Reliability of sensores based on nanowire networks

Reliability of sensores based on nanowire networks

Journal: IIE Transactions

Manuscript ID: UIIE-2643

Manuscript Type: Regular Paper

Focus Issue: Quality and Reliability Engineering

Keyword:

Auto Industry < Applications, Manufacturing < Applications, Reliability Engineering < QUALITY AND RELIABILITY, Reliability Prediction < Reliability Engineering < QUALITY AND RELIABILITY, Statistics and Applied Probability < QUALITY AND RELIABILITY

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Reliability of Sensors Based on Nanowire Networks

Nader Ebrahimi and Kristin McCulloughDivision of Statistics, Northern Illinois University, DeKalb, IL 60115

Zhili XiaoDepartment of Physics, Northern Illinois University, DeKalb, IL 60115

Abstract

Nanowires have a great potential in many industrial applications, including elec-

tronics and sensors. Palladium nanowire network based hydrogen sensors have been

found to outperform their counterparts which consist of an individual nanowire or pal-

ladium thin or thick films. However, the reliability issues still need to be addressed. In

this paper, we consider hydrogen sensors based on a nanowire network with a square

lattice structure. We provide a general model for describing the reliability behavior

of this network. Our findings reveal that one can improve the reliability function by

considering a network of nanowires rather than a single nanowire. Among many other

applications, our results can also be used to assess the reliability of any nanosystem/

nanodevice where our proposed model is a reasonable choice. What distinguishes our

work from related work are the unique difficulties that the nano components introduce

to the evaluation of reliability and the way we define reliability over cycles of hydrogen

gas.

Keywords: percolation, site percolation, bond percolation, Bernoulli random variable,

reliability function.

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1 Introduction

Sensors, particularly gas detecting sensors, made entirely of nanocomponents are becoming

the topic of more and more research as recent advances in nanotechnology have provided

the opportunity to significantly enhance their performance. Some current applications of

gas sensors include carbon monoxide detectors, breath analyzers, controls for microwave

ovens, anesthesia monitors, respirators, car ventilation systems, environmental monitors,

etc. It is common for gas sensors to be made from metal or metal oxides, which detect gas

through a change in resistivity in the presence of the gas.

The sensor’s performance is directly related to its surface area, since the electrical

conductivity of the metal or metal oxide changes as the gas is absorbed onto its surface.

Because an inverse relationship exists between surface area and particle size, incorporating

nanocomponents, with an extremely high surface area to volume ratio, is highly desirable.

The availability of nanoscale metal or metal oxides, due to newly developed nanofabrication

techniques, offers tremendous opportunities for manufacturers to increase the sensitivity

of their sensors. However, reliability issues still need to be addressed. Current commercial

gas nanosensors made of thin or thick, conductive films of metal, are effective yet lack

stability. For example, thin film sensors made of palladium (Pd), used to detect hydrogen

gas, can buckle during the desorption process, rendering them useless. Another problem is

that oxide films operate at a high temperature (150◦ − 600◦ C) which must be optimized

for both the metal oxide and the gas. This means that the sensors need to be internally

heated, which can degrade the sensor over time. So researchers are moving away from

using the film design and focusing on nanowires.

Due to their large surface area to volume ratio and available space for making electrical

contacts, individual nanowires have been widely considered as interconnects or sensing

elements in nanodevices (Dai, Wong, and Lieber 1996; Hernandez et al. 2007; Jeon et

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al 2009; Liu et al. 2004; Yang, Taggart, and Penner 2009). For example, hydrogen gas

sensors based on individual palladium nanowires show much improved performance in

speed, sensitivity, and ultra-low power consumption in comparison to their counterparts

based on palladium thin films, and thinner nanowires perform better than thick nanowires

(Jeon et al. 2009; Yang, Taggart, and Penner 2009). Hydrogen gas is very important as it is

used in scientific research and industry extensively, notably in glass and steel manufacturing

as well as in the refining of petroleum products. It is hoped to be the next clean energy

carrier for vehicles. Hydrogen gas is also very volatile, and the quick and accurate response

of a hydrogen sensor is necessary for the safe usage of all hydrogen-based applications.

Palladium is a clear choice for a hydrogen gas sensor due to the natural interaction of the

two elements. At room temperature palladium will absorb vast amounts of hydrogen.

