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Thermal diffusivity and specific of linear alkylbenzene

Wenjie Wu1,2†, Guolei Zhu3†, Qingmin Zhang4, Xiang Zhou1‡, Yayun Ding5, Haoxue Qiao1 and Jun Cao5 1 Hubei Nuclear Physics Key Laboratory, Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education, and School of Physics and Technology, Wuhan University, Wuhan 430072, China 2 Key Laboratory of Particle & Radiation Imaging (Tsinghua University), Ministry of Education, Tsinghua University, Beijing 100084, China 3 School of Marine Science and Technology, Northwestern Polytechnical University, Xian 710072, China 4 Department of Nuclear Science and Technology, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China 5 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China E-mail: [email protected]

Abstract. We report the measurements of the thermal diffusivity and the isobaric specific heat capacity of linear alkylbenzene at about 23 ◦C with the standard atmosphere, which are measured for the first . The conductivity, , and of were derived from our measurements based on thermodynamic relations. The dependence of heat capacity ratio and isobaric specific heat capacity were calculated and relevant results were discussed.

Keywords: neutrino detector, linear alkylbenzene, specific heat capacity, thermal diffusivity, speed of sound

Submitted to: Phys. Scr. arXiv:1904.12147v1 [physics.ins-det] 27 Apr 2019

† These authors contributed equally to this work ‡ Corresponding author Thermal characteristics of LAB 2

1. Introduction 10, 11]. LAB is a mixture of organic compounds with the formula of C6H5CnH2n+1 (n = 10 ∼ 13) [12]. scintillator (LS) is applied to neutrino detection Alkylbenzenes with different numbers of alkyls have since the first observation of neutrino in 1956 [1]. It varied properties. Thus LABs in two batches could plays an important role in various neutrino researches have different performance. For constructing a gigantic such as the solar neutrino problem [2], precise detector, a large quantity of LAB is required. The measurements of the third mixing angle θ13 [3, 4, temperature difference between the experimental hall 5], and the detection of neutrino-less double beta and the detector would cause a large amount of decay [6]. In order to collect enough signals, the heat transfer. Hence the heat capacity of LAB is fiducial mass of the neutrino target should be as needed to estimate the requirements to the filling and large as possible, since neutrinos are interacting with cooling processes for controlling the temperature of the matter via the weak interaction only. Meanwhile, the detector. In addition, the hydrodynamic motion inside radioactive backgrounds of LS are required to be as the detector should be well-studied for possible physics low as possible to gain sensitivity. For a monolithic analysis. The thermal diffusivity is one of the input gigantic detector, the thermal equilibrium condition for parameters to do the computational fluid dynamics LS is difficult to achieve. A nonuniform distribution of simulation. temperature could cause the convection of LS inside In this paper, we report the measurements of the detector. The convection is an obstacle to the the thermal diffusivity and the isobaric specific heat detection of low energy and rare events since it would capacity of LAB at about 23 ◦C with 1 atm. We have bring peripheral backgrounds, such as 210Pb, to the deduced the , heat capacity ratio central volume of the detector. and speed of sound based on these measurements. Recently, the non-equilibrium phenomenon is observed by Borexino, which appeares as an annual 2. Experimental method modulation in the spatial distribution of 210Po inside the detector filled with about 280 tonnes LS [7]. The 2.1. The thermal diffusivity correlation between environmental temperature and background distribution indicates that the migration of The thermal diffusivity α was measured by a dynamic backgrounds is driven by the fluid transportation due light scattering (DLC) method, and the schematic to thermal convection. One of the main goals of the diagram can be found in literature [13]. The light last operation phase of Borexino is the observation of emitted from a laser was split in two. The more intense the CNO solar neutrinos whose dominant radioactive one was guided to transmit the sample and the weaker background is 210Bi. The purified LS after phase one was used as a reference to make the apparatus one and the thermal insulation of the detector working at the heterodyne mode. Mixed light of the give a possibility to achieve this goal. Precise scattered light and the reference light was limited by determination of 210Bi concentration is essential to two pin holes to make sure that only light scattered by the sensitivity of CNO neutrinos. The β− decay of a small volume was detected by the photon counting − 2 is indistinguishable from the νe-e elastic head. The time autocorrelation function G (τ) was scattering. Conversely, the α decay of polonium measured by a digital correlator in the cross correlation 2 can be effectively picked out through Pulse-Shape mode. Based on a least-squares fitting of G (τ), the Discrimination techniques. Thus the concentration of decay constant τR can be determined and the thermal 2 210Bi can be determined through the selection of 210Po diffusivity was calculated by α = 1/(q τR), where while the secular equilibrium is reached. A stable q is the modulus of scattering vector which can be condition of 210Po is of importance to the high precision derived from the and incident angle of 210Bi concentration. Many efforts have been devoted incident light [13]. The of the sample was to monitor and control the thermal environment of controlled by a pump system and was monitored by Borexino [7]. Other LS detectors could also be suffered a pressure transmitter (Rosemont 3051S). An electric from the stability problem of backgrounds caused by heater was deployed in the sample container and a thermal convection. Therefore, the thermal quantities resistance thermometer was used to measure of LS are necessary to be measured for studying the the temperature. The relative uncertainties are 3.76 × −5 performance of the detector. 10 , 0.001, 0.015 for the measurements of wavelength, LS usually consists of a solvent and a scintillating incident angle and the decay constant respectively. solute. Linear alkylbenzene (LAB) is one kind of The relative uncertainty of thermal diffusivity is ±1.5% solvent which is used in Daya Bay, RENO and which was derived through error propagation [13]. SNO+ [3, 4, 6, 8]. It is taken into consideration for the next generation neutrino experiments as well, such as JUNO, Jinping Neutrino Experiment and LENA [9, Thermal characteristics of LAB 3

