Foundations of Molecular-Flow Networks for Vacuum System Analysis [1 ed.] 0128186879, 9780128186879

Table of contents :
FOUNDATIONS OF MOLECULAR-FLOW NETWORKS FOR VACUUM SYSTEM ANALYSIS
Copyright
Biography
Profile of author
Books by Nagamitsu Yoshimura
Preface
Introduction
Acknowledgments
1 . Physical basis of molecular-flow networks: adsorption, desorption, diffusion, and outgassing/pumping
Key-point
Keywords
1. Introduction
2. Modeling of outgassing
3. Pressure analysis using equivalent vacuum circuit
4. Conclusion
References in the paper[1-1]
Reviewed paper
Reviewed paper
1. Introduction
2. Characteristic values of a solid material as a gas source
3. Differential pressure-rise method
A Principle
B Measurement of net outgassing rates of solid materials
C Estimation of the characteristic values PX and K0 of Viton O-rings
4. Vacuum circuit composed of the characteristic values of constituent elements
5. Conclusion
References in the paper[1-3]
Reviewed paper
2. Degassing time at constant temperature
3. Outgassing rate with variable temperature
References in the paper[1-4]
Reviewed paper
2. Analysis
4. First pump-down
5. Second pump-down
6. Summary
References in the paper[1-5]
Reviewed paper
2 . Applications of the resistor-network simulation method to complicated ultrahigh vacuum systems
Key-point
Keywords
[1] Resistor network simulation method
[2] Long history of works on molecular-flow networks
1. Introduction
2. Concept of a vacuum system
A New concept of a vacuum system and its components
3. Simulation of the high vacuum system of an electron microscope
A Procedures to design a simulator circuit
B The simulator circuit
C Application results
4. Discussion
5. Summary
Acknowledgments
References in the paper [1-1]
Reviewed paper
1. Introduction
2. Linear vacuum circuit
5. Application to practical high vacuum systems
A Pressure distribution along an outgassing pipe
B Pressure distribution in an electron microscope high vacuum system
6. Conclusion
References in the paper[1-2]
Reviewed paper
4. Pressure distribution in a complex-chambers-system with homogeneous walls
Reviewed paper
References in the paper [2-1]
Reviewed paper
Introduction
1. Equivalent Networks
References in the paper[2-2]
Reviewed paper
Reviewed paper
References in the paper[2-4]
Reviewed paper
Reviewed paper
Reviewed paper
2. Model
A Transformation rules
References in the paper [2-7]
Reviewed paper
2. Vacuum pump and conduit pipe as a network component
3. Kirchhoff’s law in a network of vacuum components
Reviewed papers
5. Electrical analogues
6. Discussion
References in the paper[2-9]
Reviewed paper
1 . Methods for measuring outgassing rates
Key-point
Keywords
2. Principle of the differential pressure-rise method
Method 1
Method 2
Measurement of pumping speed of a B-A gauge
Reviewed paper
1. Introduction
2. Three-point-pressure method
A Principle
B Optimization of the measuring system
C Measurement of gas flow rates
3. Discussion
4. Conclusion
Appendix
References in the paper[1-2]
Reviewed paper
1. Introduction
2. Two-point pressure method
A Principle
B Measurement of the outgassing rate
C Validity of the one-point pressure method
3. Measurement by the orifice method
4. Discussion and conclusions
How to calibrate the relative sensitivities between the vacuum-gauges
References in the paper[1-3]
Reviewed paper
Reviewed paper
2 . Molecular-flow conductance
Key-point
Keywords
Quoted book
Reviewed paper
Reviewed paper
3 . Total and partial pressure gauges for ultrahigh-vacuum use
Key-point
Keywords
[1] Bayard-Alpert gauge and extractor gauge
[2] Partial pressure gauges
4. Summary and conclusion
References in the paper[1-1]
Reviewed paper
4. The extractor gauge
5. Comparison of pressure indicators between Bayard-Alpert gauge and extractor gauge
References in the paper[1-2]
Reviewed paper
2. Experiment
References in the paper[1-3]
Reviewed paper
Reviewed paper
3. Current PPAs (partial pressure analyzers)
A Magnetic sector
B Quadrupole (monopole)
References in the paper[2-1]
Reviewed paper
11. Calibration
12. Conclusions
References in the paper[2-2]
Reviewed paper
Index

Citation preview

FOUNDATIONS OF MOLECULAR-FLOW NETWORKS FOR VACUUM SYSTEM ANALYSIS NAGAMITSU YOSHIMURA

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright Ó 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-818687-9

For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Mara Conner Acquisition Editor: Fiona Geraghty Editorial Project Manager: Gabriela D. Capille Production Project Manager: R.Vijay Bharath Cover Designer: Miles Hitchen Typeset by TNQ Technologies

Biography Profile of author 1965:

1985:

1995:

1998e2003: 2000: 1965e2003:

Graduated from Osaka Prefecture University, Engineering Division; Electronics. Entered the Division for Development and Research Laboratory in JEOL Ltd. Engaged in research and development of vacuum technology in electron microscopes for about 30 years at JEOL Ltd. Doctorate in Engineering from Osaka Prefecture University. Doctoral dissertation: Research and development of the high-vacuum system of electron microscopes (written in Japanese). Received license of “Consultant Engineer” (Applied Science: Vacuum) No. 32211 from the Science and Technology Agency. Took the leadership for production of JEOL-Sputter-Ion Pumps in the JEOL-related production company. Retired at the age limit from JEOL Ltd. Member of AVS (American Vacuum Society), Vac. Soc. Japan and JVIA (Japan Vacuum Industry Association) (Now retired).

Books by Nagamitsu Yoshimura (1) “Vacuum Technology in Micro-Nano Electron-Probe Analytical Instruments” (in Japanese). Nagamitsu Yoshimura, Supervised by Tatsuo Okano, December 19, 2003, NTS Co. Tokyo, Japan. (2) “Vacuum Technology: Practice for Scientific Instruments”, Nagamitsu Yoshimura, January 09, 2007, Springer-Verlag, Berlin, Heidelberg. (3) “Historical Evolution Toward Achieving Ultrahigh Vacuum in JEOL Electron Microscopes”, Nagamitsu Yoshimura, 2014, Springer Briefs in Applied Science and Technology, Springer Tokyo Heidelberg New York. Dordrecht, London.

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Biography

(4) “Review: Progress in Ultra-High Vacuum Technology: Make use of molecular-flow networks for system analysis” (in Japanese), Nagamitsu Yoshimura, April 24, 2017, NTS Co. Tokyo, Japan. (5) “A-Review: Ultrahigh-Vacuum Technology for Electron Microscopes”, Nagamitsu Yoshimura, October 01, 2019, Elsevier Co. Ltd.

Preface

Introduction The first reviewed paper (in Chapter 1 of Part 1) is “Modeling of outgassing or pumping functions of the constituent elements such as chamber walls and high-vacuum pumps” (N. Yoshimura, 2001). The physical basis of the molecular-flow networks is clearly described in the abstract of the first reviewed paper, as follows: “Chamber walls, subjected to ‘in situ’ baking, sometimes show a pumping function for a high vacuum, while a highvacuum pump sometimes shows outgassing in an ultrahigh vacuum. Such functions of the system elements can be represented by the internal pressure PX. All the system elements, such as chamber walls, pumps, and pin holes through a pipe wall, can be replaced by a pressure generator with the internal pressure PX and the internal flow impedance RX in the equivalent vacuum circuit. The internal pressure PX of the chamber wall varies depending on the wall history under high vacuum. The equivalent vacuum circuit composed of many characteristic values (PX, RX) and flow impedances R can represent the gas flows in the original high-vacuum system.” It is useful to present the relationship among desorption, adsorption, and outgassing/pumping of the chamber wall surfaces. For the chamber wall surfaces, when Qdesorption rate > Qadsorption rate, Qoutgassingrate ¼ Qdesorptionrate  Qadsorptionrate and when Qdesorption rate < Qadsorption

rate,

Qpumpingrate ¼ Qadsorptionrate  Qdesorptionrate . Similarly, for the device called “vacuum pump,” when the ultimate pressure of the pump, Pultimate, is lower than the pressure, Pchamber, in the chamber to which the pump is attached,

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Preface

that is, Pultimate < Pchamber, for the pump with pumping speed Spump, Qpumping rate ¼ Spump  ðPchamber  Pultimate Þ; and when Pultimate > Pchamber, Qoutgassing rate of

pump

Spump is usually Cpump

¼ Spump  ðPultimate  Pchamber Þ;

opening.

Therefore,

Qpumping rate ¼ Cpump opening  ðPchamber  Pultimate Þ. It is well known that the anodic oxide layer of aluminum shows extremely large outgassing rate when placed inside the high vacuum chamber. The anodic oxide layer of aluminum is well known to have thick, porous layers as shown in Fig. 1, showing large outgassing rate in high vacuum.

Figure 1 Scanning electron microscope image of anodic oxide layer of aluminum. Courtesy of JEOL Ltd Courtesy of JEOL Ltd

Preface

Porous surfaces as shown in Fig. 1 work as outgassing sources in high vacuum. When such porous surfaces have been degassed at high temperature, they show an effective pumping function under high vacuum. Molecular-flow networks represent such function mentioned above.

xvii

Acknowledgments We, co-workers and I, have carried out a great deal of work on reasonable ultrahigh vacuum systems for JEOL Electron Microscopes for a long years in the Vacuum Technology Laboratory in Research/Development Division. Engineers I worked together were: Dr. Hisashi Oikawa, Mr. Haruo Hirano, Mr. Yoshio Ishimori, Mr. Kenji Ohara, Mr. Ichiro Ando. I would like to say thank you very much.

1 Physical basis of molecularflow networks: adsorption, desorption, diffusion, and outgassing/pumping Key-point On the surfaces of the chamber wall in ultrahigh vacuum, the following relationship exists: Qoutgassing ¼ Qdesorption eQadsorption . Similarly, for a sorption pump in ultrahigh vacuum, the similar relationship exists: Qpumpingesorptionepump ¼ Qadorptionesorption pump eQdesorptionesorption pump .

Keywords Adsorption; Adsorption rate; Desorption; Desorption rate; Outgassing; Outgassing rate; Pumping; Pumping rate; Pumping speed

Foundations of Molecular-Flow Networks for Vacuum System Analysis. https://doi.org/10.1016/B978-0-12-818687-9.00001-1 Copyright © 2020 Elsevier Inc. All rights reserved.

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Chapter 1 Physical basis of molecular-flow networks

Reviewed papers

[1-1] Modeling of outgassing or pumping functions of the constituent elements such as chamber walls and high vacuum pumps (N. Yoshimura, 2001) N. Yoshimura presented the paper[1-1], “Modeling of outgassing or pumping functions of the constituent elements such as chamber walls and high-vacuum pumps.” In this book the paper[1-1] is fully introduced.

Abstract Chamber walls, subjected to “in situ” baking, sometimes show a pumping function for a high vacuum, while a high-vacuum pump sometimes shows outgassing in an ultrahigh vacuum. Such functions of the system elements can be represented by the internal pressure PX. All the system elements, such as chamber walls, pumps, and pinholes through a pipe wall, can be replaced by a pressure generator with the internal pressure PX and the internal flow impedance RX in the equivalent vacuum circuit. The internal pressure PX of the chamber wall varies depending on the wall history under high vacuum. The equivalent vacuum circuit composed of many characteristic values (PX, RX) and flow impedances R can represent the gas flows in the original highvacuum system.

Keywords Internal pressure; Pressure generator; Vacuum circuit; Outgassing; Pumping

Chapter 1 Physical basis of molecular-flow networks

1. Introduction In an ultrahigh-vacuum system, the chamber wall, subjected to “in situ” baking, sometimes shows a pumping effect, and the vacuum pump sometimes shows an outgassing phenomenon. Alpert[1] reported on an interesting ultrahigh-vacuum system, where 2  10e10 Torr was kept in the “sealed-off” portion for a long period of time after the portion was isolated from the diffusion pump (DP), while the pressure in the portion connected to the diffusion pump (DP), rose gradually and reached a saturated pressure of 1  10e7 Torr, as shown in Fig. 1. Alpert[1] attributed

Figure 1. (A) Pressure vs. time in the sealed-off portion of vacuum system; (B) pressure vs. time in the ion gauge in contact with the diffusion pump (DP). From Alpert.[1]

5

6

Chapter 1 Physical basis of molecular-flow networks

the pressure rise with time to the gradual deterioration of the pumping function of the ion gauge, IG-B. Narushima and Ishimaru[2] reported the “inverted pressure distribution” in an aluminum alloy pipe system, as shown in Fig. 2. That is, after “in situ” baking, the pressure at the position far from the turbo-molecular pump (TMP) became lower than the pressure near the TMP. They attributed the “inverted pressure distribution” to the pumping function of the pipe wall.[2] We shall here represent the “footprint” of pressure in Fig. 1[1] and the “inverted pressure distribution” in Fig. 2[2] by using the characteristic values of the system elements.

2. Modeling of outgassing Yoshimura[3] defined the characteristic values of the outgassing source (the internal pressure PX and the free outgassing rate[4] Figure 2. Pressure variation at 130 C, 24 h bake. (B): Pressures far from the TMP (turbomolecular pump); (C) pressures near the TMP; QMF: quadrupole mass filter. From Narushima and Ishimaru.[2]

Chapter 1 Physical basis of molecular-flow networks

per unit surface area K0), and developed the theory of the equivalent vacuum circuit composed of the characteristic values of the constituent elements.[3] A high-vacuum system is recognized as a system composed of various walls. The chamber wall prevents molecules in the atmosphere from flowing into the high-vacuum field. Besides this shielding function, the chamber wall makes outgassing in high vacuum. This is another function of the wall. A pin hole through the pipe wall causes the leakage of gas, and this resembles outgassing. Outgassing can be represented by a pressure generator with the internal pressure PX and flow impedance RX under high-vacuum pressure P. A high-vacuum pump is recognized as a kind of wall in a high vacuum. The gas compression function of the transfer-type pump is recognized as a wall function, because it shields the gases in the fore-vacuum side. The getter pump has ultimate pressure PU, which varies according to the pump history under vacuum. The pumping (or outgassing) function of a vacuum pump (so called) is represented by a pressure generator with internal pressure PU and internal flow impedance 1/S (the reciprocal of its pumping speed). The pressure in the vacuum chamber is influenced by various walls, such as the conduit and the pump used. The chamber wall is subjected to the incidence of residual gas molecules, resulting in a functional correlation appearing among many constituent elements. When the internal pressure PX of the chamber wall is higher than the pressure P in the chamber, the chamber wall shows outgassing. On the other hand, when PX is lower than P, the chamber wall shows pumping. The vacuum pump generally has a very low internal pressure PU and a small internal flow impedance 1/S, compared with the corresponding respective characteristic values of the chamber wall. Pressures and gas flows in the high-vacuum system can be analyzed by the equivalent vacuum circuit composed of the characteristic values of the system elements.[5],[6] There are many gas species in a high-vacuum field. Internal pressure PX is the sum of the partial pressures. Therefore, pressure and gas flows in a high-vacuum system should preferably be analyzed for the respective gas species.

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Chapter 1 Physical basis of molecular-flow networks

3. Pressure analysis using equivalent vacuum circuit The pressure distributions shown in Figs. 1 and 2 can be described by the internal pressures of the system elements, which vary according to the history of the elements under vacuum. Consider the “sealed-off” portion isolated from the diffusion pump in Fig. 1.[1] The manifold wall and the pipe wall were made of the same kind of glass and were both treated by in situ baking at 420 C, and so the internal pressures of the respective walls would be likewise 2  10e10 Torr after in situ baking. The internal pressure PIG of the ion gauge scarcely varied with time in the “sealed-off” portion, which might be a little lower than 2  10e10 Torr. The equivalent vacuum circuit for the “sealedoff” portion is presented in Fig. 3A. Next, consider the portion connected to the diffusion pump. At an elapsed time after in situ baking (420 C), the internal pressure PX of the pipe wall must be as low as 2  10e10 Torr. However, residual gas molecules, mainly coming from the DP, were adsorbed onto the pipe wall, and the PX of the pipe wall gradually rose to a saturated pressure of 1  10e7 Torr, i.e., the ultimate pressure PU of the DP used. The equivalent vacuum circuit corresponding to the portion connected to the diffusion pump, under Figure 3. Equivalent vacuum circuits corresponding to the systems of Fig. 1 at the saturated condition. In (A) R ¼ 1/F, where F is the flow conductance of the manifold, and R 0 ¼ 1/F 0, where F 0 is the flow conductance of the pipe. RIG ¼ 1/SIG, where SIG is the pumping speed of IGA. In (B) PU and 1/S correspond to the characteristic values PX and RX of the pump, respectively.