The utilization of an individual wire faces challenges in nanofabrication, manipulation,

and achieving ultra-small transverse dimensions. Furthermore, a nanowire can have a

high probability of being broken in an application environment, as demonstrated by the

destruction of Pd nanowires after a few cyclings of exposure to hydrogen and nitrogen

gasses (Yang, Taggart, and Penner 2009). Thus, the reliability of a nanosensor based on

an individual nanowire can be an application issue. Recently Zeng et al. presented a

new fabrication approach that takes advantage of individual nanowires for application in

nanodevices while eliminating their nanofabriciation obstacles: they utilized commercially

available nanoporous membranes to form networks of ultra-small palladium nanowires and

used them as sensing elements to hydrogen gas (“Hydrogen Gas Sensing” 2011). Figure 1

shows a top-view scanning electron microscopy (SEM) image of a resulting network. The

nanowire network based hydrogen sensors outperform their counterparts consisting of an

individual nanowire in both sensitivity and response time. This new type of sensor also

seems to be robust and does not breakdown in response to cyclings of exposure to hydrogen

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and nitrogen gases (Zeng et al., “Hydrogen Gas Sensing” 2011; Zeng et al., “Networks of

Ultrasmall” 2011). In this paper we focus on the reliability of this nanosensor, modeling and

testing it through computer simulations as well as analytically. The notions and results

developed in this paper are applicable to other nanodevices/ nanosystems consisting of

nanocomponents that match the structure of this nanosensor. Throughout this paper we

will use the terms nanosensor, nanodevice, and nanosystem interchangeably.

There has been a great deal of research on network reliability in both the opera-

tions research and graph theory communities. See Elperin, Gertsbakh, and Lomonosov

(1991); Ball, Colbourn, and Provan (1992); Meyer (1992); Michelena and Papalambros

(1994); Muppala, Ciardo, and Trivedi (1994); Ma and Trivedi (1999); Kuo and Zuo (2003);

Leskovec, Kleinberg, and Faloutsos (2005); Zuo, Tian, and Huang (2007); and Wainwright

and Jordan (2008) for more details. What distinguishes our work from related work is

that it is impossible to know the exact structure of the network, the exact number of

components, and the exact number or location of broken components. Another important

distinction is the way that we the define reliability under these unique conditions on each

cycle. We know that each cycle of hydrogen gas will potentially damage the network, and

here we are interested in the number of times that the system can withstand this dam-

age. So for our setup reliability is defined in terms of several checks for percolation on the

network where each check occurs after a cycle of hydrogen gas.

This paper is organized as follows. In Section 2, we propose a general model for the

structure of this nanosensor and define its reliability function based on “site percolation.”

In Section 3, we consider a specific case of the proposed model and use it to obtain the

reliability function analytically. In Section 4, we give bounds for the reliability function

as well as the average number of cycles (average lifetime) based on the general model. In

Section 5, we provide an alternate way of describing the reliability function which is based

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Figure 1: SEM micrograph of network of Pd nanowires with a nominal thickness of 7 nm

on “bond percolation.” In Section 6, we propose an algorithm to numerically calculate the

reliability function as well as the expected lifetime for our general model given in Section

2. In Section 7, we obtain an “optimal network” which is based on optimizing a certain

objective function. Finally, concluding remarks are given in Section 8.

2 Network description (General Model)

Our network is composed of ultra-small palladium nanowires where each nanowire is a com-

ponent of the network. Alternately, each bond between nanowires could be a component

of the network. Our network’s purpose is to detect hydrogen gas by changing resistivity

in its presence. The resistivity is measured by sending an electrical current through the

network. The electrical current moves in one direction. Basically the interior of a metal

wire is full of unattached electrons, and when an electrical force is applied on the wire’s

opposite ends, the unattached electrons rush in the direction of the force. Thus the created

electrical current will move forwards, up, or down but not backwards through the network

in order to follow the direction of the applied force.

During testing, the network is exposed to hydrogen gas, premixed with nitrogen gas to a

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desired concentration, then the hydrogen gas is flushed out of the network by pure nitrogen

gas. This is referred to as a cycle of hydrogen gas (not to be confused with the normal

definition of a cycle in graph theory). Now when purging nitrogen gas is introduced, the

hydrogen goes through a desorption process, during which there is a chance that a nanowire

may be damaged. See Zeng, et al. for more details (Hydrogen Gas Sensing 2011). Each

cycle the electrical current may have to take a different path depending on which nanowires

break. It is not possible to tell how many wires break or their location without using a high

powered microscope such as a SEM. This would of course be costly and inefficient, and the

network would be destroyed in the process. Therefore, during each cycle our network has

the possibility of becoming damaged, and this damage is irreversible. As described above,

we have neither a directed nor an undirected network in terms of standard graph theory.

