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An impurity model for LAr detectors Aiwu Zhang, Craig Thorn, Yichen Li, Milind Diwan, Steve Kettle, Xin Qian, Jim Stewart, Chao Zhang CPAD Instrumentation Frontier Workshop Dec 8-10, 2019 Outline Motivation Model description

slide-1
SLIDE 1

An impurity model for LAr detectors

Aiwu Zhang, Craig Thorn, Yichen Li, Milind Diwan, Steve Kettle, Xin Qian, Jim Stewart, Chao Zhang CPAD Instrumentation Frontier Workshop Dec 8-10, 2019

slide-2
SLIDE 2
  • Motivation
  • Model description
  • Measurement of Henry’s coefficients for oxygen
  • Determination of impurity leak rate
  • Summary

Outline

2/14

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SLIDE 3

Motivation

  • Impurities in LAr (O2, H2O, etc.) reduce charge and light signals
  • Ultra-high purity LAr (<1 ppb) required for long drift distances (> 3.6 m)
  • A model is desired for understanding the dynamics of impurities in LAr
  • Important for detector optimization and operation

3/14

Q (fC) Oxygen concentration (ppm) Ref.: NIM 135 (1976) 151

slide-4
SLIDE 4

Model description – overview

4/14

Seven processes are considered for the impurity dynamics

LAr GAr 𝑙𝑒𝑗𝑑 𝑙𝑒𝑓𝑀 𝑠

π‘“π‘€π‘ž

β‘  β‘‘ 𝑠𝑑𝑗𝑠,π‘š β‘’ 𝑠

π‘“π‘€π‘ž

β‘£ π‘ π‘šπ‘“π‘™ β‘€ 𝑠

𝑑𝑏𝑛

β‘₯ 𝑙𝑏𝑒𝑑/𝑙𝑝𝑣𝑒 ⑦

slide-5
SLIDE 5

5/14

LAr GAr 𝑙𝑒𝑗𝑑 𝑙𝑒𝑓𝑀 𝑠

π‘“π‘€π‘ž

β‘  β‘‘ 𝑠𝑑𝑗𝑠,π‘š β‘’ 𝑠

π‘“π‘€π‘ž

β‘£ π‘ π‘šπ‘“π‘™ β‘€ 𝑠

𝑑𝑏𝑛

β‘₯ 𝑙𝑏𝑒𝑑/𝑙𝑝𝑣𝑒 ⑦

  • Ordinary differential equations for each process
  • E.g., for process #1:

π‘’π‘œπ‘—,𝑕 𝑒𝑒 = π‘œπ‘• βˆ’π‘‘π‘—,𝑕𝑙𝑒𝑗𝑑 + 𝑑𝑗,π‘šπ‘™π‘’π‘“π‘€ + 𝑑𝑗,𝑕 βˆ™ π‘’π‘œπ‘• 𝑒𝑒 , π‘’π‘œπ‘—,π‘š 𝑒𝑒 = βˆ’ π‘’π‘œπ‘—,𝑕 𝑒𝑒

π‘œπ‘—,π‘š, π‘œπ‘—,𝑕: amount of impurity in liquid, gas π‘œπ‘•: amount of argon in gas 𝑑𝑗,π‘š, 𝑑𝑗,𝑕: concentration in liquid, gas 𝑙𝑒𝑗𝑑, 𝑙𝑒𝑓𝑀: dissolution, devolution rates

Model description – overview

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SLIDE 6

The prediction from the model

  • The full model is constructed by summing up all processes
  • Concentrations are in non-linear 3rd order differential equations
  • By reducing the sampling (#6) and outgassing (#7) processes, analytical solutions:

6/14

𝑑𝑗,π‘š 𝑒 = 𝑑𝑑𝑑,π‘š + 𝐷1π‘“βˆ’π‘™πΊπ‘’ + 𝐷2π‘“βˆ’π‘™π‘‡π‘’, 𝑑𝑗,𝑕 𝑒 = 𝑑𝑑𝑑,𝑕 + 𝐷3π‘“βˆ’π‘™πΊπ‘’ + 𝐷4π‘“βˆ’π‘™π‘‡π‘’