Chapter 1 Physical basis of molecular-flow networks

the saturated condition, is presented in Fig. 3B, where the internal pressure PX of the pipe and IG are the same as the PU of the DP. There is not a gas flow in (B), which is recognized merely as a vacuum box of pressure PU of 1  10e7 Torr. Now, consider the “inverted pressure distribution” of Fig. 2.[2] Assume that the pumping function of the Bayard-Alpert gauge was negligibly small. According to the proposed model of outgassing, the internal pressure of the pipe wall, during “in situ” baking, was higher than the ultimate pressure (the same as the internal pressure) of the TMP, and so the pressure at the position far from the TMP was higher than the pressure near the TMP. On the other hand, after “in situ” baking, the internal pressure of the pipe wall became lower than the ultimate pressure of the TMP, resulting in the pressure far from the TMP becoming lower than that near the TMP. The equivalent vacuum circuit corresponding to the system of Fig. 2 is presented in Fig. 4.

4. Conclusion Functions of all the constituent elements in a high-vacuum system as an outgassing source can be represented by a pressure generator with an internal pressure PX and flow impedance RX. A high-vacuum system is defined as a system with more than two different walls with different internal pressures PX from each other. Pressures and gas flows in the high-vacuum system can be analyzed by the equivalent vacuum circuit.[5],[6] Lewin[7] discussed the theme “how to characterize the vacuum environment,” where he claimed what is visualized is the particle flux F. He proposed that the gas density n and the flux F should be

0

Figure 4. Vacuum circuit corresponding to the system of Fig. 2. (A) PX, PX > PU, during in situ baking. (B) PX < PX0 < PU , variable with time, after in situ baking.

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Chapter 1 Physical basis of molecular-flow networks

employed instead of the pressure P and the throughput Q. On the contrary, the present author claims that the pressures including internal pressures PX of elements could be regarded as kinds of potential governing net gas flows in the entire system under high vacuum.[3] This paper demonstrates that the pressures characterize the vacuum environment.

References in the paper[1-1] [1]

D. Alpert, J. Appl. Phys. 24 (7), 860e876 (1953).

[2]

K. Narushima, H. Ishimaru, J. Vac. Soc. Jpn. 25 (4), 172e175 (1982).

[3]

N. Yoshimura, J. Vac. Sci. Technol. A 3 (6), 2177e2183 (1985).

[4]

B. B. Dayton, in Proceedings of the AVS 6th. Vacuum Symposium Transactions, p. 101 (1959).

[5]

S. Ohta, N. Yoshimura, H. Hirano, J. Vac. Sci. Technol. A 1 (1), 84e89 (1983).

[6]

H. Hirano, Y. Kondo, N. Yoshimura, J. Vac. Sci. Technol. A 6 (5), 2865e2869 (1988).

[7]

G. Lewin, Vacuum 41 (7e9), 2048e2049 (1990).

Reviewed paper [1-1]

N. Yoshimura, “Modeling of outgassing or pumping functions of the constituent elements such as chamber walls and high-vacuum pumps,” Appl. Surf. Sci. 196/170, pp. 685e688 (2001).

Chapter 1 Physical basis of molecular-flow networks

[1-2] Testing performance of diffusion pumps (M. H. Hablanian and H. A. Steinherz, 1962) M. H. Hablanian and H. A. Steinherz (1962) presented the paper[1-2], “Testing performance of diffusion pumps.” This paper is composed of the following sections: 1. Introduction 2. Ultimate vacuum 3. Fore-pressure tolerance 4. Back-streaming 5. Throughput 6. Test consistency 7. Conclusions In this book Fig. 1 in Section 2. Ultimate vacuum is introduced.

Abstract Measurement techniques and experimental results are presented for the major performance criteria of oil diffusion pumps. Special emphasis is placed on new developments including the new diffusion pump oils and zeolite traps. Statistical variations observed in testing groups of pumps of the same design are presented. Comments by Yoshimura: Fig. 1 presents that porous zeolite balls, when degassed sufficiently, work as an ultrahigh-vacuum pump. However, zeolite balls work as gas sources in the extreme high-vacuum chamber evacuated by the well-designed sputter-ion pump.

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Chapter 1 Physical basis of molecular-flow networks

Figure 1. Diffusion pump (using DC704 oil) with zeolite trap and test dome.

Reviewed paper [1-2]

M. H. Hablanian and H. A. Steinherz, “Testing performance of diffusion pumps,” Trans. 8th Vac. Symp. and 2nd Inter. Congr. p. 333 (1962).

Chapter 1 Physical basis of molecular-flow networks

[1-3] A differential pressure-rise method for measuring the net outgassing rates of a solid material and for estimating its characteristic values as a gas source (N. Yoshimura, 1985) N. Yoshimura (1985) presented the paper[1-3], “A differential pressure-rise method for measuring the net outgassing rates of a solid material and for estimating its characteristic values as a gas source.” This paper presented the foundation of the molecular-flow networks. In this book the paper is almost completely introduced.

Abstract A differential pressure-rise method for measuring net outgassing rates of solid materials is introduced, which has an outstanding advantage in that the error due to chamber walls and the vacuum gauge is much reduced. The differential method was successfully applied to measuring net rates of solid materials. The net rate K per unit surface area of a solid material at a pressure P could be practically expressed as K ¼ K0(1  P/Px), where the characteristic values Px and K0 are estimated by measuring two net rates at two different pressures.

1. Introduction A high-vacuum system is generally composed of chambers containing several parts, pumps, and conducting pipes. The apparent function of their walls is to prevent the outside gases from flowing into the vacuum field. The other functions are well known as gas evolution and gas sorption.[1] Vacuum pumps also evolve gases into a high-vacuum field,[1],[2] though they effectively pump out gases in many cases. The compression function of

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Chapter 1 Physical basis of molecular-flow networks

a transfer pump could be regarded as a wall function, because it apparently prevents gases at the fore-line from flowing into the high-vacuum side. As has been seen above, a high-vacuum system in a steady state of pressure could be regarded as a system composed of various types of walls evolving and sorbing gases. It is therefore essentially important to clarify what characterizes the functions of walls (gas evolution and gas sorption). A lot of work[1],[3]e[15] has been reported on gas evolution and gas sorption, taking into account the moleculesewall interaction. Dayton[1] has discussed the fundamentals of outgassing, and has developed a detailed theory on gas evolution and gas sorption of solids. Practically, an outgassing rate of a solid material has been widely used for analyzing pressures in a high-vacuum system, apart from the mechanism involved. A lot of work[1],[5],[6],[16]e[45] has been reported on methods for measuring outgassing rates of various solid materials under various conditions. Elsey[46] has reviewed some of these methods but not a differential pressurerise method[16] written in Japanese by Yoshimura et al. in 1970. The differential method[16] has an outstanding advantage that the effect of outgassing of chamber walls is strictly eliminated and that the error induced by the pumping function of a vacuum gauge is much reduced. The present paper reports that the net outgassing rate Q of a solid material at a pressure P could be expressed by Q ¼ Q0(1  P/Px), where the values Px and Q0 can be practically estimated by measuring two different values of net outgassing rates at different pressures.

2. Characteristic values of a solid material as a gas source Let us consider a leak-free chamber of a unit volume consisting of homogeneous walls, which is evacuated by a pump to a sufficiently low pressure P1. After the chamber has been isolated from the pump, the pressure P gradually rises with elapsed time[16] and finally reaches a saturated pressure Px, as shown in Fig. 1A. The curve could be extrapolated to zero pressure, as shown by the broken line. The gradient (dP/dt)p is the net outgassing rate Q of the chamber wall at P, and (dP/dt)0 at zero pressure is called as the free outgassing rate[1] Q0 of the wall.

Chapter 1 Physical basis of molecular-flow networks

Figure 1. (A) Typical pressure-rise curve, and (B) net outgassing rate Q depending on pressure P.

The recorded pressure P as a function of the elapsed time t could be assumed to be expressed as P ¼ Px ½1  expfð Q0 = Px Þtg.

(1)

Differentiating Eq. (1) with respect to t, ðdP=dtÞp ¼ Q0 expfðQ0 = Px Þtg.

(2)

From Eqs. (1) and (2), Q defined as (dP/dt)p could be expressed as[47] Q ¼ Q0 ð1  P = Px Þ.

(3)

The net outgassing rate Q as a function of pressure P is represented by a straight line, as shown in Fig. 1B. The extrapolated line cuts the horizontal P axis and the vertical Q axis at PX and Q0, respectively. The function of the chamber wall as a gas source could be characterized by PX and Q0. The wall effectively evolves gases in the pressure range lower than PX, where the rate Q is reduces with the increase of P. On the other hand, it effectively sorbs (the word “sorb” includes both “adsorb” and “absorb”) gases in the range of pressure higher than PX. Eq. (3) can be rewritten as Q ¼ PX =ZX  P=ZX ;

(4)

if Q0 is replaced by PX/ZX. ZX with a dimension of flow impedance is also a characteristic value of the wall. The first term of Eq. (4) represents the free outgassing rate Q0, whereas the second term means the free sorption rate Qs at pressure P.[1] The effective function of the chamber wall as a gas source at a pressure P could be represented by that of a pressure generator with internal pressure PX and internal flow impedance ZX for a field of P.[47] The pressure

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Chapter 1 Physical basis of molecular-flow networks

generator introduces gases into the field of P at the net rate Q represented by the difference between the free outgassing rate Q0 and the free sorption rate Qs. Now, in a steady state of pressure, the amount of molecules adsorbed on the surface must be almost saturated. Under such a saturated condition of the wall surface, the rates of adsorption on the surface and desorption from the surface are almost balanced with each other. Even under such a steady state of pressure, gas molecules are effectively evolved from the wall surface through diffusion of molecules in the bulk wall.[1] The diffusion rate of molecules must be closely related to the values PX and ZX of the wall. The net outgassing rate Q of a wall is generally lowered slowly with the time during which the wall is exposed to a high vacuum. A wall, well pretreated by a vacuum bakeout, generally has a very low PX. Solid materials as gas sources placed inside a vacuum chamber could be treated as kinds of walls, and so the net outgassing rate K per unit surface area of a solid material could be expressed by the following equation analogous to Eq. (3): K ¼ K0 ð1  P=PX Þ.

(5)

The characteristic values, internal pressure PX, and free outgassing rate K0 per unit surface area of a solid material depend on the history of the solid under vacuum. The values PX and K0 could be estimated by two different K values, K1 and K2, measured at two different pressures P1 and P2, as PX ¼ ðK1 P2  K2 P1 Þ=ðK1  K2 Þ;

(6)

K0 ¼ ðP1 K2  P2 K1 Þ=ðP1  P2 Þ.

(7)

In the actual estimation of the characteristic values, PX and K0, of sample materials, the following notices will be useful: (1) For a material with a sufficiently high PX, K measured at P much lower than PX could be treated as K0 with only a small error.[1] (2) It is desirable that a test chamber itself is made of the sample material to be tested.[17] On the other hand, for sample pieces placed in the test chamber with walls of materials different from the sample, the effect of outgassing of the chamber walls must be eliminated in measuring net outgassing rates of the sample. A differential pressure-rise method[16] is recommended for this purpose.

Chapter 1 Physical basis of molecular-flow networks

3. Differential pressure-rise method A. Principle Notes: The principle of the differential pressure-rise method is fully described in Chapter I [3-1] Differential pressure-rise method. Therefore, Section A. Principle is omitted here.

B. Measurement of net outgassing rates of solid materials Notes: The contents of this section are presented in details in CHAP-I, so these description are omitted here.

C. Estimation of the characteristic values PX and K0 of Viton O-rings As has been described in Section II, the characteristic values PX and K0 of a solid material can be estimated from Eqs. (6) and (7) using two different K values K1 and K2, measured at different pressures P1 and P2. For example, the characteristic values, PX and K0 of the untreated Viton O-rings after 4 h evacuation were estimated as follows: The pressure-rise rates, (dP/dt)1 and (dP/dt)2, at 2  10e5 Torr of P1 were calculated from Fig. 3A as about 3.7  10e6 Torr/s and 3.3  10e7 Torr/s, respectively. Therefore, K1 at 2  10e5 Torr was

Figure 3. Pressure-rise curves for Viton O-rings. Sample surface areas and pretreatments are: (A) 8.7 cm2, without pretreatment, (B) 43 cm2, exposure to dried air (0.5 h) following vacuum bakeout (100 C, 3 h), and (C) 17.5 cm2, vacuum bakeout (100 C, 3 h). The parametric figure on each curve shows evacuation times t. Solid lines show pressure-rise curves for the combined domes with the sample and broken lines ones for the empty dome TD1.

17

18

Chapter 1 Physical basis of molecular-flow networks

calculated from Eq. (14) as 2.6  10e6 Torr L/(s cm2). Other rates (dP/dt)1 and (dP/dt)2 at 1  10e4 Torr of P2 were also calculated from Fig. 3A as about 2.0  10e6 Torr/s and almost zero Torr/s, respectively. Therefore, K2 at 1  10e4 Torr was calculated in the same way as 1.5  10e7 Torr L/(s cm2). And finally, PX and K0 of the untreated Viton O-rings after 4 h evacuation were calculated from Eqs. (6) and (7) using 2.6  10e7 Torr L/(s cm2) of K1, at 2  10e5 Torr of P1 and 1.5  10e7 Torr L/(s cm2) of K2, at 1  10e4 Torr of P2 as 2.1  104 Torr and e7 2 2.9  10 Torr L/(s cm ), respectively.
 K ðtÞ ¼ ðV1 þ V2 ÞðdP=dtÞ1  fðA1 þ A2 Þ = A1 gV1 ðdP=dtÞ2 =As ; (14) where a small error term (A2/A1)PrSg/As is omitted. The values PX and K0 of the Viton O-rings subjected to the pretreatment (B) after 5 h evacuation were estimated in the same way as 9.5  10e5 Torr and 3.6  10e9 Torr L/(s cm2), respectively. The values PX and K0 of the Viton O-rings subjected to the pretreatment (C) after 4.2 h evacuation were estimated as 4.5  10e7 Torr and 3.1  10e11 Torr L/(s cm2), respectively. The dependence of the net outgassing rates K of such Viton Orings upon pressure after selected evacuation times are presented in Fig. 6AeC, with their characteristic values PX and K0 estimated above. As seen in Fig. 6, the values PX and K0 depend on histories of Viton O-rings under vacuum. Viton O-rings pretreated by a

Figure 6. Net outgassing rate K of Viton O-rings depending on pressure P. Evacuation times and pretreatments are: (a) 4 h, without pretreatment, (b) 5 h, exposure to dried air (0.5 h) following vacuum bakeout (100 C, 3 h), and (c) 4.2 h, vacuum bakeout (100 C, 3 h).

Chapter 1 Physical basis of molecular-flow networks

vacuum bakeout (100 C, 3 h) after 4.2 h evacuation have a very low PX as 5.4  10e7 Torr. As has been seen in this section, the differential pressure rise method is suited for estimating the characteristic values PX and K0 of a sample material from two net outgassing rates measured at two different pressures. Characteristic values PX and K0 of various materials under various conditions are essentially important to design a vacuum circuit, as will be seen in the next section.

4. Vacuum circuit composed of the characteristic values of constituent elements As has been discussed in Section 2, the net outgassing rate of a wall changes depending on the pressure to which the wall is exposed. The pressure is generally governed by functions of pumps, conducting pipes and other walls, as well as by the outgassing characteristics of the wall. Effective functions of constituent elements are thus correlated with each other, resulting in giving equilibrium pressures in a vacuum system under a steady state of vacuum. A high-vacuum pump is generally characterized by the pumping speed S and the ultimate pressure Pu. When a pump with S and Pu is directly connected to a vacuum field of pressure P, the net gas flow Q from the pump into the field could be expressed by the following equation analogous to Eq. (4):[48]e[50] Q ¼ ðPu  PÞS ¼ PU =S1  P=S1

(16)

Comparing Eq. (4) with Eq. (16), Pu and S of the pump as a gas source correspond to Px and Z1 x of a wall, respectively. Therefore, the effective function of a pump with S and Pu for a field of pressure P could be represented by that of a pressure generator with internal pressure Pu and internal flow impedance 1/S,[47],[51] applied to a field of P, as in the case of a wall. Lubricative oil or grease is a typical vapor source. The function of a vapor pressure Pv could be represented by that of a pressure generator with Pv and zero flow impedance. The function of the flow route of a pipe or an orifice could be represented by a flow impedance Z, a reciprocal of conductance F.[52] Now, net gas flows and pressures in a high-vacuum system under a steady state of pressure could be analyzed by a vacuum

19

20

Chapter 1 Physical basis of molecular-flow networks

circuit[47] composed of the characteristic values of constituent elements, as described below. Let us consider the simple system of Fig. 7A, where a chamber with PX and Q0 is evacuated by a pump with ultimate pressure Pu and speed S through a pipe with PX0 , Q00 and conductance C. The vacuum circuit corresponding to system (A) is obtained by converting the system elements into the circuit elements, as shown in Fig. 7B. Both PX and PX0 of the walls are usually much higher than Pu of the pump and pressure P in the chamber. Therefore, the vacuum circuit can be simplified without an appreciable error under several conditions as follows. When PX0 can be considered to be the same as PX, the vacuum circuit (B) can be simplified to the circuit (C). If Pu is negligibly low, the circuit (C) can be further simplified to the circuit (D). If Q00 can be treated as zero, the circuit (C) can be simplified to the circuit (E). Finally, the circuit (F) is the simplest one acceptable under all the requirements described above. The circuits (B)e(F) can be analyzed by Kirchhoff’s laws.[51]e[55]

Figure 7. High-vacuum system and vacuum circuits. (A) Original system, (B) vacuum circuit corresponding to (A), and (C), (D), (E), and (F) simplified vacuum circuits under several conditions.