2.2. The isobaric specific heat capacity where βP is the isobaric thermal expansion coefficient and κ is the isothermal . β and κ The isobaric specific heat capacity c was measured T P T P were analyzed based on the variance of with by a flow calorimeter, and the schematic diagram can respect to temperature or pressure [15]. be found in literature [14]. The apparatus consists The heat capacity ratio γ for LAB is defined as of a plunger type pump, a preheater, a calorimeter c and a container for the measured sample. The pump γ = P = 1.182 ± 0.005. (5) c (Scientific Systems, Series 1500 HPLC Pump) and V the preheater were used to provide samples with The uncertainty of γ is dominated by the measure- controlled flow rate, pressure and temperature. A ments of βP , ρ and κT since the correlated uncertainty micro-heater was deployed inside the calorimeter as from cP are canceled. The relative uncertainty of γ a heat source. The temperature increment of the reaches 0.4%. sample entering and exiting the calorimeter ∆T was The adiabatic compressibility κS can be derived measured by two platinum resistance thermometers as κT −4 −1 (PRT, Fluke Corporation). Due to the inefficiency κS = = (6.553 ± 0.039) × 10 MPa , (6) of the heat absorption of the sample, a fraction of γ the heat was absorbed by the calorimeter. The where κT is the isothermal compressibility which has sample collected by the container was weighed to been measured in the literature [15]. calibrate the flow rate with an analytical balance According to the Newton-Laplace formula, the (ME204, Mettler Toledo). Then the isobaric heat thermodynamic speed of sound in LAB can be capacity can be determined by the relative difference calculated as of the thermal power of the micro-heater with respect 1 −1 vsound = √ = 1336.34 ± 3.97 m · s . (7) to the variance of the flow rate once ∆T is fixed. κSρ High accuracy was reached by keeping the calorimeter adiabatic from the environment. The experimental 3.3. Estimation of the temperature dependence of γ uncertainties are ±0.0001 W, ±0.001 g/s and ±0.01 K and cP for the measurement of thermal power, flow rate and Different experiments may work at different tempera- temperature respectively. The relative uncertainty of tures. The temperature dependence of γ and c can isobaric heat capacity is ±0.98% which was derived P be useful. According to a previous research [15], the through error propagation [14]. density at different with 0.1 MPa was described as 3. Results ρ(t) = ρ0[1 − β0(t − t0)], (8) ◦ 3.1. Measurement results where t is the temperature, t0 is 23 C, ρ0 is The thermal diffusivity α was measured by the DLC the density at t0 with 0.1 MPa. And the isobaric method at 22.98 ◦C with 1 atm to be thermal expansion coefficient at different temperatures was described as α = (7.245 ± 0.109) × 10−8 m2 · s−1. (1) β0 The isobaric specific heat capacity cP was βP (t) = , (9) 1 − β0(t − t0) measured by the flow method at 23.15 ◦C with 1 atm to be where β0 is the isobaric expansion coefficient at t0 with −1 ◦ −1 0.1 MPa. The isothermal compressibility at different cP = 2304 ± 23 J · Kg · C . (2) temperatures were fitted with an empirical equation 0 3.2. Thermal conductivity, isochoric specific heat κT (t) = κ0 + κt(t − t0), (10) capacity, heat capacity ratio and thermodynamic speed where κ0 is the isothermal compressibility at t0, and 0 of sound κt is an empirical coefficient. The speed of sound at different temperatures was described by a linear Thermal conductivity λ is the property of a material function as to transfer heat which is defined as 0 −1 ◦ −1 vsound(t) = v0 + vt(t − t0), (11) λ = αρcP = (0.1426 ± 0.0026) W · m · C , (3) 0 where ρ is the density of LAB which has been measured where v0 is the speed of sound at t0 and vt is at different temperatures and [15]. the gradient of speed of sound with respect to the The isochoric specific heat capacity can be temperature. obtained based on thermodynamic relations as [16] The heat capacity ratio at 0.1 MPa can be derived 2 as T βP −1 ◦ −1 cV = cP − = 1950 ± 24 J · Kg · C , (4) γ(t) = ρ(t)κ (t)v2 (t) (12) ρκT T sound Thermal characteristics of LAB 4 from (6)(7) while ρ(t), κT (t) and vsound(t) are known. shows that alkylbenzenes with numbers of alkyls have Similarly, the isobaric specific heat capacity at 0.1 MPa different sound [18]. Since the LAB samples can be derived as used in two measurements come from different batches, 2 2 different composition could affect the speed of sound. vsound(t)βP (t)t cP (t) = 2 (13) Assuming that the temperature dependences of sound ρ(t)κT (t)v (t) − 1 sound speed for two batches of LAB are the same, regardless from (4)(5)(12) while ρ(t), βP (t), κT (t) and vsound(t) of the absolute value of sound speed. Then the speed ◦ are known. ρ(t), βP (t) and κT (t) from 4 to 23 C has of sound in LAB of this study can be written as been reported [15]. The parameters of (8)(9)(10) are −1 0 (11) with v0=1336.34 m·s and v remains the same. listed in table 1. t Therefore, γ(t) and cP (t) can be calculated from (12) and (13) respectively. Figure 2 and figure 3 show Table 1. Parameters of (8)(9)(10) obtained from [15]. that γ decreases and cP increases respectively when temperature increases. Parameter Value