Ultimate concentrations

Fast Component (~ secs) Slow Component (hrs)

𝐼 ≑ 𝑑𝑑𝑑,𝑕 𝑑𝑑𝑑,π‘š 𝐼 = π·π‘šπ‘“π‘π‘œπ‘—π‘œπ‘• 𝑠𝑏𝑒𝑓 𝑝𝑔 π‘π‘ π‘•π‘π‘œ πΉπ‘€π‘π‘žπ‘π‘ π‘π‘’π‘—π‘π‘œ 𝑠𝑏𝑒𝑓 𝑝𝑔 π‘π‘ π‘•π‘π‘œ

Definition of the Henry’s coefficient (at equilibrium)

  • Analyzing 𝑙𝑇:

Heating power to the LAr

The model predicts a way to measure the Henry’s coefficient

𝑑𝑑𝑑,π‘š, 𝑑𝑑𝑑,𝑕, 𝑙𝐺, 𝑙𝑇 are functions of the model parameters

slide-7
SLIDE 7
  • For studying basic properties of LAr: measured longitudinal diffusion of electrons

(NIMA 816 (2016) 160)

  • Gas purification only
  • Additional heating power can be varied 0-150 W
  • Oxygen and water concentrations measured by sampling LAr into gas analyzers

(0.2 ppb precision)

The BNL 20-L LAr test stand

Details: JINST 16 06 t06001

7/14

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SLIDE 8

Henry’s coefficient for oxygen (πΌπ‘ƒπ‘¦π‘§π‘•π‘“π‘œ)

  • Data used for analysis selected based on

slow control data (LAr level, heater temperature, etc.)

8/14

Oxygen concentration data (Feb. 2016 data set)

  • πΌπ‘ƒπ‘¦π‘§π‘•π‘“π‘œ = 0.84 Β± 0.04, consistent with literature
  • Cleaning rates measured at different

heating powers

𝐼 = π·π‘šπ‘“π‘π‘œπ‘—π‘œπ‘• 𝑠𝑏𝑒𝑓 𝑝𝑔 π‘π‘ π‘•π‘π‘œ πΉπ‘€π‘π‘žπ‘π‘ π‘π‘’π‘—π‘π‘œ 𝑠𝑏𝑒𝑓 𝑝𝑔 π‘π‘ π‘•π‘π‘œ = 𝑠

π‘‘π‘šπ‘œ

𝑠

π‘“π‘€π‘ž

slide-9
SLIDE 9

Understanding the water data …

  • The water case is more complicated
  • outgassing process (#7) can’t be ignored
  • adsorption on surfaces may explain the fast cleaning observed in data

9/14

Water concentration data (Feb. 2016 data set)

𝐼π‘₯𝑏𝑒𝑓𝑠 = 3 Γ— 10βˆ’9 from NIST REFPROP

  • from equation of state calculation
  • Water vapor pressure ~10βˆ’22 bar (at 90 K)

(extrapolated from empirical equations)

  • More data are needed
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SLIDE 10

Another application - Numerical fit to the data

10/14

  • The leak rate can be determined:
  • ~ 5Γ—10-6 mole/h with purification off;
  • ~10-7 mole/h with purification on;
  • It is further reduced when heating

power is increased.

  • The full model is numerically fitted to the data
  • The measured Henry’s coefficient is used;
  • The purification off regions also fitted

100W 0W 30W 100W 0W Purification off

  • The model fits the data very well
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SLIDE 11

Keeping impurities away from the LAr

  • The dependence of leak rate on the input heating power can be explained by a

simple diffusion model:

𝑠

π‘“π‘€π‘ž the evaporation rate

π‘Š

𝑛 the mole volume of GAr,

𝐡𝑑 the cross sectional area perpendicular to the flow direction 𝐸 the diffusion coefficient of the impurity

𝑑𝑗,𝑕 𝑦 = 𝐷 βˆ™ 𝑓

βˆ’ π‘ π‘“π‘€π‘žβˆ™π‘Š

𝑛

πΈβˆ™π΅π‘‘ βˆ™π‘¦

  • The larger 𝑠

π‘“π‘€π‘ž (higher heating power), or the smaller cross sectional area (𝐡𝑑),

  • The smaller the concentration in the gas (𝑑𝑗,𝑕).
  • Adding a baffle in the GAr near the top region is expected

to help keeping impurities from reaching the LAr surface.