Chapter 1 Physical basis of molecular-flow networks

It should be noted that the vacuum circuit naturally involves the relationship among the effective functions of constituent elements as gas sources in an original system. In a high-vacuum system, the element with the lowest internal pressure PX always works as an effective pump, whereas the element with the highest PX always works as an effective gas source. The element with intermediate PX works as a pump when its PX is lower than the pressure P in the vacuum field, and works as a gas source when PX is higher than P in the field. Pressures including internal pressures PX of elements could be regarded as kinds of potential governing net gas flows in the entire high vacuum system. Analyses on net gas flows and pressures must be conducted on the individual gas species separately since both the conductance of a pipe and pumping speed of a pump depend on the gas species. Partial pressures Pi in a high-vacuum system are analyzed by a vacuum circuit for individual gas species “i.” Total pressures P in a molecular flow region are obtained as P [53] P ¼ Pi . Thei electrical simulation technique is convenient and powerful for analyzing a high-vacuum system under a steady state of pressure.[47] Much work[3],[4],[47],[51]e[61] on the subject under a steady state of pressure has been carried out over several decades, much of which has been reviewed by Kendall.[62],[63] Pressure distributions due to net gas flows through impedances have been considered in this section. Other types of pressure distributions, or special pressure distributions in spacious chambers exit due to a beaming effect of gases emitted from local gas sources and due to a flow pattern.[64]e[66]

5. Conclusion The net outgassing rate K per unit surface area of a solid material under a pressure P could be practically expressed using the characteristic values PX and K0 as K ¼ K0(1  P/PX). The values PX and K0 could be estimated by measuring two different net rates K1 and K2 at different pressures P1 and P2. The differential pressure-rise method has an outstanding advantage that the effect of outgassing of chamber walls is strictly eliminated and that the error induced by the pumping function of a vacuum gauge is much reduced. Net gas flows and pressures in a high-vacuum system under a steady state of pressure can be analyzed by a vacuum circuit composed of the characteristic values of constituent elements.

21

22

Chapter 1 Physical basis of molecular-flow networks

References in the paper[1-3] [1]

B. B. Dayton, AVS 6th Vacuum Symposium Transactions (1959), p. 101.

[2]

D. Alpert, J. Appl. Phys. 24, 860 (1953).

[3]

W. Teubner, Vak. Tech. 16, 69 (1967).

[4]

W. Teubner, Vak. Tech. 16, 95 (1967).

[5]

A. Schram, Le Vide No. 103, 55 (1963).

[6]

R. Jaeckel, AVS 8th Vacuum Symposium Transactions (1961), p. 17.

[7]

J. Neubert, Vak. Tech. 13, 19 (1963).

[8]

M. D. Malev, Vacuum 23, 43 (1973).

[9]

K. W. Rogers, AVS 10th Vacuum Symposium Transactions (1963), p. 84.

[10] B. B. Dayton, AVS 9th Vacuum Symposium Transactions (1962), p. 293. [11] A. Schram, AVS 9th Vacuum Symposium Transactions (1962), p. 301. [12] T. Kraus, AVS 10th Vacuum Symposium Transactions (1963), p. 77. [13] F. Pagano, AVS 13th Vacuum Symposium Transactions (1966), p. 103. [14] H. Mizuno and G. Horikoshi, Abstract for 25th Annual Meeting of the Vacuum Society of Japan (1984), p. 79. [15] W. G. Perkins, J. Vac. Sci. Technol. 10, 543 (1973). [16] N. Yoshimura, H. Oikawa, and O. Mikami, J. Vac. Soc. Jpn. 13, 23 (1970). [17] Y. Ishimori, N. Yoshimura, S. Hasegawa, and H. Oikawa, J. Vac. Soc. Jpn. 14, 295 (1971). [18] R. M. Zabel, Rev. Sci. Instrum. 4, 233 (1933). [19] S. H. Cross, Vacuum 10, 86 (1960). [20] B. D. Power and F. C. Robson, AVS 8th Vacuum Symposium Transactions (1961), p. 1175. [21] R. Geller, Le Vide No. 74, 71 (1958). [22] J. Blears, E. J. Greer, and J. Nightingale, Advances in Vacuum Science and Technology (Pergamon, Oxford, 1960), Vol. 2, p. 473. [23] G. Carter, D. A. Armour, and L. de Chernatony, Vacuum 20, 643 (1970).

Chapter 1 Physical basis of molecular-flow networks

[24] R. Caland G. Lewin, Br. J. Appl. Phys. 18, 1459 (1967). [25] J. R. Young, J. Vac. Sci. Technol. 6, 398 (1969). [26] G. Moraw, Vacuum 24, 125 (1974). , R. Geller, and G. Mongodin, Le Vide No. 69, 195 (1957). [27] R. Barre [28] B. D. Power and D. J. Crawley, Transactions of the 1st International Vacuum Congress (1958), p. 206. [29] J. C. Boulassier, Le Vide No. 74, 71 (1958). [30] G. S. Grossart, Transactions of the 3rd. International Vacuum Congress (1965), p. 89. [31] F. J. Schittko, Vacuum 13, 525 (1963). [32] D. J. Crawley and L. de Chernatony, Vacuum 14, 7 (1964). [33] B. H. Colwell, Vacuum 20, 481 (1970). [34] R. Nuvolone, in Proceedings of the 7th. International Vacuum Congress (1977), p. 219. [35] C. W. Oatley, J. Appl. Phys. 5, 358 (1954). [36] R. P. Henry, Le Vide No. 82, 226 (1959). [37] R. P. Henry, Le Vide No. 144, 316 (1969). [38] A. Berman, I. Hausman, and A. Roth, Vacuum 21, 373 (1971). [39] F. Markley, R. Roman, and V. Vosecek, AVS 8th Vacuum Symposium Transactions (1961), p. 78. [40] W. Beckmann, Vacuum 13, 349 (1963). [41] R. S. Barton and R. P. Govier, J. Vac. Sci. Technol. 2, 113 (1965). [42] R. S. Barton and R. P. Govier, Vacuum 20, 1 (1970). [43] R. J. Elsey, P. E. Gear, and E. B. Iberson, Internal Memorandum Nimbus F series, Rutherford Laboratory (1970). [44] G. Messer and N. Treiz, in Proceedings of the 7th. International Vacuum Congress (1977), p. 223. [45] S. Komiya and Y. Sugiyama, J. Vac. Sci. Technol. 16, 689 (1979). [46] R. J. Elsey, Vacuum 25, 347 (1975).

23

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Chapter 1 Physical basis of molecular-flow networks

[47] S. Ohta, N. Yoshimura, and H. Hirano, J. Vac. Sci. Technol. A 1 (1), 84 (1983). [48] S. Dushman, Scientific Foundations of Vacuum Technique. 2nd ed. (Wiley, New York, 1962), p. 121. [49] T. Tom and B. D. Janes, J. Vac. Sci. Technol. 6, 304 (1969). [50] N. A. Florescu, Vacuum 12, 259 (1962). [51] D. W. Stops, Br. J. Appl. Phys. 4, 350 (1953). [52] M. H. Hablanian, J. Vac. Sci. Technol. 7, 237 (1970). [53] B. R. F. Kendall, J. Vac. Sci. Technol. 5, 45 (1968). [54] B. R. F. Kendall and R. E. Pulfrey, J. Vac. Sci. Technol. 6, 326 (1969). [55] B. R. F. Kendall and G. Englehart, Rev.Sci. Instrum. 41, 1623 (1970). [56] S. Dushman, Production and Measurement of High Vacuum (General Electric, New York, 1922). [57] J. M. Aitken, Br. J. Appl. Phys. 4, 188 (1953). [58] A. D. Degras, Le Vide No. 64, 155 (1956). [59] J. Delafosse and G. Mongodin, Les Calculs De La Technique Du Vide, 18  Par La Socie te Francaise Des Inge nieurs Et Techniciens Du Vide, (Édite 1961). [60] W. Teubner, Exp. Tech. Phys. 10, 279 (1962). [61] H. Pingel, in Proceedings of the 14th. International Vacuum Congress (1968), p. 251. [62] B. R. F. Kendall, J. Vac. Sci. Technol. 9, 247 (1972). [63] B. R. F. Kendall, J. Vac. Sci. Technol. A 1, 1881 (1983). [64] B. B. Dayton, AVS 3rd Vacuum Symposium Transactions (1956), p. 5. [65] W. Steckelmacher, Vacuum 16, 561 (1966). [66] I. Arakawa, M. Kim, and Y. Tuzi, J. Vac. Sci. Technol. A 2, 168 (1984).

Reviewed paper [1-3]

N. Yoshimura, “A differential pressure-rise method for measuring the net outgassing rates of a solid material and for estimating its characteristic values as a gas source,” J. Vac. Sci. Technol. A 3 (6), pp. 2177e2183 (1985).

Chapter 1 Physical basis of molecular-flow networks

[1-4] The effect of bake-out on the degassing of metals (B. B. Dayton, 1962) B. B. Dayton (1962) presented the paper[1-4], “The effect of bakeout on the degassing of metals.” This paper is composed of the following sections: 1. Introduction 2. Degassing time at constant temperature 3. Outgassing rate with variable temperature. In this book Sections 2. Degassing time at constant temperature and 3. Outgassing rate with variable temperature are introduced.

Abstract Equations are derived for the effects of a bake-out period on the rate of outgassing of hydrogen and other gases dissolved in metals. It is shown that van Liempt’s formula for the time to remove 95% of the gas is incorrect, and a corrected formula is derived. The equations are compared with available experimental data.

2. Degassing time at constant temperature In the following y will represent the fraction of the original total gas content which has been removed by degassing for time t (in seconds). It can be shown that the equation derived by van Liempt[5] for y < 0.5 is a good approximation of the result given by the exact solution of the Fick’s law equations for diffusion of gas out of a plane sheet. However, on page 782 of his paper he made the incorrect assumption that for y > 0.5 the “apparent outgassing depth” would continue to move inwards from each surface with a velocity proportional to (Dm/t)1/2, where Dnm is the diffusion coefficient (in cm2/sec), even when the concentration

25

26

Chapter 1 Physical basis of molecular-flow networks

in the middle of the sheet has fallen below the initial concentration, c0. This results in the incorrect equation 2

t ¼ pL2m =256Dnm ð1  yÞ ;

ðy > 0:5Þ;

(1)

where Lm is the thickness of the sheet (in cm). In van Liempt’s method the distribution curve for the gas concentration as a function of distance x from the exposed surface is approximated by a straight line through the origin with a slope c0/ anm, where c0 is the initial concentration and anm is the “apparent outgassing depth,” which moves inward with time. In his first paper[10] van Liempt derives the formula   1=2 anm ¼ 4 = p1=2 ðDnm tÞ (2) by equating the area under the curve of concentration as a function of the dimensionless parameter x/2(Dnmt)1/2for the special case of diffusion in a semi-infinite medium with the area of the right triangle whose altitude is the initial concentration, c0, and whose base is anm/2(Dnmt)1/2. This is equivalent to defining anm so that the total quantity of gas removed in time t during degassing into a high vacuum through unit area of the plane surface of a semi-infinite medium is accurately given by c0anm/2. However, there is another method of defining anm which gives a slightly different result but follows logically from van Liempt’s assumption that the distribution curve for the gas concentration in the solid can be approximated by a straight line with slope c0/anm. The outgassing rate will always be given by K ¼ ðG0 =AÞðdy=dtÞ;

(3)

where G0 is the total initial gas content and A is the exposed surface area (in cm2). From Fig. 1, which shows the assumed linear distribution of concentration as a function of the distance, it is evident that for anm < Lm/2 the total gas removed through unit area of one exposed surface will be equal to the triangular area c0anm/2. Hence, yG0 =A ¼ C0 anm =2:

(4)

From Eqs. (3) and (4) the outgassing rate through this surface is K ¼ ðc0 =2Þðdanm =dtÞ;

(5)

Chapter 1 Physical basis of molecular-flow networks

Figure 1. Inward motion of the apparent outgassing depth.

as can be seen from Fig. 1, where the moving line with slope c0/anm sweeps out a triangular area (C0/2)danm in the time dt. The outgassing rate will also be given by K ¼ Dnm ðvc=vxÞx¼0 ;

(6)

where (vc/vx)x¼0 is the concentration gradient at the exposed surface. From Fig. 1 it is evident that this concentration gradient is c0/ anm in van Liempt’s diagram, and therefore K ¼ Dnm ðC0 =anm Þ.

(7)

Combining Eqs. (5) and (7) gives anm ¼ 2ðDnm tÞ1=2 .

(8)

Since 4/p ¼ 2.26, Eq. (8) gives a slightly lower value of anm than Eq. (2). However, the correct equation for the outgassing rate of a semi-infinite solid is obtained by substituting Eq. (2) rather than Eq. (8) in Eq. (5). The advantage of determining anm by the method which led to Eq. (8) will become apparent later when we consider the case of y > 0.5. From Fig. 1 it is evident that the fraction of gas removed is 1/2

y ¼ anm =Lm

ðy < 0:5Þ

(9)

27

28

Chapter 1 Physical basis of molecular-flow networks

when the sheet is exposed on both sides. Substituting Eq. (9) in Eq. (2) gives van Liempt’s formula for the degassing time for y < 0.5: t ¼ py 2 L2m =16 Dnm .

(10)

t1=2 ¼ pL2m =64Dnm .

(11)

For y ¼ 0.5 we have

We have no criticism of Eqs. (10) and (11), for it can be shown that they give correct results. When anm > Lm/2 corresponding to y > 0.5 for a sheet exposed on both sides, Eq. (5) must be replaced by K ¼ ðLm =4Þðdb=dtÞ

(12)

K ¼ Dnm ðc0  bÞ=ðLm =2Þ;

(13)

and Eq. (7) by

where b is the difference between the initial concentration, c0, and the actual concentration at the middle of the sheet (x ¼ Lm/2) as shown schematically by Fig. 2 (cf. Fig. 4 in van Liempt’s paper[5]). The area of the shaded triangle in Fig. 2 is ((Lm/4)db), and the

Figure 2. Decline of concentration at midpoint.

Chapter 1 Physical basis of molecular-flow networks

concentration gradient is (c0  b)/(Lm/2). Combining Eqs. (12) and (13) gives Z Z b 8Dnm t db ; (14) dt ¼ 2 Lm t1=2 0 c0  b or

  t ¼ t1=2 þ L2m = 8Dnm ln½c0 =ðc0  bÞ.

(15)

From Fig. 2 it can be shown that the fraction of gas removed in time t is y¼

c0 Lm  ðLm =2Þðc0  bÞ ; c0 Lm

(16)

Thus ðc0  bÞ=c0 ¼ 2ð1  yÞ.

(17)

Substituting Eqs. (11) and (17) in Eq. (15) gives the correct equation for the degassing time (in sec) for y > 0.5: t ¼

pL2m L2 1 . þ m ln 64Dnm 8Dnm 2ð1  yÞ

(18)

Instead of Eq. (18) van Liempt obtained the incorrect Eq. (1) because he assumed incorrectly that Eq. (2) would hold for anm > Lm/2. From Fig. 2 it is evident that anm ¼ ðLm =2Þc0 =ðc0  bÞ.