−3 ρ0 854.6±0.1 kg·m −4 ◦ −1 β0 (8.894±0.094)×10 C 1.35 −4 −1 κ0 (7.738±0.027)×10 MPa κ0 (3.227±0.259)×10−6 MPa−1 · ◦C−1 t 1.3

1.25

1420 heat capacity ratio 1.2 1415 1410 1.15 1405 1400 1.1 1395 4 6 8 10 12 14 16 18 20 22 ° speed of sound [m/s] 1390 temperature [ C] 1385 Figure 2. Temperature dependence of heat capacity ratio γ(t) 1380 1375 1370 16 17 18 19 20 21 22 23 C)]

° °

temperature [ C] ⋅

2400 Figure 1. Temperature dependence of speed of sound vsound(t). Black dots are measured data and the red line is fitting result with (11). Fitting result gives v =1382 m·s−1 and v0 =−3.92 0 t 2200 m·s−1 · ◦C−1

◦ ◦ vsound(t) from 16 C to 23 C was measured 2000 by a commercial sound velocity profiler (Valeport miniSVP). The speed of sound was calculated from 1800 the time taken to travel a known distance for a single pulse of sound whose is 2.5 MHz. The 1600 measured result is shown as the black points in figure 1. isobaric [J/(kg 4 6 8 10 12 14 16 18 20 22 Equation (11) was used to describe the temperature temperature [°C] dependence of the speed of sound and extrapolate the lower limit of temperature to 4 ◦C. The fitting result is Figure 3. Temperature dependence of isobaric specific heat capacity cP (t) shown as the red line in figure 1 and it gives v0=1382 −1 0 −1 ◦ −1 m·s and vt=−3.92 m·s · C . There is a 3.4% difference between v0 and (7). Equation (7) is the thermodynamic speed of sound which is the limitation 4. Conclusions and discussions of v0 when the frequency of sound approaching zero. Nevertheless, two speed of sound are indistinguishable We measured the thermal diffusivity and the isobaric ◦ while the critical frequency is at GHz magnitude which specific heat capacity of LAB at about 23 C with 1 is much larger than 2.5 MHz [17]. There is a study atm. The thermal conductivity, heat capacity ratio Thermal characteristics of LAB 5 and the speed of sound were calculated. Results are [3] F. P. An, J. Z. Bai, A. B. Balantekin, et al. Observation summarized as following: of -Antineutrino Disappearance at Daya Bay. Physical Review Letters, 108(17):171803, April 2012. • α = (7.245 ± 0.109) × 10−8 m2 · s−1; [4] RENO Collaboration, J. K. Ahn, S. Chebotaryov, −1 ◦ −1 et al. Observation of Reactor Electron Antineutrinos • cP = 2304 ± 23 J · Kg · C ; Disappearance in the RENO Experiment. Physical • λ = (0.1426 ± 0.0026) W · m−1 · ◦C−1; Review Letters, 108(19):191802, May 2012. [5] Double Chooz Collaboration, Y. Abe, C. Aberle, et al. Indi- • γ = 1.182 ± 0.005; cation of Reactor νe Disappearance in the Double Chooz −1 Experiment. Physical Review Letters, 108(13):131801, • vsound = 1336.34 ± 3.97 m · s . March 2012. The 3.4% deviation of speed of sound between two [6] S. Andringa, E. Arushanova, S. Asahi, et al. Current Status batches of LAB suggests that the final LAB used for and Future Prospects of the SNO+. Advances in High Energy Physics, 2016. the experiment should be measured in situ while the [7] D. Bravo-Bergu˜no,R. Mereu, P. Cavalcante, et al. The composition is fixed. Based on the assumption that Borexino Thermal Monitoring & Management System 0 and simulations of the fluid-dynamics of the Borexino two batches of LAB have the same vt, the temperature dependence of heat capacity ratio γ(t) and isobaric detector under asymmetrical, changing boundary condi- tions. Nuclear Instruments and Methods in Physics Re- specific heat capacity cP (t) were estimated. search Section A: Accelerators, Spectrometers, Detectors Considering the nominal value of cP , the heat and Associated Equipment, 885:38–53, March 2018. capacity per kiloton LS can be estimated as 2.3 × [8] Vincent Fischer. Search for neutrinoless double-beta 109 J · ◦C−1. It’s a large amount of heat which should decay with SNO+. arXiv:1809.05986 [hep-ex, physics:physics], September 2018. be taken into account at the LS filling and cooling [9] Fengpeng An, Guangpeng An, Qi An, et al. Neutrino stage due to the temperature difference of storage physics with JUNO. Journal of Physics G: Nuclear and and operation. For a typical structure of neutrino Particle Physics, 43(3):030401, 2016. [10] John F. Beacom, Shaomin Chen, Jianping Cheng, et al. experiments, LS is separated from the buffer which is Physics prospects of the Jinping neutrino experiment. used to shield external radioactivity. For example, the Chinese Physics C, 41(2):023002, 2017. container of LS for JUNO is made of acrylic and [11] Michael Wurm, John F. Beacom, Leonid B. Bezrukov, is chosen as the buffer [9]. The conductivity of acrylic et al. The next-generation liquid-scintillator neutrino −1 ◦ −1 observatory LENA. Astroparticle Physics, 35(11):685– is about 0.2 W·m · C and that of water is about 732, June 2012. −1 ◦ −1 0.6 W·m · C which are larger than LAB [19]. [12] Yayun Ding, Zhiyong Zhang, Jinchang Liu, et al. A new Therefore heat is easier transferred from LAB to water, -loaded liquid scintillator for reactor neutrino and the residual heat in water could be piled up at detection. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detec- the peripheral region of acrylic which would induce the tors and Associated Equipment, 584(1):238–243, January convection of LS. Therefore, it is important to maintain 2008. the thermodynamic equilibrium as long as possible [13] Sheng Wang, Ying Zhang, Maogang He, Shi Zhang, and which is a challenge for future neutrino experiments. Xiong Zheng. Thermal diffusivity of di-isopropyl ether (DIPE) in the temperature range 298–530K and pressure up to 10MPa from dynamic light scattering (DLS). Fluid Acknowledgments Phase Equilibria, 376:202–209, August 2014. [14] Maogang He, Chao Su, Xiangyang Liu, Xuetao Qi, Nan Lv. Measurement of isobaric heat capacity of pure water up This work was supported by the Major Program of to supercritical conditions. The Journal of Supercritical the National Natural Science Foundation of China Fluids, 100:1–6, May 2015. (Grant No. 11390381), the Strategic Priority Research [15] X. Zhou, Q. M. Zhang, Q. Liu, et al. , Program of the Chinese Academy of Sciences (Grant isobaric thermal expansion coefficients and isothermal of linear alkylbenzene. 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