Tsuji, R. Mihara, T. Saito, S. Hagihara, T. Oizumi, N. Takeda, T. Ohgi, T. Yanagisawa, S. Akao, N.
Nakaso, K. Yamanaka, Mater. Trans. 55, 1040 (2014)
15. C.S. Ewing, S. Bhavsar, G. Veser, J.J. McCarthy, J.K. Johnson, Langmuir 30, 5133 (2014)16. S.L. Warring, D.A. Beattie, A.J. McQuillan, Langmuir 32, 1568 (2016)123