(19)

Combining Eqs. (15) and (19) gives the correct formula for anm when anm > Lm/2:  
 (20) anm ¼ ðLm =2Þexp 8Dnm =L2m t  t1=2 . Eq. (20) is quite different from Eq. (2), and the quantity anm no longer has any important physical significance when anm > Lm/2. We may now compare the correct Eq. (18) with the exact result as obtained from Eq. (4.18) in Crank’s book.[4] The fraction y becomes Mt/MN in Crank’s symbols. The time required to remove the fraction y of the gas dissolved in a sheet exposed on both sides can also be determined from Fig. 4.7 on page 57 of Crank’s book where Crank’s symbol l equals Lm/2. When the outgassing rate is not limited by surface phenomena but depends only on the diffusion coefficient, the curve marked N in Crank’s Fig. 4.7 should be used. This curve shows that for y ¼ 0.5

29

30

Chapter 1 Physical basis of molecular-flow networks



4Dnm t=L2m

1=2

¼ 0:45

(21)

or t ¼ 0:0506L2m =Dnm ;

(22)

where t is in seconds. This is in close agreement with Eq. (18) which for y ¼ 0.5 reduces to Eq. (11), corresponding to van Liempt’s formula, where p/64 ¼ 0.0491. For y ¼ 0.95, Crank’s Fig. 4.7 gives  1=2 ¼ 1:09 (23) 4Dnm t=L2m or t ¼ 0:297L2m =Dnm ;

(24)

whereas the incorrect Eq. (1) for y ¼ 0.95 gives t ¼ 4:91L2m =Dnm ;

(25)

while the correct Eq. (18) gives t ¼ 0:337L2m =Dnm

(26)

which is much closer to the exact Eq. (24). Thus van Liempt’s approximate formulas are not very accurate above y ¼ 0.5, and in this region Crank’s Fig. 4.7 or Eq. (18) should be used. For a sheet exposed on only one side the right side of Eqs. (10), (11), (18), (22), (24), (25), and (26) should be multiplied by 4. Similarly, it can be shown that van Liempt’s equation for the degassing time of a long cylindrical rod of radius R (in cm) for the case anm > R or y > 2/3 is not correct. His equation for y < 2/3 is also not a good approximation except for t < 0.1R2/ Dnm. By applying Eqs. (3) and (6) the author has derived the following corrected equations for the degassing time of a long cylindrical rod at constant temperature: k j  1=2 t ¼ 3pR2 y=32Dnm 1  ð1  4y=3Þ ðy < 2 = 3Þ; (27a)   t ¼ pR2 =24Dnm ½1  ð4=pÞln 3ð1  yÞ;

ðy > 2 = 3Þ;

(27b)

Calculations based on these equations agree very well with the more complicated exact formulas corresponding to Eq. (5.23) on page 66 and Eq. (5.49) on page 73 of Crank’s book[4].

Chapter 1 Physical basis of molecular-flow networks

From Eq. (27b)


  y ¼ 1  ð1=3Þexp ðp=4Þ  6Dnm t=R2 ;

(28)

which is convenient for estimating the fraction of gas removed at time t when y > 2/3. Substituting Eq. (28) in Eq. (3) and using G0 =A ¼ Rc0 =2; the outgassing rate for t > (p/24)R2/Dnm will be
  K ¼ c0 ðDnm =RÞexp ðp=4Þ  6Dnm t=R2

(29)

(30)

in units of cm3 (s.t.p.) sece1 cme2, or


 (31) Knm ¼ ð760 Dnm = 273 RÞ 103 c0  exp ðp=4Þ  6Dnm t=R2

in unit of Torr L at the temperature T (K) per second per square centimeter of exposed surface, where c0 is the initial gas concentration in cm3 (s.t.p.) per cm3 of solid. When 0 < t < (p/24)R2/Dnm, the following approximate formula can be used for the outgassing rate of a wire: i h 1=2 (32) Knm ¼ ð760T =273Þ103 c0  ðDnm =ptÞ  ðDnm = 2RÞ Euringer[11] and Edwards[12] have measured the degassing rate as a function of time for a nickel wire which had been previously degassed and then saturated with hydrogen so that the initial gas content was uniformly distributed. The degassing rateetime curves plotted on a logelog graph were in agreement with the theoretical curve for diffusion of hydrogen out of a metal wire without any interference by rate-determining interactions at the surface. The theoretical curve was based on the formula K ¼ 2c0 ðDnm =RÞ

N X

  exp  Dnm b2i t=R2 ;

(33)

i¼1

where K is in units of cm3 (s.t.p.) sece1 cme2 and bi represents the roots of the equation J0 ðxÞ ¼ 0

(34)

in which J0(x) is the Bessel function of the first kind of order zero. The first four roots of Eq. (34) are b1 ¼ 2.405, b2 ¼ 5.520, b3 ¼ 8.654, b4 ¼ 11.792.

31

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Chapter 1 Physical basis of molecular-flow networks

When t > R2/3Dnm, the terms for i > 1 in the series in Eq. (33) are negligible compared to the first term and Eq. (33) reduces to
  K ¼ c0 ðDnm =RÞexp ln 2  b21 Dnm t=R2 . (35) Eq. (35) is nearly identical to Eq. (30) since ln 2 ¼ 0.69 while p/4 ¼ 0.78 and b21 ¼ 5:77. Integration of Eq. (33) with respect to time and multiplying by A/G0 as given by Eq. (29) yields the exact formula for the fraction, y, of gas removed from a long cylinder in time t: y ¼ 14

N X

  2 2 b2 i exp  Dnm bi t=R ;

(36)

i¼1

where use is made of the relation N X

4=b2i ¼ 1.

i¼1

[13]

checked the validity of Eq. (36) for the degassing of Lawson nickel wire but found discrepancies attributable to the fact that the initial gas content was not uniformly distributed but was concentrated near the surface of the wire. Eborall and Ransley[14] present the following formula for the fraction y of gas removed in t seconds by diffusion from a cylinder of finite length Lc (in cm) and radius R (in cm): ( ) N X   2 2 2 bi exp  Dnm bi t=R y ¼ 1 4 i¼1

( 



8=p

2

N X

) h i 2 2 2 ð2j  1Þ exp  Dnm ð2j  1Þ p t=Lc ; 2

(37)

j¼1

where bi represents the successive roots of Eq. (34). In Eq. (37) we have corrected in Eborall and Ransley’s article[14]  2 a misprint  2 where p t Lc is given as pt L2c . Eborall and Ransley measured the accumulated hydrogen evolved from cylindrical samples of an aluminum-magnesium alloy at temperatures in the range from 300 to 500 C. They obtained good agreement between the experimental curve and the theoretical curve based on Eq. (37) when allowance was made for the initial time lag for the specimen to attain the temperature of the furnace.

Chapter 1 Physical basis of molecular-flow networks

3. Outgassing rate with variable temperature When the outgassing rate is not limited by processes occurring at the exposed surface but depends only on the diffusion coefficient, Dnm, as indicated by Eq. (6), the effect of suddenly raising the temperature, Tm, of a metal plate from room temperature, Ta, to a bake-out temperature, Tb, at the time tb will be a sudden increase in the outgassing rate by the factor Db/Da, where Db is the value of Dnm at Tb and Da is the value at Ta. This follows from Eq. (6) when it is noted that the concentration gradient vc/vx is not changed by a sudden change in temperature. However, when the metal is then maintained at the bake-out temperature, the concentration gradient at x ¼ 0 begins to decrease rapidly from its value at time th ¼ tb to a value at th > tb given by ðvc=vxÞx¼0 ¼ c0 =xnm ;

(38)

where xnm is given by 1=2

xnm ¼ 60p1=2 ½Da tb þ Db ðth  tb Þ

(39)

in which th is the total pumping time in hours and th < t1/2/3600 where t1/2 is given by Eq. (11). This follows from the procedure explained in Section 9.1 on page 147 of Crank’s book[4] since Dnm ¼ Da

ðth < tb Þ

(40a)

Dnm ¼ Db

ðth > tb Þ

(40b)

and Eq. (9.3) in Crank’s book[4] becomes tx ¼ 3600½Da tb þ Db ðth  tb Þ;

(41)

where tx (in seconds) is equivalent to Crank’s symbol T, while th is the pumping time in hours. Corresponding to Crank’s Eq. (9.4) we have vc=vtx ¼ v2 c=vx2 .

(42)

For a semi-infinite slab, Eq. (42) gives h i 1=2 c ¼ c0 erf x=ð4tx Þ

(43)

and ðvc=vxÞx¼0 ¼ c0 =ðptx Þ

1=2

.

(44)

33

34

Chapter 1 Physical basis of molecular-flow networks

Substituting Eq. (41) in Eq. (44) leads to Eqs. (38) and (39). Substituting Eqs. (38), (39), and (40b) in Eq. (6) gives an equation for the outgassing rate after bake-out begins (th > tb): K ¼

ðDb =3600pÞ

1=2

c0

(45)

1=2

½th  ð1  Da =Db Þtb 

where K is in units of cm3 (s.t.p.) sece1 cme2 and c0 is the initial concentration in cm3 (s.t.p.) of gas per cm3 of metal. In units of Torr L sece1 cme2 the outgassing rate for th > tb is Knm ¼

103 ð760T =273ÞðDb =3600pÞ ½th  ð1  Da =Db Þtb 

1=2

c0

1=2

.

(46)

If bake-out does not begin until after th ¼ 1 h, then the outgassing rate at th ¼ 1 h is K1 ¼ 103 ð760T =273ÞðDa =3600pÞ1=2 c0 ;

(47)

and after bake-out begins (th > tb) Knm ¼

ðDb =Da ÞK1 ½ðDb =Da Þðth  tb Þ þ tb 

1=2

.

(48)

This equation indicates that the outgassing rate at th ¼ tb, just after the temperature has been quickly raised to Tb, will be Db/Da times the rate just before the increase in temperature, and as th increases beyond tb the outgassing rate falls rapidly and approaches asymptotically a rate which is (Db/Da)1/2 times what the rate would have been if the temperature had been maintained at Ta corresponding to no bake-out. This is shown graphically in Fig. 3 by the solid line curve in the region 1 to 10 hours (and the dotted curve beyond th ¼ 10) where K1 has been chosen equal to 108 Torr L sece1 cme2 and Db/Da ¼ 100 as convenient values to illustrate the equation. Eq. (45) through Eq. (48) apply only to the case in which the thickness, Lm, of the plate is so large that the outgassing rate at th is equivalent to that calculated for a semi-infinite solid. The van Liempt outgassing depth is given by   anm ¼ 60 4=p1=2 ½Da tb þ Db ðth  tb Þ1=2 (49) and a plate of thickness Lm exposed to vacuum on only one side will behave like a semi-infinite solid until anm ¼ Lm, while for a plate exposed on both sides the outgassing depth reaches the midpoint and exponential decay of the outgassing rate begins

Chapter 1 Physical basis of molecular-flow networks

35

when anm ¼ Lm/2. If th ¼ tnm when anm ¼ Lm, then Eq. (49) gives for a plate exposed on only one side tnm ¼

pL2m þ ½1  ðDa =Db Þtb . 16  3600Db

(50)

Hence, for a plate exposed on only one side, Eqs. (46) and (48) hold for the time period tb < th < tnm, where tnm is given by Eq. (50). When the metal is maintained at the steady bakeout temperature, Tb, from th ¼ tb to th ¼ tc and then suddenly cooled to room temperature, Ta, and maintained at Ta after th ¼ tc, then for th > tc we have in place of Eq. (41) tX ¼ 3600½Da tb þ Db ðtc  tb Þ þ Da ðth  tc Þ;

(51)

and Dnm in Eq. (6) is again replaced by Da so that by combining Eqs. (6), (40a), (44), and (51) we have Knm ¼

103 ð760T =273ÞðDa =3600pÞ

1=2

fth þ ½ðDb =Da Þ  1ðtc  tb Þg

c0 1=2

.

(52)

in units of Torr L sece1 cme2.

Figure 3. Theoretical curve for effect of bake-out. Ta, ambient temperature and Tb, bake-out temperature.

36

Chapter 1 Physical basis of molecular-flow networks

Using Eq. (47) we can also write Knm ¼

K1 fth þ ½ðDa =Da Þ  1ðtc  tb Þg

1=2

.

(53)

Comparing Eqs. (48) and (53) it may be noted that for th ¼ tc the outgassing rate just after reducing the temperature suddenly from Tb to Ta will be Da/Db times the rate just before this decrease in temperature. When th is small compared to [(Db/Da)  1](tc  tb), Eq. (53) indicates that the outgassing rate is nearly constant for a while after the temperature has been lowered to room temperature. As th increases and becomes large compared to [(Db/Da)  1](tc  tb), the outgassing rate for a “semi-infinite” slab of metal approaches as a limit the ordinary rate for room temperature outgassing without bake-out, 1=2

Knm ¼ K1 =th .

(54)

This behavior is illustrated in Fig. 3.

References in the paper[1-4] [4]

J. Crank, The Mathematics of Diffusion, Oxford University Press, London (1957).

[5]

J. A. M. van Liempt, Rec. Trav. Chim. d. Pays-Bas 57, 871e882 (1938).

[10] J. A. M. van Liempt, Rec. Trav. Chim. 51, 114 (1932). [11] G. Euringer, Z. f. Physik 96, 37e52 (1935). [12] A. G. Edwards, Brit. J. Appl. Physics 8, 406e409 (1957). [13] R. W. Lawson, Brit. J. Appl. Physics 13, 115e121 (1962). [14] R. Eborall and C. E. Ransley, J. Inst. Metals LXXI, 525e552 (1945).

Reviewed paper [1-4]

B. B. Dayton, “The effect of bake-out on the degassing of metals,” Transactions of the 9th National Vacuum Symposium, 1962 (Macmillan, New York, 1963), pp. 293e300.

Chapter 1 Physical basis of molecular-flow networks

[1-5] The variation in outgassing rate with the time of exposure and pumping (K.W. Rogers, 1963) Rogers (1963) presented the paper[1-5], “The variation in outgassing rate with the time of exposure and pumping.” The paper [1-5] is composed of the following sections: 1. Introduction 2. Analysis 3. Typical history 4. First pump-down 5. Second pump-down 6. Summary. In this book Sections 2. Analysis, 4. First pump-down, 5. Second pump-down, and 6. Summary are introduced.

Abstract Past experiments have shown that the pump-down time of a vacuum chamber varies with the exposure to moist air. This paper presents a theoretical analysis of the variation. It is shown that when the vapor outgassing dominates the pumpdown, the slope of the log pressure versus log pumping time curve will vary from 1/2 to 3/2 for a semi-infinite plate. This is in contrast to the theoretical slope of 1/2 that is found for uniformly distributed gas in a semi-infinite plate.

2. Analysis In this analysis it will be assumed that the wall outgassing is governed by the usual diffusion equations, with a constant diffusion coefficient. vC v2 C ¼D 2 vt vX

(1)

37

38

Chapter 1 Physical basis of molecular-flow networks



vC N ¼ D vX

 (2) X ¼0

D; length2/time C; molecules/length3 N; molecules/length2 time It is first necessary to determine suitable initial conditions and boundary conditions for use in solving these equations. The initial conditions will depend upon the exposure history of the chamber walls. If the chamber has just been thoroughly outgassed, there will be a negligible concentration of molecules in the wall and the appropriate initial condition will be C ¼ 0 throughout the wall. If the chamber has just been exposed to a moist atmosphere for a long period of time, the concentration will be nearly uniform at a value C0 which will depend upon the wall temperature and the partial pressure of the water vapor in the moist atmosphere. Due to surface condensation, the value of C0 to be associated with the wall surface may be many times the concentration of water molecules in the atmosphere. This occurs even when the surface temperature is well above the temperature at which the pure water vapor would condense and results from the greater attraction between the water molecule and the wall than between water molecules. While these represent two simple initial conditions, in the general case the initial conditions will be more complex. In the typical operation of a chamber, the walls may be exposed alternately to both moist atmospheres and vacuums. If these exposure times are short compared to the time required to establish a uniform concentration of molecules within the sorption layer, the outgassing rate will depend upon the exposure history of the chamber over several of the preceding cycles. Since there are wide variations of possible exposure histories, it will be useful to analyze a typical history that demonstrates the effects of these variations.