Some of the nanowires are directed. However for some nanowires the electrical current may

travel one direction through them during a given cycle and the opposite way during the

next cycle.

We measure the reliability of the network in terms of percolation. A percolation process

is the mathematical model of the randomness of the spread of fluid through a medium

where the terms “fluid” and “medium” can be interpreted broadly. For our network of

nanowires we are referring to the flow of an electrical current through the nanowires.

Through percolation theory we are able to relate the probabilistic and topological properties

of the behavior of connected open clusters in a random graph. The basic mathematical

construction of such a nanosystem of nanowires would be as follows. Take a countably

infinite lattice, say L, with a vertex set V and an edge set E . Two types of percolation, bondand site, need to be considered for such a nanosystem. For bond percolation we assume the

vertices are independently either open or closed and that the current can only flow through

edges that are connected by two open vertices. For site percolation we assume directly that

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all edges are independently either open or closed. Every site percolation problem on lattice

L can be expressed as a bond percolation problem on the covering lattice of L. This is

intuitive since each edge can be expressed in terms of two vertices. The converse however

is not true. For more information see Rue and Held (2005).

For our purposes, we concentrate on site percolation on a square lattice of size n×m (m

rows and n− 1 columns of horizontal wires connected by vertical wires at the ends). The

square lattice is assumed to be a good approximation for the structure of our nanosensor.

That is, we simplify the nanosensor to be a square lattice structure consisting of n · mnanobonds and n∗ = n · (m − 1) + (n − 1) · m nanowires. Here we let the nanowires be

treated as nanocomponents of the nanosystem or nanodevice. A Bernoulli random variable

is applied to each nanowire to determine if it is open (functioning) or closed (broken). Let

Xn×m be the random variable that represents the number of cycles of hydrogen exposure

that the nanosystem will last through before it no longer percolates. We refer to Xn×m

as the “lifetime of the nanosystem.” Also, Rn×m(x) = P (Xn×m > x) for x = 0, 1, 2, . . .

and Rn×m(0) = 1. Rn×m(x) and E(Xn×m) are referred to as the reliability function and

the expected lifetime/ expected number of cycles, respectively. It should be noted that

contrary to many other classical situations, the lifetime here is a discrete random variable

which takes values 1, 2, . . . . For more details about classical situations see Lawless (2003).

For our setup percolation occurs from left to right, and recall that the percolation path of

the electrical current can never travel backwards. Figure 2 shows an example of a 4 × 4

network for site percolation, and figure 3 shows an example of a 4 × 4 network for bond

percolation. Closed wires are not drawn, and closed vertices are marked with an x. Both

examples percolate.

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Figure 2: 4× 4 Site percolation

x

x

x

x

x

Figure 3: 4× 4 Bond percolation

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Figure 4: 2× 5 Network

3 2×m Networks

To begin we examine a square lattice of size 2 × m. So the network would consist of m

horizontal wires stacked in a column, connected by vertical wires on each side. In other

words it would look like a ladder. Figure 4 is an example of a 2×5 network which percolates.

In this section percolation refers to site percolation.

Let m = 1. Then we have a single horizontal wire. It is clear that the network will

percolate until the wire breaks. So X2×1 follows the geometric distributions, and thus we

know R2×1(x) = px and E(X2×1) =1

1−p , where p is the probability of the wire not breaking

and 0 ≤ p < 1.

Now suppose that the probability of a wire not breaking, p, remains the same from wire

to wire. Also assume that wires break independently of each other. It is important to note

that even though we assume independence between nanowires, we have dependence from

cycle to cycle. That is, the behavior of the nanosensor in a given cycle strongly depends

on the state of the nanosensor on the previous cycles. Then the following result gives the

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reliability function of a of 2×m network, R2×m(x), for m ≥ 2.

Theorem 1: The reliability function of a 2×m network for m ≥ 2 is:

R2×m(x) = P (X2×m > x) = 1− (1− px)m. (3.1)

For proof see the appendix. In Theorem 1, if p → 1, then P (X2×m = ∞) = 1. That is, the

network never fails.