11/14

Leak from the top Diffusion LAr Evaporation x

Ref: K. W. Reus et al., Diffusion coefficients in flowing gas. I.

𝑠

π‘“π‘€π‘ž

π‘„β„Žπ‘“π‘π‘’π‘—π‘œπ‘•

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SLIDE 12

Future work on impurities

  • Understand water impurity with

more data; all other impurities

  • Verification of the baffle idea
  • Electron attachment rate
  • Electron lifetime
  • vs. impurity concentration
  • vs. E-field

12/14

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SLIDE 13

The tool under developing – LArFCS

  • Mainly for field response in LArTPCs
  • Contains ~250-L LAr
  • LAr purity can achieve < 1 ppb level in ~1 week, with continuous gas purification and one

time liquid purification in the LAr filling line

13/14

  • An ideal place for

further studying the impurity performances

55.5” ID 22”

  • Expected cryogenic operation

and purity demonstration soon

  • More details, please refer to
  • Dr. Yichen Li (yichen@bnl.gov)

who is also attending this workshop

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SLIDE 14

Summary

  • A mathematical model for impurities in LAr is constructed and validated
  • It predicts a way of measuring the Henry’s coefficient for an impurity in argon.
  • The measured Henry’s coefficient for oxygen is 0.84Β±0.04, which is

consistent with literature;

  • It suggests adding a baffle will help in reducing impurity concentrations in the

detector.

  • More studies are expected to come about with the LArFCS.

14/14

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SLIDE 15

Backup …

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SLIDE 16

The full model

(1) Impurity exchange at liquid-gas surface π‘’π‘œπ‘—,𝑕 𝑒𝑒 = π‘œπ‘• βˆ’π‘‘π‘—,𝑕𝑙𝑒𝑗𝑑 + 𝑑𝑗,π‘šπ‘™π‘’π‘“π‘€ + 𝑑𝑗,𝑕 βˆ™ π‘’π‘œπ‘• 𝑒𝑒 , π‘’π‘œπ‘—,π‘š 𝑒𝑒 = βˆ’ π‘’π‘œπ‘—,𝑕 𝑒𝑒 (2) Evaporation of LAr π‘’π‘œπ‘—,π‘š 𝑒𝑒 = βˆ’π‘‘π‘—,π‘šπ‘ 

π‘“π‘€π‘ž,

π‘’π‘œπ‘—,𝑕 𝑒𝑒 = βˆ’ π‘’π‘œπ‘—,π‘š 𝑒𝑒 (3) Purification of LAr in liquid phase π‘’π‘œπ‘—,π‘š 𝑒𝑒 = βˆ’π‘‘π‘—,π‘šπ‘ 

𝑑𝑗𝑠,π‘š

π‘’π‘œπ‘—,𝑕 𝑒𝑒 = 0 (4) Purification of GAr and its condensation π‘’π‘œπ‘—,π‘š 𝑒𝑒 = 1 βˆ’ πœ—π‘„ 𝑑𝑗,𝑕𝑠

𝑑𝑗𝑠,𝑕,

π‘’π‘œπ‘—,𝑕 𝑒𝑒 = βˆ’π‘‘π‘—,𝑕𝑠

𝑑𝑗𝑠,𝑕,

𝑠

𝑑𝑗𝑠,𝑕 = 𝑠 π‘“π‘€π‘ž =

π‘„π‘—π‘œ βˆ†πΌπ‘“π‘€π‘ž (5) Leakage of impurities from outside source π‘’π‘œπ‘—,π‘š 𝑒𝑒 = 0, π‘’π‘œπ‘—,𝑕 𝑒𝑒 = 𝑠