4. First pump-down Considering only the pump-down immediately following the initial moist atmosphere exposure, two limiting conditions can be noted. If the equation is evaluated when the pumping time is short compared to the exposure time, the (t  tm1)1/2 term dominates. Since t  tm1 is the pumping time, this equation is the usual diffusion equation. On the other hand, if the equation is evaluated after long pumping times compared to the exposure

Chapter 1 Physical basis of molecular-flow networks

39

Figure 1. Variation in outgassing rate with exposure time for an initially outgassed system. (Any consistent time units.)

time, a (t  tm1)3/2 term dominates and the equation no longer resembles the usual diffusion equation. This result is shown in Fig. 1 where the relative outgassing rate has been calculated for a series of exposure times. In this and the following figures, the outgassing rate has been ratioed to the parameter (D/p)1/2C0. Included in this plot are some experimental data from Hayashi[1] and Power and Crawley.[2] These results were determined by calculating the average wall outgassing rate from the ratio of the chamber wall area to the pump mass flow, and using a value near the start of pump-down to evaluate C0(D/p)1/2. While the average outgassing rate should depend on the type of “O” ring seals, etc., and the parameter C0(D/p)1/2 would be expected to vary from chamber to chamber, for the three chambers considered the variation was only about a factor of 4 and the level was about 5  1015 molecules/cm2 sec1/2. In Fig. 1, the experimental level was adjusted to coincide with the theory at one point, so the method can only be used to check the general slope of the experiment and theory. The range of the slope in the Power and Crawley data[2] results from determining the outgassing rate by two methods. In the first method the pumping speed was assumed constant and in the second method the speed was varied according to their estimated pumping speed as a function of pressure. The resulting variation in outgassing rate shows the importance of accurately determining the pumping speed curve. The

40

Chapter 1 Physical basis of molecular-flow networks

variation in slope of the Hayashi[1] data resulted from changes in wall temperature. While the experimental and theoretical slopes show the approximate agreement, it appears specific experiments in a single chamber will be required to determine the validity of the theoretical approach. Power and Crawley[2] did present data for a short exposure time along with the data for the 17-hour exposure. The pump-down time required after the short exposure was much longer than the theoretical time, but since that experimental pump-down time was comparable to the time required after an extended dry air exposure, it is concluded that other factors contributed to the long pump-down time. If the system has a pumping speed that is independent of pressure, Fig. 1 can be used to estimate the relationship between the exposure time and the pumping time required to reach a specific pressure level. When this is done for a typical range of pumping and exposure times, it is found that the required pumping time is proportional to the exposure time raised to the 1/2 to 2/3 power. Thus doubling the exposure time should increase the pump-down time by about 50%.

5. Second pump-down If the initial moist atmosphere exposure time was very long compared to the time of interest, the concentration can be considered uniform, and the first pump-down will exhibit the square root relationship between outgassing rate and pumping time. If the system is now exposed to a moist atmosphere, and again pumped down, the resulting outgassing variation will depend upon both the previous pump-down time and the following moist atmosphere exposure time. Examples of the resulting outgassing variations are shown in Fig. 2 for a range of initial pumping times which have been followed by a unit exposure time. The general pattern of the outgassing curve is to follow the thoroughly outgassed curve until the outgassing rate is of the same order of magnitude as that at the end of the prior pumpdown at which point the curve flattens out approaching the prior pump-down curve, but always lagging behind it by a time interval equal to the sum of the exposure time and the present pumping time. This return to the square root relationship serves to emphasize that the unit exposure and subsequent pump-down represent only a relatively small variation in the general pump-down of a semi-infinite plate. This can be seen in Fig. 3 where the curve with a 100 unit initial pump-down time has been replotted so

Chapter 1 Physical basis of molecular-flow networks

Figure 2. Variation in outgassing rate with prior pumping time for a system with a uniform initial concentration. (Any consistent time units.)

Figure 3. Variation in outgassing rate with total elapsed time. (Any consistent time units.)

the time scale includes the initial pump-down and exposure. Here it is seen that the unit exposure and the subsequent pump-down occupy only a small fraction of the logarithmic plot, and it is also evident that the pump-down will continue along the original curve, despite additional minor exposures. This illustrates the

41

42

Chapter 1 Physical basis of molecular-flow networks

general problem in attempting to outgas the walls of a chamber when the slow diffusion rate makes the walls essentially semiinfinite. Each succeeding pump-down will produce an outgassing rate only slightly below that of the previous pump-down, and since the pressure is directly related to the outgassing rate, enormous pumping times would be required to significantly reduce the system base pressure. In practice this difficulty is overcome by baking the chamber, thus increasing the diffusion coefficient enough to remove the gas from the finite walls. Once the system has been baked and cooled, the slow diffusion rate will delay any significant in-gassing, and the analysis for the thoroughly outgassed system will apply. This emphasizes that the outgassing characteristics of a system are dependent upon the complete exposure history of the system, and not just the conditions immediately preceeding the pump-down. Notes (by N. Yoshimura): In Figs. 1e3, C; concentration in molecules/length3; C0; initial concentration; D; diffusion coefficient in length2/time.

6. Summary The results of a theoretical analysis have demonstrated that for a diffusion-controlled process, the outgassing rate depends upon the exposure history of the system. It has been shown that the slope of the log outgassing rate versus log pumping time curve will vary from nearly 0 to 3/2, if the layer is thick enough to satisfy the criterion L  2(Dt)1/2.

References in the paper[1-5] [1]

Hayashi, Trans 4th Nat. Vac. Symp. 1957, p. 13, Pergamon Press, New York (1958).

[2]

Power, B. D., and Crawley, J. E., Advances in Vacuum Sciences and Technology, (Namur Congress), 1958, pp. 206e211, Pergamon Press, Oxford (1960).

Reviewed paper [1-5]

K. W. Rogers, “The variation in outgassing rate with the time of exposure and pumping,” Transactions of the 10th National Vacuum Symposium, 1963 (Macmillan, New York, 1964) pp. 84e87.

2 Applications of the resistornetwork simulation method to complicated ultrahigh vacuum systems Key-point As described in the Introduction to Part 1, the following relationship is the foundation of the molecular-flow networks. Qoutgassing ¼ Qdesorption eQadsorption : As for the vacuum pumps, a similar relationship exists: Consider the sorption pump [ultimate pressure Pu (Pa), pumping speed S (L/s), which is applied to the vacuum chamber of pressure P (Pa)]. Then, Qpumpingsorptionpump ¼ Qadorptionsorption pump eQdesorptionsorption pump . When the sorption pump with Pu (Pa) and S (L/s) is applied to the chamber of pressure Pchamber (Pa), which is evacuated with a large sputter ion pump (SIP), the sorption pump acts as an outgassing source.

Keywords [1] Resistor network simulation method Qoutgassing ¼ Qdesorption  Qadsorption Molecular-flow networks Resistor network simulation method Characteristic values as a gas source K ¼ K0 ð1  P=PxÞ

Foundations of Molecular-Flow Networks for Vacuum System Analysis. https://doi.org/10.1016/B978-0-12-818687-9.00002-3 Copyright © 2020 Elsevier Inc. All rights reserved.

43

44

Chapter 2 Applications of the resistor-network simulation method

Vacuum circuit Designing a simulator circuit Electric circuit of the simulator Internal pressure Pressure generator Matrix calculation of pressures Digital computer Pressure distribution along an outgassing pipe

[2] Long history of works on molecular-flow networks Dushman’s well known textbook Electrical analogue to a high vacuum system Negative pressure generators of e760 Torr Vacuum metrics and electrical metrics

Chapter 2 Applications of the resistor-network simulation method

[1] Resistor network simulation method Reviewed paper

[1-1] "Resistor network simulation method for a vacuum system in a molecular flow region'' (S. Ohta, N. Yoshimura, and H. Hirano, 1983) S. Ohta, N. Yoshimura, and H. Hirano (1983) presented the paper,[1e3] “Resistor network simulation method for a vacuum system in a molecular flow region.” This paper is discussed in detail in this book. Note: The same paper was published in “JEOL Product Information in Electron Microscope.” The figures in JEOL Product Information[1-2-1] are more clear and, as a result, the figures in the JEOL Product Information are utilized in this book.

Abstract A simulation method to obtain the pressure distribution in a vacuum system has been proposed. The method introduces a new concept with regard to the function of each component of the vacuum system. The vacuum pump is regarded as a “vacuum resistor” connected at one side to the perfect vacuum. The gas source is regarded as a pressure generator connected to the vacuum. A connecting pipe is regarded as a vacuum resistor between the above elements. The vacuum sides of the pump

45

46

Chapter 2 Applications of the resistor-network simulation method

elements and the gas source elements are assumed to be connected by an imaginary route, and thus the vacuum system can be regarded as a closed vacuum circuit network. Such a vacuum circuit network may be replaced by an electric circuit network of a simulator for the vacuum system. The simulator was employed for the high vacuum system of an electron microscope in order to obtain the pressure distribution. The results obtained were in good agreement with the measured pressure distribution.

1. Introduction It is often difficult to estimate the pressure distribution in a vacuum system composed of many gas sources and several pumps. In order to overcome this difficulty, an electric circuit network simulator was developed to obtain the pressure distribution in the system in the molecular flow region. The simulation method is summarized as follows. (1) A vacuum pump and conducting pipe are converted into vacuum resistors. A gas source is converted into a pressure generator with vacuum resistance. The vacuum system is thus converted into a vacuum circuit network composed of these elements. (2) Vacuum resistors and pressure generators in a vacuum circuit are regarded as resistors and voltage generators in an electric circuit. Thus, the vacuum circuit itself is regarded as an analogous electric circuit. (3) Finally, the voltage distribution found in the electric circuit network corresponds to the pressure distribution in the vacuum system. This simulation method was applied to a vacuum system in the molecular flow region, and the pressure distribution in the system was successfully simulated.

2. Concept of a vacuum system In a molecular flow region, high vacuum pumps retain an almost constant pumping speed. The conductance of the conducting pipe is independent of the high vacuum pressure. The gassing rate[1] of the gas sources is considered to be almost constant. (*The term gassing rate includes outgassing rate and leakage rate.) In such a vacuum system, Ohm’s law is held; that is, P ¼ Q (1/ S þ 1/C) in relation to pressure P, gas load Q, pumping speed S,

Chapter 2 Applications of the resistor-network simulation method

and conductance C, where P is considered to correspond to voltage V, Q to current I, and 1/S or 1/C to resistance R in the electric circuit. The above concept suggests the possibility of simulating a vacuum system by an electric resistor network.

A. New concept of a vacuum system and its components An actual vacuum field is treated as an appositive pressure field above the perfect vacuum. The new concept of the vacuum system and its components are schematically presented in comparison with the conventional concepts in Fig. 1AeD. The gassing rate Q (Pa.L/s) can be defined by Q ¼ PQ /RQ. That is, the gas source with the gassing rate Q (Pa.L/s) is assumed to be replaced by a serial combination of a pressure generator with generating pressure PQ (Pa) and a vacuum resistor with resistance RQ ¼ PQ/Q (s/L). The negative terminal of the series is connected to the perfect vacuum in Fig. 1A. In addition, for constant Q, PQ must be large enough compared to the vacuum pressure, and RQ large enough compared to the resistances of other vacuum resistors described below. A vacuum pump with pumping speed S (L/s) is assumed to be a vacuum resistor with resistance RS ¼ 1/S (s/L), which is connected to the perfect vacuum, as shown in Fig. 1B. A conducting pipe with conductance C (L/s) is assumed to be a vacuum resistor with resistance RC ¼ 1/C (s/L), as shown in Fig. 1C. Here, the concept of an imaginary route is newly introduced to form a closed circuit. This imaginary route is a common route with zero resistance which connects the negative terminal of the pressure generator with PQ and the perfect vacuum side of the vacuum resistor with RS, as shown in Fig. 1D. The idea of introducing the imaginary route could be considered to be equivalent to the idea that pumped gases are reintroduced into the system through gas sources. According to the conventional concept, the pressures Pm and Pn in the vacuum system shown in Fig. 1D, are derived as follows: Pm ¼ Q=S;

(1)

Pn ¼ Pm þ Q=C ¼ Qð1=S þ 1=CÞ:

(2)

0 Pm

Pn0

The pressures and in the proposed vacuum circuit shown in Fig. 1D are derived as follows: 0 ¼ PQ RS =ðRQ þ RC þ RS Þ; Pm

(3)

47

48

Chapter 2 Applications of the resistor-network simulation method

Figure 1. Comparison between conventional and proposed concepts for vacuum elements and vacuum systems. The outlined part shows a high vacuum region. Ground potential corresponds to the perfect vacuum (0 Pa). 0 Pm ¼ Q=S;

(3’)

Pn0 ¼ PQ ðRC þ RS Þ=ðRQ þ RC þ RS Þ;

(4)

Pn0 y Qð1=S þ 1=CÞ;

(4’)

because, RQyPQ/Q, RQ[RS, and RQ[RC.

Chapter 2 Applications of the resistor-network simulation method

Comparison of Eqs. (1) and (3’) shows that Pm is almost equal 0 in the proposed vacuum circuit corresponding to the presto Pm sure Pm in the conventional vacuum system. This relation is also correct for Pn and Pn0 . Hence, the new concept is naturally acceptable. 0 , and between P The very small difference between Pm and Pm n 0 and Pn , will be discussed in detail in the later section (4. Discussion) on the dependence of the gassing rate on pressure. The above vacuum circuit will be simulated by the corresponding electric circuit network. In the vacuum circuit, pressure P, gas load Q, and resistance R are measured in units of Pa, Pa.L/s, and 1/L, respectively. These units correspond to volt (V), ampere (A), and Ohm (U), respectively, in the electric circuit network. However, in a practical simulation, the values of circuit elements are multiplied by a certain factor to construct a practical simulator. At first, the generated pressure of a gas source is tentatively set at 105 Pa, which is large enough as compared with the vacuum pressure required above. Full simulation procedures are presented in the following section.

3. Simulation of the high vacuum system of an electron microscope A contamination-free electron microscope was designed in which the vacuum pressure and vacuum quality around the specimen were essential. Fig. 2 is a typical example of an electron microscope with two pumps and five evacuation pipes used to evacuate the whole column. Photographic films in the camera chamber evolve a large amount of gas under a high vacuum. In order to minimize the adverse effect of this gas on the lens column, a differential pumping system is effectively provided by a small orifice inserted between the lens column and the camera chamber. The simulation method was applied to the vacuum system in Fig. 2, to obtain a reasonable vacuum distribution in the whole system. At the start of simulation, the conductance of every conducting pipe and the amount of gas of every part of the system have been estimated. The gas load Q generally consists of outgassing from materials and leakage. In a high vacuum system, the amount of leakage

49

50

Chapter 2 Applications of the resistor-network simulation method

Figure 2. The arrangement of the high vacuum system of an electron microscope: GC; gun chamber, MLC; minilab chamber, ACD; anticontamination device, SC; specimen chamber, IA; intermediate aperture, OR; orifice, CC; camera chamber, MPL; main pumping line, SIP; sputter ion pump, TMP; turbo-molecular pump.

must be made negligibly small compared with the amount of outgassing from the component materials. The outgassing rates (Pa.L.s1.cm2) from the component materials were measured by the pressure-rise method.[2] The typical values measured under several specific treatment conditions for the component materials are tabulated in Table 1. The treatment conditions in the table were selected so as to be consistent with the practical conditions in the vacuum system of this microscope. As seen in the table, the degassing treatment given by Ref. [2] is effective for components, especially Viton O rings. The amount of gas at every part of the system was calculated using the outgassing rate data in the table.

Chapter 2 Applications of the resistor-network simulation method

51

Table 1 Outgassing rates (Pa・L・s-1・cm-2) of the component materials (Ohta et al., 1983).

Evacuation time

30 h following bakeouta

200 h

Stainless steel Steel, Ni coated Casting materials of camera chamber Alumina (fine in structure) Viton-A O-ringb EM film

3.3  108 3.3  108

6.7  108 6.7  108

1.3107

1.3  10 1.3  106

24 h

1.3  106

-10

a

One-week vacuum bakeout at about 60 o C. One-week vacuum bakeout at about 100 o C before assembly.

b

A. Procedures to design a simulator circuit A typical procedure for the simplest system in Fig. 1D is shown in Fig. 3AeE instead of the procedure for our system. (a) First, let us estimate Q, C, and S of the system as 10-3 Pa$L/s, 10 L/s, and 100 L/s, respectively, as shown in Fig. 3A. (b) Draw the corresponding vacuum circuit (b) by “the proposed concept” shown in Fig. 1. Note that the value PQ is set to105 Pa which is large enough as compared with the vacuum pressure. (c) Convert the vacuum circuit into the corresponding electric circuit (c), that is, Pa into V (volts) and s/L into U (Ohms). In this circuit, the voltage generator of 105 V corresponds to the pressure generator of 105 Pa, and the electric resistors correspond to the vacuum resistors.

Figure 3. Procedures for designing a simulator circuit of a vacuum system.