Our next result gives E(X2×m) (the expected lifetime of a 2 ×m network) as well as

the variance of X2×m.

Theorem 2:

(a)

E(X2×m) =m∑k=1

(−1)k+1

(m

k

)1

1− pk, (3.2)

(b)

V (X2×m) = 2

m∑k=1

(−1)k+1

(m

k

)pk

(1− pk)2+

m∑k=1

(−1)k+1

(m

k

)1

1− pk

− [

m∑k=1

(−1)k+1

(m

k

)1

1− pk]2.

(3.3)

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See the appendix for the proof.

Due to the alternating form of E(X2×m) and V (X2×m) in (3.2) and (3.3) it is difficult

to evaluate them for large values of m, say m > 50. For those situations it is easier to use

an approximation. If we take λ = −log(p), then we can write

E(X2×m) =∑∞

x=0 P (X2×m > x) =∑∞

x=0 1− (1− px)m =∑∞

x=0 1− (1− e−xλ)m.

Using the inequality

∫∞0 (1− (1− e−xλ)m) dx <

∑∞x=0 1− (1− e−xλ)m < 1 +

∫∞0 (1− (1− e−xλ)m) dx,

we then have

1λ

∑mk=1

1k <

∑∞x=0 1− (1− e−xλ)m < 1 + 1

λ

∑mk=1

1k .

Therefore one choice for an approximation of E(X2×m) is

E(X2×m) ≈ 1

2+

1

λ

m∑k=1

1

k. (3.4)

To see how good the approximation (3.4) is, the percolation was simulated 10,000 times

for different values of m and p, and the expected value was approximated using the mean

of the simulations. A general algorithm to compute E(Xn×m) is described in section 6.

The approximation in (3.4) was then compared to our simulated results. Figure five shows

a graph of the log of the expected value of the number of percolations for different values

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0.0 0.2 0.4 0.6 0.8 1.0

01

23

45

6

Probability of a Wire Breaking

Log

of E

xpec

ted

Num

ber o

f Per

cola

tions

2x22x32x5

2x102x252x50

2x2002x500

2x2000

Figure 5: log[E(X2×m)] for site percolation calculated using our algorithm from section 6for various values of m

of m and p. Throughout this paper log stands for the natural logarithm. The results are

graphed in terms of the probability of a wire breaking, 1− p, and the log of the expected

number of times the network will percolate before it no longer functions, log[E(X2×m)].

For example, if p = .90 (so the probability of a wire breaking is .10) and m = 5, then from

figure 5 the log[E(X2×5)] = 3.1088. Thus E(X2×5) is 22.3949. From the equation (3.4),

E(X2×5) ≈ 22.30 which is very close to the number we obtained through simulation. We

calculated this for different values of m and p, and we conclude that the approximation in

(3.4) works quite well.

To approximate the variance it is clear that

E(X22×m) = 2

∑∞x=0 xP (X2×m > x)+E(X2×m). Now, 2

∑∞x=0 xP (X2×m > x) = 2

∑∞x=0 x(1−

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(1− e−xλ)m).

Using the inequality

∫∞0 2x(1− (1− e−xλ)m) dx < 2

∑∞x=0 x(1− (1− e−xλ)m) < 1+

∫∞0 2x(1− (1− e−xλ)m) dx,

one can show that∫∞0 2x(1− (1− e−xλ)m) dx ≈ 1

λ2 [∑m

k=11k2

+ (∑m

k=11k )

2].

Approximating 2∑∞

x=0 x(1− (1− e−xλ)m) ≈ 12 + 1

λ2 [∑m

k=11k2

+ (∑m

k=11k )

2],

we have E(X22×m) ≈ 1 + 1

λ2

∑mk=1

1k2

+ 1λ2 (

∑mk=1

1k )

2 + 1λ

∑mk=1

1k .

Then E(X22×m)−E2(X2×m) ≈ 1+ 1

λ2

∑mk=1

1k2+ 1

λ2 (∑m

k=11k )

2+ 1λ

∑mk=1

1k−(12+

1λ

∑mk=1

1k )

2.

Thus an approximation for the variance is

V (X2×m) ≈ 3

4+

1

λ2

m∑k=1

1

k2. (3.5)

The approximation (3.5) has also been compared to the simulated results, and we can

conclude that it is a good approximation.