π‘šπ‘“π‘™,

(6) Sampling of Ar π‘’π‘œπ‘—,π‘š 𝑒𝑒 = βˆ’π‘‘π‘—,π‘šπ‘ 

𝑑𝑏𝑛 1 + πœ€πœ ,

π‘’π‘œπ‘—,𝑕 𝑒𝑒 = 𝑑𝑗,π‘šπ‘ 

π‘‘π‘π‘›πœ€πœ,

πœ€πœ = πœπ‘• πœπ‘š βˆ’ πœπ‘• (7) Outgassing of impurities 𝑒𝑑𝑗,𝑑 𝑒𝑒 = 𝑑𝑗,𝑕𝑙𝑏𝑒𝑑 𝑑𝑗,𝑑

𝑑𝑏𝑒 βˆ’ 𝑑𝑗,𝑑 βˆ’ 𝑙𝑝𝑣𝑒𝑑𝑗,𝑑, π‘’π‘œπ‘—,𝑑 𝑒𝑒 = 𝐡𝑏𝑒𝑑 𝑒𝑑𝑗,𝑑 𝑒𝑒 ,

π‘’π‘œπ‘—,𝑕 𝑒𝑒 = βˆ’ π‘’π‘œπ‘—,𝑑 𝑒𝑒 , π‘’π‘œπ‘—,π‘š 𝑒𝑒 = 0

  • π‘œπ‘—,π‘š, π‘œπ‘—,𝑕: amount of impurity in moles in liquid

and gas

  • π‘œπ‘š, π‘œπ‘•: amount of LAr and GAr in moles
  • 𝑑𝑗,π‘š, 𝑑𝑗,𝑕: concentration in liquid and gas, in mole
  • f impurity per mole of argon
  • 𝑠

π‘“π‘€π‘ž, 𝑠 𝑑𝑗𝑠,𝑕, 𝑠 𝑑𝑗𝑠,π‘š, 𝑠 π‘šπ‘“π‘™, 𝑠 𝑑𝑏𝑛: rates (in mole/s) for

LAr evaporation, GAr circulation, LAr circulation, impurity leakage, LAr sampling

  • 𝑠

π‘“π‘€π‘ž = 𝑠 𝑑𝑗𝑠,𝑕 = π‘„π‘—π‘œ βˆ†πΌπ‘“π‘€π‘ž, with π‘„π‘—π‘œ being the total

heat (in W) into the LAr, including the heat power leakage, and βˆ†πΌπ‘“π‘€π‘ž = 161.14𝐾/𝑕: LAr heat of vaporization

  • 𝑙𝑒𝑗𝑑, 𝑙𝑒𝑓𝑀: rates (in sβˆ’1) for dissolution and

devolution at the liquid-gas surface

  • 𝑙𝑒𝑓𝑀 = 𝐼𝑦𝑦𝑙𝑒𝑗𝑑, with 𝐼𝑦𝑦 describing the

Henry’s coefficient for the impurity in argon

  • 𝑙𝑏𝑒𝑑, 𝑙𝑝𝑣𝑒: rates (in sβˆ’1) for impurity adsorption

and outgassing

  • πœπ‘•, πœπ‘š: number density of GAr and LAr,

πœ€πœ~0.005 for LAr at 90K

  • 𝑑𝑗,𝑑 ∢ impurity concentration on outgassing

surface per unit area; 𝑑𝑗,𝑑

𝑑𝑏𝑒: the adsorbed

impurity could be saturated

  • πœ—π‘„ the efficiency of the GAr purifier

π‘œπ‘—,π‘š = 𝑑𝑗,π‘šπ‘œπ‘š, π‘œπ‘—,𝑕 = 𝑑𝑗,π‘•π‘œπ‘•

16

  • Ref. for (1): G. M. Nathanson et. al., β€œDynamics and Kinetics at the Gas-

Liquid Interface”, J. Phys. Chem., 100(31):13007-13020, 1996.