1.3  105 2.7  107

52

Chapter 2 Applications of the resistor-network simulation method

In Fig. 3C, 105 V of the voltage generator is too large and 10 and 10e1 U of resistors are too small to construct a practical simulation circuit. Therefore, the following modifications are made as shown in Fig. 3D and E. (d) Modify the electric circuit as follows: (1) Multiply the voltage of the generator 105 V by 10e4. As a result, the reconverting factor k1 from voltage to pressure becomes 104 Pa/V. (2) Multiply the resistances of all resistors in Fig. 3C by 103. In this case the reconverting factor k1 remains 104 Pa/V. It should be noted that the relative distribution of the voltage in the circuit Fig. 3D is not changed by these modifications. In this circuit, however, 1011 U for the resistor is too large to construct a simulation circuit. (e) Further modify the circuit into the simulator circuit Fig. 3E in which 104 U of the resistor is used in place of 1011 U. The resistor of 104 U is easily obtainable. Note that 104 U is still large enough compared with other resistances, i.e., 102 and 10 U. Therefore, the relative distribution of the voltage in the circuit is kept constant with an error of one percent. As a result, the reconverting factor k2 from V to Pa in Fig. 3E is given by e2

k2 ¼ 107  k1 ¼ 103 Pa=V. The procedure for designing the simulator circuit of the high vacuum system of the electron microscope shown in Fig. 2 is the same as the one described above, though the high vacuum system has many elements. Fig. 4 presents the derived simulator circuit of the system in which the reconverting factor is the same as that described, 10e3 Pa/V.

B. The simulator circuit In the simulator circuit of Fig. 4, the outlined part corresponds to the whole column of the electron microscope. The resistors along the line between the gun chamber (GC) and the camera chamber (CC) correspond to the resistance of an orifice, conducting pipes, apertures, etc. The resistors located on the left side of the outlined part correspond to the outgassing rates of each part of the whole column. The resistors located above the outlined part correspond to the outgassing rates of evacuation pipes connecting the pumps to the column. These resistors related to the outgassing rates are to be connected to the voltage generator which supplies 10 V.

Chapter 2 Applications of the resistor-network simulation method

53

Figure 4. Electric circuit of the simulator. The outlined part is the whole column. The values of the resistors are represented in U and the symbols K and M used for the resistors mean, respectively.

The resistor of 10 U connected to switch 1 (SW1) corresponds to the cryo-pumping function of an anticontamination device (ACD) consisting of liquid-nitrogen-cooled fins. The pumping speed of this device is estimated to be 100 L/s. The voltage generator has a variable voltage range of 0e1 kV, is connected to CC and switch 2 (SW2) corresponds to the camera chamber with a variable pressure range of 0e1 Pa.

C. Application results Fig. 5 shows the simulated pressures in some parts of the microscope column. In Fig. 5, the vertical axis shows the reconverted pressures from voltages measured at GC, MLC, .CC in the simulator with and without ACD.

54

Chapter 2 Applications of the resistor-network simulation method

Figure 5. Pressure distribution in the electron-microscope column.

The pressure distribution curves indicate the minimum pressure at the specimen chamber (SC) in both cases. The ACD is proved to be effective for the specimen chamber. Fig. 6 shows changes of the simulated pressure in the specimen chamber PSC and the pressure in the “minilab” chamber (see Fig. 2) PMLC, depending on the pressure in the camera chamber PCC, from 0 to 1 Pa. The pressure PSC and PMLC gradually increase with varying PCC in the range from 0 to 1 Pa; however, their values are kept almost constant in the practical PCC range up to 10-2 Pa.

Figure 6. Changes of simulated pressures in the specimen chamber and the minilab chamber, depending on the pressure in the camera chamber. PSC is the pressure in the specimen chamber, PMLC the pressure in the minilab chamber, and PCC the pressure in the camera chamber.

Chapter 2 Applications of the resistor-network simulation method

Table 2 Comparison between simulated pressures (Pa) and measured pressures (Pa).

Simulated pressures (Pa)

Minilab chamber Sputter ion pump

Measured pressures (Pa)

SW1 open

SW1 close

ACD not cooled

ACD cooled

1.8  105 7.1  106

1.2  105 4.5  106

2.1  105 8.5  106

1.1  105 4.7  106

Table 2 shows a comparison between the simulated pressures and the measured pressures in the minilab chamber and the sputter ion pump. The measured pressures in the former were measured by a BayardeAlpert gauge, while the pressures in the latter were derived from its ion currents. The simulated pressures show good agreement with the measured pressures in both measurement points.

4. Discussion The simulation method presented has been successfully applied to the vacuum system of an electron microscope. However, the following two points remained unsolved: (1) In our development, we assume that the intrinsic ultimate pressure of a high vacuum pump is 0 Pa, and therefore ignore the fact that high vacuum pumps generally have their own intrinsic ultimate pressures. (2) The dependence of gassing rates of gas sources upon the high vacuum pressure has not yet been discussed. Regarding the first problem, when a vacuum pump with a pumping speed S (L/s) and an intrinsic ultimate pressure Pu (Pa) is used to evacuate gas load Q (Pa. L/s), the pressure just above the pump is Q/S þPu (Pa). This means that a high vacuum pump with S (L/s) and Pu (Pa) can be replaced by the vacuum resistor of 1/S (s/L) and the pressure generator of Pu connected in series, where the negative terminal of the pressure generator is connected to the 0 Pa region. The second problem is discussed in detail. According to conventional vacuum system analysis the gassing rates of gas sources are considered to be independent of the vacuum level. On the other hand, the gassing rates are treated in this method as being dependent on the vacuum pressure as described below.

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There are two kinds of gas sources: one is leakage and the other is outgassing. In the case of a leakage source, the following assumptions can be accepted. (1) The leakage rate into the perfect vacuum Q0 is maximum and that into the atmospheric pressure is zero. (2) The leakage rate at any vacuum pressure Q is smaller than Q0 by the amount which is proportional to pressure P. Under the above assumptions, the dependence of the leakage rate upon pressure is given as
  (5) Q ¼ Q0 1  P 105 : In the case of the outgassing source, similar assumptions are introduced. That is, (3) the outgassing rate reaches the maximum value Q0 in the perfect vacuum, 0 Pa, and becomes zero at a critical pressure P’. (4) The outgassing rate at any vacuum pressure Q is smaller than Q0 by the amount which is proportional to pressure P. Under these assumptions, the dependence of the outgassing rate upon pressure is given as Q ¼ Q0 ð1  P=P0Þ:

(6)

This results in a more accurate vacuum circuit than that described before. The modified vacuum system and the corresponding vacuum circuit are proposed in Figs. 7A and B in place of the one in Fig. 1D.

Figure 7. Modified high vacuum system and vacuum circuit version in the proposed concept. (A) High vacuum system, and (B) proposed vacuum circuit.

Chapter 2 Applications of the resistor-network simulation method

In Fig. 7B, Q0 denotes the leakage rate into the perfect vacuum. Q of the vacuum circuit shown in Fig. 7B can be easily found to satisfy Eq. (5). In the case of the outgassing source, 105 Pa in Fig. 7B should be changed to the critical pressure P ‘(Pa) at which the outgassing rate becomes zero. As is clear in the above discussion, Eqs. (5) and (6) can be naturally introduced from our proposed concept. The slight differences between Eqs. (1) and (3) and between Eqs. (2) and (4) can be also understood from the above arguments. The simulator circuit of Fig. 4 was designed on the basis of Eqs. (3) and (4). In a vacuum system such as the one used for a long-tube accelerator, where gas sources and conducting pipes coexist, the concept of the distributed constant circuit[3] can be employed for its pressure analysis.

5. Summary A simulation method for the vacuum system by the electric simulator has been established. The fundamental concept of the vacuum circuit is represented in Fig. 1AeD. In this study, the vacuum circuit was converted into the corresponding electric circuit, and the electric circuit was modified to form the simulator. It would be desirable to simulate the vacuum system by the vacuum circuit without modification. Further improvement of computer technology will make it possible to directly simulate a complex vacuum system by a vacuum circuit. Note: The term gassing rate includes outgassing rate and leakage rate.

Acknowledgments We thank the JEOL staff for their helpful assistance extended to our experiments.

References in the paper

[1-1]

[1]

Term gassing rate includes outgassing rate and leakage rate.

[2]

N. Yoshimura, H. Oikawa, O. Mikami, J. Vac. Soc. Jpn. 13, 23 (1970).

[3]

S. Ohta, presented at the 4th Annual Meeting of the Vacuum Society of Japan, Abstract for 5th Annual Meeting of the Vacuum Society of Japan, 21a3, 1963.

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Chapter 2 Applications of the resistor-network simulation method

Reviewed paper [1-1]

S. Ohta, N. Yoshimura, and H. Hirano, “Resistor network simulation method for a vacuum system in a molecular flow region,” J. Vac. Sci. Technol. A 1 (1), pp. 84e89 (1983).

Chapter 2 Applications of the resistor-network simulation method

[1-2] "Matrix calculation of pressures in high-vacuum systems'' (H. Hirano, Y. Kondo, and N. Yoshimura, 1988) H. Hirano, Y. Kondo, and N. Yoshimura (1988) presented the paper,[1-4] “Matrix calculation of pressures in high-vacuum systems.” The paper is composed of the following sections: 1. Introduction 2. Linear Vacuum Circuit 3. Matrix Analysis Method 4. Computer Analysis 5. Application to Practical High Vacuum Systems 6. Conclusion In this book Sections 1. Introduction, 2. Linear Vacuum Circuit, 5. Application to Practical High Vacuum Systems, and 6. Conclusion are introduced.

Abstract A matrix analysis method had been applied to two high vacuum systems: an outgassing pipe and an electron microscope. These vacuum systems were analyzed as linear vacuum circuits with pressure sources, current sources, and conductance using a digital computer. The pressures at six points along an outgassing pipe were calculated for two different models. A 5% difference in pressures was noted between the models. Pressures throughout an electron microscope high vacuum system were also successfully calculated by a digital computer. The pressure varied over three orders of magnitude depending on position.

1. Introduction A new concept has been recently introduced by Yoshimura[1] that various constituent elements of a high vacuum system, such as chamber walls, pumps, and pipe walls, may be regarded as a kind of gas source against the perfect vacuum. The function of a gas source for a high vacuum field of pressure P is

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represented by the function of a pressure source with a characteristic pressure and with a characteristic conductance applied to a field of P.[1],[2] As a result, a high vacuum system may be represented by a linear vacuum circuit composed of many pressure sources. The vacuum circuit naturally involves the relationship among the effective functions of constituent elements as gas sources. As is well known, a linear circuit can be analyzed by the matrix analysis method.[3] This paper introduces a matrix analysis method for calculating pressures in general high vacuum systems. The matrix method makes it possible for a digital computer to calculate the pressures in very complex vacuum circuits with many constituent elements. The mathematical procedure and computer calculated results are presented in this report for an outgassing pipe and for an electron microscope.

2. Linear vacuum circuit The following equation has been already introduced and discussed[1],[2] for expressing the relationship between the net outgassing rate of a gas source Q and the field pressure of high vacuum P. Q ¼ Q0 ð1  P=Px Þ ¼ Q0  Cx P;

(1)

where Cx¼Q0/Px. Clearly, this equation involves the assumption of the constant sticking probability of impinging gas molecules upon the wall surface under high vacuum. The net outgassing rate Q reduces with the increase in the field pressure P from the maximum rate Q0 by the rate CxP. In other words, the characteristic pressure Px of a gas source is regarded as a critical pressure at which its net outgassing rate Q reaches zero. Based on Eq. (1), a gas source at a field pressure P can be represented by a pressure source with a characteristic pressure Px and with a characteristic conductance Cx applied to a vacuum field of pressure P, as shown in Fig. 1A. The pressure source introduces gases into the field of P at the net rate Q which is given by Q0(1P/Px). The pressure source naturally sorbs gases effectively in a field of pressure P higher than Px. A high vacuum pump is also represented by a pressure source with a characteristic pressure Px and a characteristic conductance Cx. Such characteristic values of a pump, Px and Cx, correspond to its intrinsic ultimate pressure Pu and its pumping speed S, respectively. The function of a pump is the same in essence at that of a gas source.[1] The function of the flow route itself can be represented by the flow conductance C.

Chapter 2 Applications of the resistor-network simulation method

Figure 1. (A) Gas sources with Px and Q0; (B) gas sources with a current source Q.

A high vacuum system composed of many constituent elements (gas sources, including pumps, and flow routes) can be analyzed by the corresponding linear vacuum circuit composed of many pressure sources and conductances. However, it is generally difficult to estimate the Px and Q0 values of gas sources, the reason for which is the lack of data for the characteristic values Px and Q0 of various system materials under various conditions. On the other hand, net outgassing rates Q of system materials under a high vacuum have been measured and reported by many investigators. Under such a situation, handling a gas source as a current source [Fig. 1B] is also convenient practically, though the pressure dependence of the net outgassing rate is not reflected in the analysis. Now, let us consider a simple vacuum system [Fig. 2A] in a steady state, where a chamber with Px and Q0 is evacuated by a pump with its intrinsic ultimate pressure Pu and with its pumping speed S through a pipe with conductance C and with a negligibly small outgassing rate. The vacuum circuit corresponding to the system in Fig. 2A is presented in Fig. 2B, reflecting the pressure dependence of the outgassing rates of gas sources. The vacuum circuit with a current source of Q instead of a pressure source for the system is given in Fig. 2C.

5. Application to practical high vacuum systems Two high vacuum systems, an outgassing pipe and an electron microscope, were analyzed by a digital computer, based on the matrix analysis method.

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Figure 2. (A) High vacuum system composed of a gas source (P, Q0, Q) and a pump (Pu, S); (B) vacuum circuit with pressure sources Px and Pu; and (C) one with a pressure source Pu and a current source Q.

A. Pressure distribution along an outgassing pipe The pressures in the outgassing pipe system in Fig. 5A were calculated, assuming the same Px regardless of the pipe position, as follows. First, the following characteristic values of each element of the system are assumed as Px ¼ 1  104 Pa; Q0 ðor QÞ ¼ 1  104 Pa;

Chapter 2 Applications of the resistor-network simulation method

63

Figure 5. Outgassing pipe system (A), and the corresponding vacuum circuits (B) and (C). Circuit (B) is composed of pressure sources only, and (C) of current sources only.

C ¼ 10 L=s S ¼ 100 L=s. The pipe was divided into five elemental pipes with conductances of 50 L/s each. The vacuum circuit corresponding to the system in Fig. 5A is presented in Fig. 5B. A gas source is represented by a pressure source with a conductance. Another vacuum circuit, assuming the same net outgassing rate regardless of the pipe position, is presented in Fig. 5C. A gas source in this case is represented by that of a current source.

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Chapter 2 Applications of the resistor-network simulation method

Figure 6. Pressure distribution along the outgassing pipe of Fig. 5A. The points (•) correspond to the circuit (B), and the points (B) to the circuit (C).

The node pressures of the circuits in Figs. 5B and C were calculated and are presented in Fig. 6. The solid line shows the pressure distribution given by the following equation: Pk ¼ 1  106 þ kð1  k=2Þ  105 Pa. A pressure distribution along an outgassing pipe has been mathematically analyzed by us[4] under the assumption that the net outgassing rate per unit surface area of the pipe is the same regardless of the position. The pressure Pk in the pipe is expressed by the following equation[4] Pk ¼ Q=S þ kð1  k=2ÞQ=C; where k is the fraction showing the position, and Q is the net outgassing rate of the whole pipe. Substituting 1  104 Pa・L/s, 100 L/s, and 10 L/s for Q, S, and C, respectively, the pressure Pk for the system of Fig. 5A is mathematically expressed as Pk ¼ 1  106 þ kð1  k=2Þ  105 Pa.

(3)

The pressure distribution calculated using Eq. (3) is also presented in Fig. 6 by a solid line.

Chapter 2 Applications of the resistor-network simulation method

Comparing the three values (•, B, calculated line) at each position in Fig. 6, one can find the following: (1) There is a difference of w 5% between the pressures at each node in the circuits in Figs. 5B and C. This difference is due to whether or not the pressure dependence of outgassing is reflected in the circuit analysis. (2) The output node pressures in the circuit in Fig. 5C show good agreement with the corresponding mathematically calculated pressures Pk of Eq. (3). The quite small difference will vanish if the number of divided elemental pipes is increased sufficiently.

B. Pressure distribution in an electron microscope high vacuum system The pressure distribution in an electron microscope highvacuum system was analyzed by this method. The vacuum circuit is shown in Fig. 7. The Q and C input values are indicated for the

Figure 7. Vacuum circuit representing the electron microscope high vacuum system. The input data are seen for the individual elements. Node (1): gun chamber (GC), (5): minilab chamber (MLC), (6): specimen chamber (SC), (8): intermediate aperture (IA), and (10): camera chamber (CC).