4 Bounds for Relaibility Function and Expected Lifetime

Under General Model

In this part we obtain several results related to Rn×m(x) and E(Xn×m) under our proposed

model including bonds for both quantities. Throughout this section and the next section it

is assumed that p remains the same from wire to wire and 0 ≤ p < 1. Also, a wire breaks

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independently from the other wires, but we assume dependence from cycle to cycle. First

let m = 1. Then we have the following theorem.

Theorem 3:

(a)

Rn×1(x) = P (Xn×1 > x) = p(n−1)x, (4.1)

(b)

E(Xn×1) =1

1− pn−1, (4.2)

(c)

V (Xn×1) =pn−1

(1− pn−1)2. (4.3)

See the appendix for the proof.

Now let m ≥ 2. Percolation in this case gets quite complicated due to the fact that the

possible percolation paths are dependent on which wires break, and the wires are breaking

randomly. However, we can construct lower bonds for Rn×m(x) = P (Xn×m > x) and

E(Xn×m). In many practical situations it is desirable or necessary to check whether or not

the reliability of a network meets a given specification when p is known. If a lower bound

is already known to meet or exceed that specification, then one knows for sure that the

network meets the specification.

Consider a simpler sub-network where paths are independent. Remove all vertical con-

nections that are not part of the boundary of the network. Now, as in the 2 × m case,

percolation must occur along one of the independent rows of horizontal wires, which gives

the following result.

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Theorem 4: (a)

(a)

Rn×m(x) ≥ 1− (1− p(n−1)x)m, (4.4)

(b)

E(Xn×m) ≥m∑k=1

(−1)k+1

(m

k

)1

1− p(n−1)k. (4.5)

The proofs are similar to those of Theorem 1 and Theorem 2(a) and are omitted. It

should be noted that when the sub-network no longer percolates, the original network may

still be functioning. This occurs since in the original network the percolation path can

move between the independent rows in the sub-network. For example, if p = .90, n = 10,

and m = 10, then using (4.4) R10×10(x) ≥ 1− (1− (.90)9x)10. We should note that using

(4.4), if p → 1, then P (Xn×m = ∞) = 1. That is, the network never fails.

Figures 6-8 are graphs of the log results for the lower bound of E(Xn×m), the logarithm

of the expected lifetime of a n × m network, calculated using (4.5) for various n × m

networks. One clear pattern that can be seen in figures 6-8 is that as n gets larger, the

network performs worse.

5 Bond Percolation

For bond percolation, the Bernoulli random variable is assigned to the vertices. A wire

would be closed if both of it’s vertices were closed. Therefore, the probability of a wire

being closed is 2q − q2, and the probability of a wire being open is 1 − (2q − q2). If we

replace p with 1 − (2q − q2) in (3.1)-(3.3), then they will hold true for bond percolation.

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0.0 0.2 0.4 0.6 0.8 1.0

01

23

45


Log

of E

xpec

ted

Num

ber

of P

erco

latio

ns

2x6

3x9

5x15

10x30

25x75

Figure 6: Lower bound for log[E(Xn×m)] for site percolation calculated for networks witha 1:3 ratio for the dimensions n : m

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4


Log

of E

xpec

ted

Num

ber

of P

erco

latio

ns

4x3

12x9

20x15

40x30

60x45

Figure 7: Lower bound for log[E(Xn×m)] for site percolation calculated for networks witha 4:3 ratio for the dimensions n : m

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0.0 0.2 0.4 0.6 0.8 1.0

01

23

4


Log

of E

xpec

ted

Num

ber

of P

erco

latio

ns

5x10

10x5

50x20

20x50

Figure 8: Lower bound for log[E(Xn×m)] for site percolation calculated for networks n×mand m× n

Similarly if we replace p with 1 − (2q − q2) in (3.4) and (3.5) they will provide us with a

good approximation of E(X2×m) and V (X2×m). Also from Theorem 4, one can obtain a

bound for E(Xn×m) by simply replacing p with 1− (2q − q2).

6 Algorithm/ Simulations

As we have mentioned in previous sections, it is very hard to obtain the reliability function

Rn×m(x) = P (Xn×m > x) and the expected value of Xn×m when both n and m are larger

then 2. In this section we provide an algorithm to find them.