  • Ref. for (7): J. Zhang, β€œPhysical insights into kinetic models of adsorption”,

Separation and Purification Technology, 229:115832, 2019.

slide-17
SLIDE 17

The full model

𝑒𝑑𝑗,π‘š 𝑒𝑒 = βˆ’ 1 π‘œπ‘š βˆ™ 𝑠

𝑑𝑗𝑠,π‘š + 𝑠 π‘“π‘€π‘ž + 𝐼𝑦𝑦𝑙𝑒𝑗𝑑

π‘œπ‘• π‘œπ‘š βˆ™ 𝑑𝑗,π‘š + 1 π‘œπ‘š βˆ™ π‘™π‘’π‘—π‘‘π‘œπ‘• + 1 βˆ’ πœ—π‘„ 𝑠

π‘“π‘€π‘ž βˆ’ 𝑠 π‘‘π‘π‘›πœ€πœ βˆ™ 𝑑𝑗,𝑕,

𝑒𝑑𝑗,𝑕 𝑒𝑒 = 1 π‘œπ‘• βˆ™ π‘œπ‘•πΌπ‘¦π‘¦π‘™π‘’π‘—π‘‘ + 𝑠

π‘“π‘€π‘ž + 𝑠 π‘‘π‘π‘›πœ€πœ βˆ™ 𝑑𝑗,π‘š βˆ’ 1

π‘œπ‘• βˆ™ π‘œπ‘•π‘™π‘’π‘—π‘‘ + 𝑠

π‘“π‘€π‘ž + 𝐡𝑏𝑒𝑑𝑙𝑏𝑒𝑑 𝑑𝑗,𝑑 𝑑𝑏𝑒 βˆ’ 𝑑𝑗,𝑑

βˆ™ 𝑑𝑗,𝑕 + 1 π‘œπ‘• 𝑠

π‘šπ‘“π‘™ + 𝐡𝑏𝑒𝑑𝑙𝑝𝑣𝑒𝑑𝑗,𝑑 , 𝑒𝑑𝑗,𝑑 𝑒𝑒

= 𝑑𝑗,𝑕𝑙𝑏𝑒𝑑 𝑑𝑗,𝑑

𝑑𝑏𝑒 βˆ’ 𝑑𝑗,𝑑 βˆ’ 𝑙𝑝𝑣𝑒𝑑𝑗,𝑑, (this equation is from the outgassing process)

π‘œπ‘š = π‘œ0,π‘š βˆ’ 𝑠

𝑑𝑏𝑛 βˆ™ 𝑒 βˆ™ 1 + πœ€πœ ,

π‘œπ‘• = π‘œ0,𝑕 + 𝑠

𝑑𝑏𝑛 βˆ™ 𝑒 βˆ™ πœ€πœ

  • Summing up all processes, the equations describing the impurity concentrations are expressed in the following

17

𝑏4 = 𝑏1π‘œ0,𝑕 + 𝑏2π‘œ0,π‘š π‘œ0,π‘•π‘œ0,π‘š , 𝑏5 = 𝑏1𝑠

π‘“π‘€π‘ž + 𝑏0𝑠 𝑑𝑗𝑠,π‘š

π‘œ0,π‘•π‘œ0,π‘š , 𝑏0,π‘š = 𝑏0𝑠

π‘šπ‘“π‘™

π‘œ0,π‘•π‘œ0,π‘š , 𝑏0,𝑕 =

𝑏1π‘ π‘šπ‘“π‘™ π‘œ0,π‘•π‘œ0,π‘š,

𝐷1 to 𝐷4 are determined by initial conditions 𝑏0 = π‘™π‘’π‘—π‘‘π‘œ0,𝑕, 𝑏1 = 𝑠

π‘“π‘€π‘ž + 𝑠 𝑑𝑗𝑠,π‘š + πΌπ‘¦π‘¦π‘™π‘’π‘—π‘‘π‘œ0,𝑕,

𝑏2 = 𝑠

π‘“π‘€π‘ž + π‘™π‘’π‘—π‘‘π‘œ0,𝑕,

𝑏3 = 𝑠

π‘“π‘€π‘ž + πΌπ‘¦π‘¦π‘™π‘’π‘—π‘‘π‘œ0,𝑕.