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Table 2 Calculated pressures at every node in an electron microscope

Node

Output pressure (Pa)

(1) GC (2) (3) (4) (5) MLC (6) SC (7) (8) IA (9) (10) CC (11) (12) (13) (14) (15) (16) (17)

2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105 2.211  105

individual circuit elements. Table 2 and Fig. 8 show the calculated pressures for each node. Note that the pressure varies almost three orders of magnitude through the microscope column. The calculated pressures at the gun chamber (GC), minilab chamber (MLC), and camera chamber (CC) agree well with the measured pressures. Especially, the pressure at MLC of 9.2  10e6 Pa (see Table 2) shows a good agreement with the measured pressure of 1.1  10e5 Pa, which is already reported elsewhere.[2]

6. Conclusion The matrix analysis method has been applied to an outgassing pipe and to an electron microscope. The matrix method makes it possible for a digital computer to calculate pressures at many positions of complex high vacuum systems. The characteristic values Px and Q0 reflect the pressure dependence of outgassing rates Q. Materials for an ultrahigh vacuum have very low Px values, showing strong pressure dependence of

Chapter 2 Applications of the resistor-network simulation method

Figure 8. Calculated pressure distribution of the electron microscope high vacuum system.

the outgassing rate. However, the values Px and Q0 of various materials are not available in general. It is indeed expected that the characteristic values Px and Q0 of various ultrahigh vacuum materials under various pretreatment conditions are to be measured and published by many researchers for accurate analysis of ultrahigh vacuum systems.

References in the paper[1-2] [1]

N. Yoshimura, J. Vac. Sci. Technol. A 3 (6), 2177e2183 (1985).

[2]

S. Ohta, N. Yoshimura, H. Hirano, J. Vac. Sci. Technol. A 1 (1), 84e89 (1983).

[3]

J. Staudhammer, Circuit Analysis by Digital Computer, Electrical Engineering Series (Prentice-Hall, Englewood Cliffs, NJ, 1975), p. 50.

[4]

H. Hirano and N. Yoshimura, J. Vac. Sci. Technol. A 4 (6), 2526e2530 (1986).

Reviewed paper [1-2]

H. Hirano, Y. Kondo, and N. Yoshimura, Matrix calculation of pressures in high-vacuum systems, J. Vac. Sci. Technol. A 6 (5), pp. 2865e2869 (1988).

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[1-3] "Analysis of pressure distribution, based on vacuum circuits'' (N. Yoshimura and H. Hirano, 1988) N. Yoshimura and H. Hirano (1988) presented the brief report,[1-4] “Analysis of pressure distributions, based on vacuum circuits,” written in Japanese. The paper is composed of the following sections: 1. Introduction 2. Pressure distribution through a high vacuum system 3. Pressure distribution along a conducting pipe 4. Pressure distribution in a complex-chambers-system with homogeneous walls 5. Closing. In this book Section 4. Pressure distribution in a complexchambers-system with homogeneous walls is introduced.

4. Pressure distribution in a complexchambers-system with homogeneous walls For a complex-chambers-system with homogeneous walls we can analyze the pressure distribution using the measured pressures at one position under the assumption that all system walls show the same outgassing rates regardless of elapsed time. Let us consider the complex-chambers-system with homogeneous walls, as shown in Fig. 3A. Also, assume that the outgassing rates per unit area of all chamber wall surfaces and pipe wall surfaces are the same, though the values vary with the elapsed time. Abbreviations are as follows: A1, A2, A3, A4; Inner-surface areas of respective chambers A01, A12, A23, A34, A41; Inner-surface areas of respective pipes F01, F12, F23, F34, F41; Conductance of respective pipes, known values S; Pumping speed; known value PX and q0; Unknown accuracy. Usually these values are set appropriately from the estimated outgassing rates

Chapter 2 Applications of the resistor-network simulation method

Figure 3. A system of combined chambers of homogeneous walls and the corresponding circuits. (A) Original vacuum system, (B) the corresponding vacuum circuit, and (C) an electric circuit analogous to circuit (B).

Vacuum circuits corresponding to the system in Fig. 3A are presented in Fig. 3B, where the ultimate pressure of the pump is assumed as 0 Pa. Take notice here that the flow resistances related to PX and q0 are much higher than the resistances related to F of the pipe and S of the pump. As a result, we can assemble the electric circuit with the relative voltage distribution which is equivalent to the relative pressure distribution in the vacuum circuit. Using appropriate values for K, K’, and V (Volt), assemble an electric circuit [Fig. 3C] analogous to the vacuum circuit [Fig. 3B]. The voltage distribution in the circuit (c) is almost the same as the pressure distribution in the vacuum circuit (b). Measure the respective voltages V0, V1, V2, V3, and V4. In the case that the pressure just above the vacuum pump is measured as P0, then pressures in the respective chambers P1, P2, P3, and P4 are measured as follows: P1 ¼ ðV 1 =V 0 ÞP0 ; P2 ¼ ðV 2 =V 0 ÞP0 ; P3 ¼ ðV 3 =V 0 ÞP0 ; and P4 ¼ ðV 4 =V 0 ÞP0 :

Reviewed paper [1-3]

N. Yoshimura and H. Hirano, Analysis of Pressure Distribution, based on Vacuum Circuits, J. Vac Soc. Japan, 27, (5), pp. 471e473, (1984) (in Japanese).

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[2] Long history of works on molecular-flow networks Reviewed papers

[2-1] "Comments on: 'Resistor network simulation method for a vacuum system in a molecular flow region' '' [J. Vac. Sci. Technol. A 1, 84 (1983)] (B. R. F. Kendall, 1983) B. R. F. Kendall (1983) presented the short report,[2-1] “Comments on: ‘Resistor network simulation method for a vacuum system in a molecular flow region’ [ J. Vac. Sci. Technol. A 1, 84 (1983)].” In this book this report is fully introduced. A recent paper by Ohta et al.[1] discusses a practical application of the theory of molecular-flow networks. Readers of this paper could be left unaware of a substantial body of related work which has been developed during the past several decades. Molecular-flow network theory is based on the similarities between the equations describing current flow in electrical networks and the equations describing the flow of gases under pure molecular flow conditions. The long history of the subject can be traced back to Dushman, who discussed the analogies between molecular-flow resistance, conductance, flow rate, and pressure on the one hand, and electrical resistance, conductance, current, and potential on the other. These ideas received limited circulation as early as 1922,[2] and appeared again in 1949 in Dushman’s well known textbook.[3] An early application of molecular flow network theory was given by Aitken[4] in 1953. He described an electrical analogue of the vacuum system of the Oxford 140 MeV synchrotron,

Chapter 2 Applications of the resistor-network simulation method

complete with current sources which could be switched to various points to simulate real and virtual leaks. This was probably the first application of an electrical analogue to the design and operation of a large, complex vacuum system. A brief note by Stops (1953)[5] and a much more detailed paper by Teubner (1962)[6] extended the theory to include chambers interconnected by flow restrictors and to nonequilibrium flow conditions. The volumes of the chambers were represented by capacitances. Both writers used the concept of a pump as a resistive sink for gas molecules. Stops included the ultimate pressure of the pump by introducing a series electromotive force. These concepts reappear in the recent work by Ohta et al. Electrical analogies were also mentioned by several other workers,[7e9] often accompanied by appropriate reminders that the theory could be expected to apply only if true molecular flow (free of beaming effects) was present. Barnes[10] used a rather different approach, based on both electrical and radiant heat transfer analogies, to describe the mass flow of gases to cooled panels in vacuum chambers. Implications of the superimposed flow of different gases in the same system were discussed in another early paper.[11] Statements of the molecular flow equivalents of Kirchhoff’s laws, venin’s theorem, and the concepts of input and output The impedance appeared shortly afterwards and were extended to various cases of nonequilibrium flow with gas mixtures.[12,13] Much of the work done on molecular flow networks up to 1971 was summarized in a review paper presented at the Fifth International Vacuum Congress.[14] The paper discussed symbols for use in network diagrams, the properties of flow-restricting materials, applications to gas analysis, and the use of electronic analogue simulators for design and leak location. The work reported by Ohta et al. represents a useful addition to the expanding list of applications of molecular flow network theory. It is to be hoped that it will increase awareness of the power of network techniques and of electronic simulation in vacuum applications. A logical direction for further research is the development of digital computer programs capable of including the gas beaming corrections in assemblies of practical components.

References in the paper [1]

[2-1]

S. Ohta, N. Yoshimura, and H. Hirano, J. Vac. Sci. Technol. A 1 (1), 84 (1983).

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[2]

S. Dushman, Production and Measurement of High Vacuum (General Electric, New York, 1922).

[3]

S. Dushman, Scientific foundations of Vacuum Technique, 1st ed. (Wiley, New York, 1949). See also 2nd ed., 1962.

[4]

M. J. Aitken, Brit. J. Appl. Phys. 4, 188 (1953).

[5]

D. W. Stops, Brit. J. Appl. Phys. 4, 350 (1953).

[6]

W. Teubner, Exp. Tech. Phys. 10, 279 (1962).

[7]

A. D. Degras, Le Vide 11, 155 (1956).

[8]

J. Delafosse and G. Mongodin, Le Vide 16, 118 (1961).

[9]

W. Steckelmacher, Vacuum 16, 561 (1966).

[10] C. B. Barnes, Vacuum 14, 429 (1964). [11] B. R. F. Kendall, J. Vac. Sci. Technol. 5, 45 (1968). [12] B. R. F. Kendall and R. E. Pulfrey, J. Vac. Sci. Technol. 6, 326 (1969). [13] B. R. F. Kendall and Gladys Englehart, Rev. Sci. Instrum. 41, 1623 (1970). [14] B. R. F. Kendall, J. Vac. Sci. Technol. 9, 247 (1972).

Reviewed paper [2-1]

B. R. F. Kendall, Comments on: Resistor network simulation method for a vacuum system in a molecular flow region, J. Vac. Sci. Technol. A 1, 84 (1983).

Chapter 2 Applications of the resistor-network simulation method

[2-2] "Theoretical analysis of a two-pump vacuum system''* (B. R. F. Kendall, 1968) B. R. F. Kendall (1968) presented the paper,[2-2] “Theoretical analysis of a two-pump vacuum system.” The paper is composed of the following sections Introduction 1. Equivalent Networks 2. Steady-State Analysis 3. Transient Response 4. Experimental Confirmation 5. Discussion. In this book Sections Introduction and 1. Equivalent networks are introduced.

Abstract Factors affecting the pressure in a vacuum chamber pumped by two dissimilar pumps are investigated with the aid of a simplified theoretical model. The possibility of one of the pumps acting as a significant source of gas or vapor is taken into account. It is shown that the minimum chamber pressure may be obtainable only with a finite pumping-line impedance in series with this pump. An experimental illustration of the phenomenon is given. Applications of the theory to the design of vacuum systems are discussed.

Introduction Two complementary pumping devices are frequently used to maintain a low pressure in a vacuum chamber. Examples are mechanical pump/adsorption traps, diffusion pump/cold traps, and ion pump/titanium getter combinations. These are all basically

*This work was supported by NASA Sustaining Grant No. NRG-39-009-015.

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pumping devices because they act as sinks for gas molecules. The members of each combination are complementary in the sense that each pumps any gases which are not efficiently pumped by the other. At least one member of any pair of complementary pumping devices may simultaneously be acting as a gas or vapor source.[1] These gases or vapors must be pumped by the other member of the pair of pumping devices. The result is a complex counter-flow of gases and vapors in different parts of the system, which is best analyzed in terms of network theory. A partial analysis of a two-pump vacuum system has already been carried out by Teubner.[2] The following treatment differs from this earlier analysis in using more detailed models to represent the pumps, and in dealing separately with the partial pressures of the two types of gases present. Pure molecular-flow conditions are assumed.

1. Equivalent Networks The network corresponding to a two-pump vacuum system contains a large number of variables in its most general form. It is, therefore, too cumbersome to analyze in detail. However, the problem can be reduced to more realistic dimensions by making appropriate simplifying assumptions. A common two-pump arrangement is shown in Fig. 1A. Pump A is connected through a pumping line with impedance R to the work chamber, while pump B is mounted in the chamber or very close to it. It can, therefore, be assumed that the impedance between pump B and the chamber is negligible. Pump A is assumed to act as a constant-pressure source of a gas or vapor pumped by pump B. Gas evolution from pump B is assumed negligible compared with the chamber outgassing rate. These assumptions are reasonably valid for mechanical pumps/zeolite trap diffusion pumps/cold traps, and certain other pump combinations. The gases and vapors present are assumed to fall into two groups. Those pumped primarily by pump A will be collectively referred to as gas A, while those pumped primarily by pump B will be collectively defined as gas B. In practice, gas A might be a permanent gas B, with gas B a condensable vapor. It is further assumed that pump B does not significantly pump gas A, and that the action of pump A has no significant effect on the equilibrium pressure of gas B above it.

Chapter 2 Applications of the resistor-network simulation method

Figure 1. (A) Common type of two-pump vacuum system; (B) and (C) equivalent networks for gases A and B, drawn separately.

Figs. 1B and C show the equivalent networks for gases A and B, respectively. The work-chamber volume is represented by a capacitance-like term V. Outgassing and leaks in the work chamber are represented by the currents QA and QB. Reciprocal pumping speeds of the two pumps are represented by the impedance RA and RB, which are “grounded” to imaginary points of zero pressure via the constant pressure differences PA and PB. Pressure PA represents the ultimate pressure of gas A which would exit above pump A after prolonged pumping on a perfectly clean sealed system. Pressure PB has the same significance for gas B and pump B. The equilibrium pressure of gas B above pump A is represented by p0B . In practice, this might be the effective vapor

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pressure of a pumping fluid, including contributions from water vapor in the case of a mechanical pump. The impedance R is assumed to be identical for the two gases. Point X represents the interior of the chamber.

References in the paper[2-2] [1]

F. Pagano, Trans. 8th AVS Vac. Symp. (The Macmillan Company, New York, 1962), p. 86.

[2]

W. Teubner, Exp. Tech. Phisik 10, 275 (1962).

Reviewed paper [2-2]

B. R. F. Kendall, Theoretical analysis of a two-pump vacuum system, J. Vac. Sci. Technol. 5 (2), pp. 45e48 (1968).

Chapter 2 Applications of the resistor-network simulation method

[2-3] "An electrical analogue to a high vacuum system'' (J. Aitkin, 1953) J. Aitkin (1953) presented a Correspondence,[2-3] “An electrical analogue to a high vacuum system.” This short report is fully described in this book. In a complex vacuum system where there are several pumping lines in parallel, or, where, as in synchrotrons and betatrons, the vacuum chamber is a toroidal tube of low pumping speed, the calculation of the effect on the pressure of altering various parts of the pumping system (e.g., speed or number of pumps, length of pumping lines) is rather tedious. So too is the estimation of the whereabouts of a leak from observed pressure gauge readings on the system. An electrical analogue can be used to obtain the required answers very quickly by simple voltage measurements. The analogue is only applicable when the pressure is small enough for the mean free path to be larger than the dimensions of the system; the rate of mass flow through a tube is then given by, Q ¼ Sðp1  p2 Þ

(1)

where S is the speed of the tube and depends only on its dimensions, p1 and p2 are the pressures at either end of the tube. Similarly, for a diffusion pump of speed S0, at the throat of which the pressure is p0, the rate of mass flow through the pump is, Q ¼ S0 p0 :

(2)

Both Eqs. (1) and (2) are analogous to Ohm’s law, so that an electrical analogue can be constructed in which voltage corresponds to pressure, current to mass flow, and resistance to the reciprocal of speed. Fig. 1 shows the analogue constructed for the vacuum system of the Oxford 140-MeV synchrotron. The procedure for finding the locality of a leak is first to measure the pressures on the five gauges on the vacuum system. The probe leak on the analogue is then connected to various points of the electrical network until one is found for which the ratios of

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Figure 1. Electrical analogue. R1¼90 U, R2¼170 U, R3¼95 U, R4¼65 U, R5¼850 U, R6¼25 kU, R7¼100 kU. V ¼ voltmeter (f.s.d. 2.5 V), mA ¼ milliammeter (f.s.d. 4 mA), S ¼ switches, A, pump throat; B, cold trap; C, gauge; D, probe leak; E, electron gun; F, neoprene sleeves; G, target.

the voltages at the points corresponding to the gauges is the same as the ratios of the actual gauge readings. The permanent leak into the system is due to diffusion of air through the eight neoprene sleeves (see Fig. 1) which join the eight sectors of the orbit tube together and the residual pressure due to this leak for various different pumping arrangements is found by connecting eight equal current sources (not shown) to the points corresponding to the sleeves; the electrical circuit is then altered to correspond to the various arrangements and the voltages observed in each case. Pressure due to evolution of vapor (e.g., from the gun heater) which is condensed on the cold trap, can be dealt with by closing the three switches shown, connecting on the probe leak at the point corresponding to the gun heater and proceeding as before. The analogue has been in use for the past year and has been found to be invaluable, particularly in the localization of leaks. There are 15 places around the orbit tube at which leaks may occur and, by use of the analogue, suspicion can in a few minutes be centered on one of these places. With this localization

Chapter 2 Applications of the resistor-network simulation method

procedure a straightforward method of leak finding (a jet of coal gas or butane in conjunction with a normal ionization gauge) has been found to be quite adequate. M.J. Aitken Clarendon Laboratory Oxford

Reviewed paper [2-3]

J. Aitkin, Correspondence: An electrical analogue to a high vacuum system, Brit. J. Appl. Phys. 4, 188, 1953.

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[2-4] "Further applications of the electrical analogue to vacuum system'' (D. W. Stops, 1953) D. W. Stops (1953) presented a Correspondence,[2-4] “Further applications of the electrical analogue to vacuum system..” This short report[2-4] is fully described in this book. The analogous aspects of the vacuum systems and certain electrical circuits have long been recognized, but apparently like the application of analogue techniques to heat and mass flow problems, the practical usefulness has taken some time to be appreciated. The letter from M. J. Aitken,[1] has indicated that the electrical analogue may play an increasingly important part in the study and design of large vacuum systems, and to these ends, a transient response type would often prove useful. Elementary considerations show that the ordinary type of vacuum system operating under free molecular gas flow conditions may be represented by an R-C electrical circuit with various sources of e.m.f.’s or currents. The related equations are: Q¼dP/W¼VdP/dt at constant volume. i¼dv/R¼Cdv/dt at constant capacity. The analogous quantities are apparent from comparison of these two sets of equations. A simple vacuum system, shown diagrammatically by the upper part of the figure (Fig. 1), has a pump of speed S, lumped elements of volume V and tabulation resistances W with leaks Q0 at various points. These leaks may be viscous in character and therefore not satisfy the linear pressure relation, but the changes in gas pressure in the system will usually be so small relative to the external pressure PA, that there will be little error in writing,  Q1 ¼ ðPA  P1 Þ W10 ; etc:; with W10 as the effective resistance of the leak. The lower part of the figure (Fig. 1) shows the analogous electrical circuit, with resistors R replacing the tubes, condensers C replacing the volumes, and voltages v corresponding to the pressures. Measurements of voltages with time will give the corresponding variations of pressures with time, and likewise the various currents i will give the appropriate values of throughputs Q.