The network L as described before is set in the cartesian plane, and the location of

the lower left lattice point (nanobond/ vertex) is placed at (1, 1). So, there will be n ·mvertices, (x, y) ∈ Z2. The edges will be the horizontal and vertical line segments with

unit length joining (x, y) to (x± 1, y) and (x, y ± 1) (Rue and Held, 2005). There will be

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n∗ = n · (m− 1)+ (n− 1) ·m edges in total representing the nanowires. The algorithm and

how to implement it are described below.

(a) A sequence of iid Bernoulli random variables, say U1, U2, . . . Un∗ are generated and

assigned to the nanowires. Since Ui ∼ Bernoulli(p),

Ui =

⎧⎪⎨⎪⎩

1 if open

0 if closed

(b) Any nanowires that break are removed from consideration, and the remaining wires

are sorted into open directional clusters. We define an open directional cluster to be

any connected component of L in which all the edges are open. But the connected

component must be formed in a manner which obeys the movement of the electrical

current.

(c) The clusters are tested for percolation. If no clusters percolate, then the algorithm is

stopped. The result is recorded, and the next simulation starts.

(d) If one of the clusters percolate, then the remaining open wires are assigned new

Bernoulli random variables and steps (b) and (c) are repeated.

(e) The algorithm is repeated several times, and the average of the simulations is used to

approximate Rn×m(x) and E(Xn×m). The results from the simulations are stored in

a vector and can also be used to approximate the variance or performs other analyses.

As an example, percolation on the n × n network, n > 2 was simulated 1,000 times

for each combination of n and p. The mean and variance of the 1,000 simulations were

used to approximate E(Xn×n) and V (Xn×n) for their respective values of n and p. Figure

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0.0 0.2 0.4 0.6 0.8 1.0

01

23

4


Log

of E

xpec

ted

Num

ber o

f Per

cola

tions

3x3

5x510x1025x25

Figure 9: log[E(Xn×n)] for site percolation calculated using our algorithm from section 6for various values of n

nine shows the graph of the log of the resulting expected value for various n× n networks

over all values of p. One can observe that as n gets larger, the n× n network does worse,

so it was not necessary to simulate very large networks. Using the simulation results, we

can also construct the mass function of Xn×n. Figure 10 shows the reliability function of

X10×10 when p = .90 using 1,000 simulations.

A separate algorithm, similar to the first, was used to model the bond percolation and

was used to simulate all of the previously discussed networks. The estimates were verified

to perform well. The results for bond percolation were similar to those for site percolation,

but bond percolation on the network resulted in lower values of E(Xn×m). Figures 11-14

summarize the findings. Again the log of the expected value is graphed.

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2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

Number of Percolations

Rel

iabi

lity

Figure 10: R10×10(x) = P (X10×10 > x), calculated using our algorithm from section 6 forsite percolation with p = .90

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0.0 0.2 0.4 0.6 0.8 1.0

01

23

45

Probability of a Bond Breaking

Log

of E

xpec

ted

Num

ber o

f Per

cola

tions

2x22x32x5

2x10

2x252x50

2x2002x500

2x2000

Figure 11: log[E(X2×m)] for bond percolation calculated using our algorithm from section6 for various values of m

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4


Log

of E

xpec

ted

Num

ber o

f Per

cola

tions

3x3

5x5

10x10

25x25

Figure 12: Lower bound for log[E(Xn×n)] for bond percolation calculated for various n×nnetworks

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0.0 0.2 0.4 0.6 0.8 1.0

01

23

4


Log

of E

xpec

ted

Num

ber o

f Per

cola

tions

2x4

3x6

5x10

10x20

25x50

Figure 13: Lower bound for log[E(Xn×m)] for bond percolation calculated for networkswith a 1:2 ratio for the dimensions n : m

0.00 0.05 0.10 0.15 0.20 0.25

0.0

0.5

1.0

1.5

2.0


Log

of E

xpec

ted

Num

ber o

f Per

cola

tions

25x2

25x10

25x2025x2525x3025x4025x50

Figure 14: Lower bound for log[E(X25×m)] for bond percolation calculated for variousvalues of m

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7 An Optimal Network

From Sections 3-5 it is clear that the 2×m network is best in terms of acheiving a higher

expected lifetime, E(Xn×m). It is also clear that by increasing m, E(X2×m) also increases.