The coefficients in the solution are

𝑑𝑗,π‘š 𝑒 = 𝑑𝑑𝑑,π‘š + 𝐷1π‘“βˆ’π‘™πΊπ‘’ + 𝐷2π‘“βˆ’π‘™π‘‡π‘’, 𝑑𝑗,𝑕 𝑒 = 𝑑𝑑𝑑,𝑕 + 𝐷3π‘“βˆ’π‘™πΊπ‘’ + 𝐷4π‘“βˆ’π‘™π‘‡π‘’

slide-18
SLIDE 18

From Folippo Resnati’s talk https://indico.fnal.gov/event/21535/contribution/4/material/slides/0.pdf

slide-19
SLIDE 19

Ref: β€œSolubility of Water in Compressed Nitrogen, Argon, and Methane” by M. Rigby and J. M. Prausnitz, Journal of Physical Chemistry, Vol 72 (1), 1968.

slide-20
SLIDE 20

Henry’s law

At a constant T, the amount of a given gas that dissolves in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid. An equivalent way of stating the law: the solubility (C, unit in mL/L e.g.) of a gas in a liquid is proportional to the partial pressure (unit in atm e.g.)

  • f the gas above the liquid:

C = 𝑙𝑄

𝑕𝑏𝑑

Henry solubility 𝐼𝐷𝑄 ≑ 𝑑𝑏/π‘ž with 𝑑𝑏 the concentration of a species in liquid and π‘ž the partial pressure of that species in gas phase. The SI unit for 𝐼𝐷𝑄 is

π‘›π‘π‘š 𝑛3𝑄𝑏 or often used as π‘›π‘π‘š π‘€βˆ™π‘π‘’π‘›.

𝐼𝐷𝑄 can be expressed as the dimensionless ratio between 𝑑𝑏 and 𝑑𝑕, the concentration in gas phase: 𝐼𝐷𝐷 ≑ 𝑑𝑏/𝑑𝑕 And 𝐼𝐷𝐷 = 𝐼𝐷𝑄 βˆ™ π‘†π‘ˆ for ideal gas with 𝑆, π‘ˆ the gas constant and temperature. Another Henry’s law solubility constant is 𝐼𝑦𝑄 ≑ 𝑦/π‘ž with 𝑦 the molar mixing ratio in the liquid. The conversion between 𝑦 and 𝑑𝑏 is 𝑑𝑏 β‰ˆ 𝑦 βˆ™ πœπ‘€/𝑁𝑀, 𝜍, 𝑁 are density and molar mass of the liquid. Therefore 𝐼𝑦𝑄 =

ML ρ𝑀 βˆ™ 𝐼𝐷𝑄. 𝐼𝑦𝑄 has an SI unit of π‘„π‘βˆ’1.

Henry solubility defined via molality: 𝐼𝑐𝑄 ≑ 𝑐/π‘ž with 𝑐 representing molality (of the solved species in liquid). 𝐼𝑐𝑄 has SI unit of π‘›π‘π‘š βˆ™ π‘™π‘•βˆ’1 βˆ™ π‘„π‘βˆ’1. If there is only one solute in the solvent, 𝑐 can be related with 𝑑𝑏 by 𝑑𝑏 = π‘πœπ‘€/(1 + 𝑐𝑁𝑀) β‰ˆ π‘πœπ‘€ (approximation is valid at very small concentration), thus 𝐼𝑐𝑄 = 𝐼𝐷𝑄/πœπ‘€. Henry volatility is defined as 𝐿𝐼

𝑄𝐷 = π‘ž 𝑑𝑏 = 1/𝐼𝐷𝑄; similarly there are other definitions of volatility terms, I ignore them here.

Ref: Atmos. Chem. Phys., 15, 4399-4981, 2015,

  • R. Sander, β€œCompilation of Henry’s law constants for water as solvent”

The Henry’s coefficient we refer to KH = 𝑑𝑕/𝑑𝑏 (volatility term)

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SLIDE 21

MICROBOONE- NOTE-1026-PUB NIMA 305 (1991) 177

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SLIDE 22

Ref: G. T. Preston et. al., β€œSolubilities of hydrocarbons and carbon dioxide in liquid methane and in liquid argon”, J Phys. Chem., 75(15):2345, 1971