Chapter 2 Applications of the resistor-network simulation method

Figure 1. Analogous vacuum and electrical systems.

The scaling factors introduced into the values of the parameters will determine the relative time scales, which in turn will usually decide the mode of presentation of the information provided by the analogue. This subject has been discussed at length by Pashkis and Baker,[2] Lawson and McGuire,[3] and others. Lawson and McGuire have also examined mathematically the accuracy attainable with a lumped element analogue of a continuously distributed system, which will be relevant in the vacuum case when the system tabulation is also the main volume. With the analogue envisaged, the effects of inserting pumps or pumping devices at various points in a vacuum line may be easily determined. By switching a metered quantity of charge into a point, the sudden evolution of known quantities of gas or vapor may be simulated. Similarly, the response of a system to mixtures of gas and/or vapors may be found by varying the values of the parameters appropriate to each component.

References in the paper[2-4] [1]

M. J. Aitken, Brit. J. Appl. Phys., 4, p. 188, 1953.

[2]

V. Pashkis and H. O. Baker, Trans. Amer. Soc. Mech. Engrs., 64, p. 105, 1942.

[3]

D. I. Lawson and J. H. McGuire, Private communication.

Reviewed paper [2-4]

D. W. Stops, Further applications of the electrical analogue to vacuum systems, Brit. J. Appl. Phys. 4, 350, 1953.

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[2-5]

"Analogies  el ementaires entre le vide et l'electricit e'' (A. D. Degras, 1956) A. D. Degras (1956) presented the paper,[2-5] “Analogies le mentaires entre le vide et l’e lectricite ,” written in French. e The paper is composed of the following sections: 1. Introduction me e le mentaire 2. Syste 3. Applications 4. Conclusion. me e  le mentaire are introduced. In this book figures in 2 Syste Abbreviations are as follows: 1/R conductance de la liaison  ne rateur 1/r conductance du ge C capacit e du condensateur E tension .m. du ge  ne rateur (Fig. 1e4) E1 f. e F admittance de la canalisation P pression P1 pression limite de la pompe S debit de la pompe servoir V volume du re

Figure 1. Systeme parfait de pompage (in section 2. Syst eme  el ementaire).

Chapter 2 Applications of the resistor-network simulation method

Figure 2. Systeme avec desorption.

Figure 3. Syst eme avec fuite reelle.

Figure 4. Systeme avec fuite reelle et desorption.

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Reviewed paper [2-5]

lementaires entre le vide et l’e lectricite , Le A. D. Degras, Analogies e Vide, No. 64, Jullet-Aout, pp. 155e162, 1956, in French.

Chapter 2 Applications of the resistor-network simulation method

[2-6] "Vacuum circuit for a parallel evacuation'' (S. Ohta, 1962) S. Ohta (1962) presented a short report,[2-6] “Vacuum circuit for analyzing a parallel evacuation system” in “Meeting on vacuum technology”(in Meeting of Vac. Soc. Japan, 1962) in Japanese. Vacuum circuit corresponding to a multiple-pump system, where vacuum pumps were converted to negative pressure generators of e760 Torr. Ohta also presented the respective vacuum circuits corresponding to a long orbit accelerator and an instrument to make a clean, pure plasma. The figures in this short report are introduced, as follows. Notes (by Yoshimura) S. Ohta designed the vacuum circuits in the field lower than 760 Torr, compared with the atmospheric pressure, as seen in Figs. 1e3. Negative batteries are highlighted with red circles by Yoshimura. This short report stimulated the present author (Yoshimura) to design the molecular-flow network for an ultrahigh vacuum electron-microscope system, based on desorption/ adsorption theory.

Figure 1. Conventional method (left) and our method (right) for analyzing vacuum systems.

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Figure 2. Vacuum circuit for QP (Plasma-Purify Instrument).

Figure 3. Vacuum circuit for Long-Orbit Accelerator Tube.

Reviewed paper [2-6]

S. Ohta, On parallel operations of vacuum systems, J. Vac. Soc. Japan, p. 21a-3 ,1962, in Japanese.

Chapter 2 Applications of the resistor-network simulation method

[2-7] "Numerical modeling of vacuum systems using electronic circuit analysis tools'' (S. R. Wilson, 1987) S. R. Wilson (1987) presented the paper,[2-7] “Numerical modeling of vacuum systems using electronic circuit analysis tools.” The paper is composed of the following sections: 1. Introduction 2. Model A. Transformation rules B. Circuit description C. Analysis D. Interpretation 3. Example 4. Conclusions. In this book Section 2. Model: A. Transformation rules is introduced.

Abstract The analysis of large and complex high vacuum systems is a tedious process, especially when distributed loads and pumps are involved. It has been suggested that by using the appropriate transformations, a vacuum system can be modeled as an electrical network. Well-developed numerical techniques exist for the computational characterization of electrical networks, often implemented in the form of a general-purpose circuit simulation code. In this paper, a method is presented for the modeling of high vacuum systems using a circuit analysis code. The method includes the capability to model outgassing, and to obtain the response of a system to transient and time-varying loads. An example is given of the analysis of a vacuum system with distributed pumps and distributed loads using the commonly available SPICE II program. The use of this technique is expected to significantly reduce the effort required to analyze such systems as accelerators, storage rings, and process lines.

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2. Model A. Transformation rules The fundamental relation describing the molecular flow of gas through a component is given by Q ¼ SP;

(1)

where Q is the gas flow rate in Torr.L/s, S is the conductance or speed of the components in L/s, and P is the instantaneous pressure in Torr across the component. For a fixed volume V in liters, Q is given by Q ¼ VdP=dt;

(2)

where dP/dt is the time rate of change of the pressure in the volume. The relation describing the flow of slowly varying electrical currents is I ¼ E=R;

(3)

where I is the current in amperes (C/s), E is the potential across the electrical component in volts, and R is the resistance of the element in ohms (V/A). Letting Y¼1/R, Eq. (3) can be written as I ¼ YE;

(4)

where Y is the admittance or conductance of the electrical element in mhos (A/V). For a capacitive element, which means that the element is capable of storing some amount of electrical charge, the following relation holds: I ¼ CdE=dt;

(5)

where C is the capacitance of the element in farads (C/V), and dE/ dt is the time rate of change of the impressed potential. To determine the relationships between the vacuum metrics and electrical metrics, we follow DeGras[2] in comparing the terms in similar equations. Matching terms in Eqs. (1) and (4), we see that Q corresponds to I, and that S corresponds to Y. Likewise, matching terms in Eqs. (2) and (5), V corresponds to C, and P corresponds to E. Observe that there is no passive vacuum element which corresponds to an electrical inductor. However, an active element, such as a flow-controlled valve, can behave as an inductor. The relationship between the electrical measure of inductance and

Chapter 2 Applications of the resistor-network simulation method

the control parameters of the flow controller can be abstractly found. The pressure across a flow-controlled element can be given as P ¼  KdQ=dt;

(6)

where K is a constant for the particular flow controller which describes its sensitivity to a change in flow rate, in s2/L. The electrical relation for an inductor is given by E ¼  LdI=dt;

(7)

where L describes the inductance of the electrical element in Henrys (V.s/A). Matching terms in Eqs. (6) and (7), we see that K corresponds to L. The physical meaning of K is seen by writing the units as s/(L/s). The idea is that K is the amount of time required to change the effective speed of the flow-controlled element by 1 L/s. The correspondences described above are summarized in Table 1. Based upon the collection of relationships, a pipe can be modeled as a combination of a resistive and capacitive element in the form of a passive integrator. A flow-controlled element would be modeled as the series combination of a resistive and an inductive element. The electrical model of a pump is obtained by observing that the pump has an intrinsic speed, which is assumed to be constant, and an intrinsic ultimate pressure, also assumed to be constant. The pump can then be represented as the series combination of a resistive element and a constant voltage source.[2] The electrical conductance of the resistive element, in mho, is numerically equal to the pump speed, while the voltage across the constant voltage source in volts is numerically equal to the ultimate pressure in Torr. This model is illustrated in Fig. 1A.

Table 1 Correspondences between vacuum metrics and electrical metrics.

Vacuum metric

Vacuum symbol

Vacuum units

Electrical metric

Electrical symbol

Electrical units

Pressure Gas flow Conductance Volume Flow controller

P Q S V K

Torr Torr.L/s L/s L S2/L

Potential Current Conductance Capacitance Inductance

E I Y C L

V A mho F H

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Figure 1. Electrical models of vacuum components: (A) electrical model for a pump, (B) segmentation model for a pipe, and (C) distributed pump model. Note that the pipe has been split into three segments.

A gas source can be presented as a constant current source, with value in amperes numerically equal to the flow rate in Torr.L/s. This is useful for modeling gas loads and other sources, including outgassing. The modeling of outgas loading on a vacuum element is achieved by dividing the element into segments. Each segment

Chapter 2 Applications of the resistor-network simulation method

is modeled as a split component with a constant gas load representing the outgassing contribution from that segment. This is illustrated in Fig. 1B. The resistances in ohms are computed from Rseg ¼ 1=2NSe ;

(8)

where N is the number of segments and Se is the conductance of the element. The segment capacitances in farads are found from Cseg ¼ Ve =N ;

(9)

where Ve is the total volume of the element, assuming axial symmetry. The segment gas loads in amperes are given by Iseg ¼ Ioutgas =N ;

(10)

where Ioutgas is the total outgas load for the element. The segmented outgassing model is essentially a finite element model for the original component. The accuracy of the computed pressures along such an element increases as N becomes larger. For most situations, letting N be about 10 gives good results, but the analyst must be the judge of the results for the particular application. Distributed pumping systems can be modeled by simply attaching pumps of the desired size at the proper places. This may require the splitting of components into segments, such as when attaching distributed pumps along the length of a pipe. This is illustrated in Fig. 1C. Several types of gas loads may be applied. The most common is the steady state or constant load. Transient gas loads may also be modeled. An example of a transient gas load is a pulsed ion source. Since transient loads are easily described as timevarying currents, the powerful techniques of Laplace and Fourier analysis can be applied, just as for electronic circuits.

References in the paper [2]

[2-7]

A. D. Degras, Vide 11, 155 (1956).

Reviewed paper [2-7]

S. R. Wilson, Numerical modeling of vacuum systems using electronic circuit analysis tools, J. Vac. Sci. Technol. A 5 (4), pp. 2479e2483 (1987).

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[2-8] "An analysis of a complex network of vacuum components and its application'' (G. Horikoshi, Y. Saito and K. Kakihara, 1990) Horikoshi et al. (1990) presented the paper,[2-8] “An analysis of a complex network of vacuum components and its application.” The paper is composed of the following sections: 1. Introduction 2. Vacuum pump and conduit pipe as a network component 3. Kirchhoff’s law in a network of vacuum components 4. Conclusion. In this book Sections 2. Vacuum pump and conduit pipe as a network component and 3. Kirchhoff’s law in a network of vacuum components are introduced.

Abstract An analysis of a complex network composed of many vacuum components was made by using an analogy between pipe conductance and electric conductance of circuit components. In the analysis, we have applied concepts of Kirchhoff’s law and Green function. The Green function matrix introduced in the analysis gives effective pumping speeds for all connection points in the network. As an example, we applied the analytic method to a network with a satisfactory result.

2. Vacuum pump and conduit pipe as a network component A vacuum pump with a pumping speed S and an ultimate pressure Pu can be considered as a network component with two terminals (Fig. 1). One of the terminals (a) is an open end where the pressure is kept constant to Pu. The other end (b) corresponds to

Chapter 2 Applications of the resistor-network simulation method

Figure 1. Vacuum pump and conduit pipe as network components.

the pump mouth and the pressure P will be determined by the gas flow Q through the pump as P ¼ Pu þ Q=S;

(2)

assuming the pump does not release gas. If we introduce a virtual “reference” point which is an ideal gas sink, we can consider that the end point of the pump is connected to the “reference” (P¼0) with a pressure jump Pu. In the next step, we consider a conduit pipe of length L and of uniform cross-section having a uniform distribution of gas generation (Fig. 1). Letting c and q be the pipe conductance per unit length and gas generation rate per unit length, respectively, we can obtain the pressure distribution P(x) along the pipe as follows:   P2  P1 q q 2 x ; þ (3) PðxÞ ¼ P1 þ x 2C 2CL L which is shown in Fig. 1. Here C ¼ c/L is the conductance of the pipe. The boundary conditions are P(0) ¼ P1 and P(L) ¼ P2. From Eq. (3) we obtain the following relations

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Q1 ¼

P1  P2 qL P1  P2 qL C  ; Q2 ¼ Cþ ; 2 2 L L

(4)

where Q1 and Q2 are the amounts of gas flowing at both ends of the pipe, x ¼ 0 and x ¼ L, respectively. The quantity qL (¼Qp) is the total amount of gas released in the pipe. These are reduced further as Q 1 þ Q2 ¼ ðP1  P2 ÞC; Q2  Q1 ¼ qL ¼ Qp : 2

(5)

The gas flow in the pipe can be divided into two categories, i.e. (1) a constant gas flow (Q1 þ Q2)/2, which streams through the pipe without any change, and (2) a gas flow Qp which is released in the pipe and leaves at both ends. The former generates a pressure difference across the pipe but the latter does not (cf. Fig. 2). Now we normalize the constant gas flow of category (1) by Qp/2 and introduce x as the normalized constant gas flow. Using x, the quantities Q1 and Q2 can be expressed as Q1 ¼ ðx  1Þ

Qp Qp ; Q2 ¼ ðx þ 1Þ ; 2 2

(6)

and the first formula in Eq. (5) becomes Qp x ¼ ðP1  P2 ÞC: 2

(7)

The sign of the flow indicates its direction, i.e., if the gas flows toward þx, the flow is positive, and vice versa.

Figure 2. Gas flow in a pipe.

Chapter 2 Applications of the resistor-network simulation method

3. Kirchhoff's law in a network of vacuum components The relation given by Eq. (7) corresponds to the one between electric current and voltage difference across a resistor. Therefore, we can write down two equations of Kirchhoff’s law as follows: (1) The sum of pressure differences across network components along an arbitrary closed loop is always zero or, in the case that the loop includes the reference point, equal to the difference between the Pu values of two vacuum pumps in the loop, i.e., X zij Qij ðÞ ¼ 0 ðor ¼ DPu Þ; Cij loop

(8)

where the summation should be done for all components in the loop in a given direction. (2) At any connection point but the reference point, the total amount of gas flowing into this point is always zero, i.e.,  1 X
(9) 1  zij Qij þ Qii ¼ 0; 2 j where Qii is the amount of gas generated at the ith point. In some cases, a vacuum chamber will be used as a connection point. Pi is the pressure at the ith point, and Cij(¼cij/Lij) and Qij(¼qijLij) are the pipe conductance and the amount of gas generation between the ith and jth points, respectively. If i