Unfortunately by increasing m, we also increase V (X2×m), which is not desirable. That is,

there are competing objectives, and we need to compromise between a higher mean and a

lower variance. For that we define the objective function

F = α1E(X2×m)− α2V (X2×m), (7.1)

where α1 and α2 are weights assigned to E(X2×m) and V (X2×m). In our objective func-

tion F , α1 units of E(X2×m) are being traded for α2 units of V (X2×m). Now to come up

with an optimal network, the goal is to find the value of m which maximizes the objective

function F . Using equations (3.4) and (3.5),

F = α1E(X2×m)− α2V (X2×m) = α1[12 + 1

λ

∑mk=1

1k ]− α2[

34 + 1

λ2

∑mk=1

1k2],

where λ = −log(p). Thus maximizing our objective function with respect to m when

α1 = 1 and α2 =12 is equivalent to maximizing 1

λ

∑mk=1

1k − 1

2λ2

∑mk=1

1k2

with respect to m.

For example if p = .90, then the optimal m is between 5 and 6. Thus, the optimal network

is a 2× 5 or 2× 6.

8 Concluding Remarks

We started with a n×m network, or a n×m nanosystem with nanowires as its nanocom-

ponents. For our proposed network we described situations where it fails and defined the

random variable Xn×m that represents the number of times that the network can be ex-

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posed to hydrogen before it will fail to percolate. We referred to Xn×m as the lifetime

of the network. Although our original development was motivated by the hydrogen gas

sensing network of ultrasmall palladium nanowires formed on a filtration membrane, the

proposed framework is applicable to any nanosystem or nanodevice of size n × m with a

square lattice structure.

For 2×m and n×1 networks we obtained the reliability functions R2×m(x) = P (X2×m >

x) and Rn×1(x) = P (Xn×1 > x) as well as E(X2×m) and E(Xn×1) analytically. We also

derived bounds for Rn×m(x) = P (Xn×m > x) and E(Xn×m). In general it is hard to

compute the reliability function Rn×m(x) and E(Xn×m) when n > 2 and m ≥ 2. We

presented an algorithm that can be used to obtain both quantities. We reveal that the

reliability can indeed be significantly improved if a nanowire network with some specific

structure is used to replace the individual nanowire.

9 Acknowledgments

This research was partially supported by the grant from the National Security Agency under

the grant number H98230-11-1-0138.1. Zhili Xiao also acknowledges financial support by

the Department of Energy (DOE) Grant No. DE-FG02-06ER46334 (nanowire network

fabrication). The United States government is authorized to reproduce and distribute

reprints notwithstanding any copyright notation herein.

References

[1] Ball, M.O., Colbourn, C.J., & Provan, J.S. (1992). Network Reliability (ISR; TR 1992-

74). College Park, MD: Digital Repository at the University of Maryland.

24

Page 24 of 29



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

[2] Dai, H.J., Wong, E.W., & Lieber, C.M. (1996). Probing electrical transport in nano-

materials: Conductivity of individual carbon nanotubes. Science, 272, 523-526.

[3] Elperin, T., Gertsbakh, I., Lomonosov, M. (1991). Estimation of Network Reliability

Using Graph Evolution Models. IEEE Transactions on Reliability, 40, 572-581.

[4] Hernandez, Sandra C., Chaudhuri, Debanshu, Chen, Wilfred, Myung, Nosand V., &

Mulchandani, Ashok. (2007). Single Polypyrrole Nanowire Ammonia Gas Sensor. Elec-

troanalysis, 19, 2125-2130.

[5] Jeon, Kye Jin, Lee, Jun Min, Lee, Eunsongyi, & Lee, Wooyoung. (2009). Individual Pd

nanowire hydrogen sensors fabricated by electron-beam lithography. Nanotechnology,

20, 135502.

[6] Kuo, Way & Zuo, Ming J. (2003). Optimal Reliability Modeling: Principles and Appli-

cations. Hoboken, NJ: John Wiley and Sons.

[7] Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data, second edition.

Hoboken, NJ: John Wiley and Sons.

[8] Leskovec, Jure, Kleinberg, Jon, & Faloutsos, Christos. (August 2005). Graphs over time:

densification laws, shrinking diameters and possible explanations. KDD ’05 Proceedings

of the eleventh ACM SIGKDD international conference on Knowledge discovery in data

mining, Chicago, IL.

[9] Liu, Haiqing, Kameoka, Jun, Czaplewski, David A., & Craighead, H.G. (2004). Poly-

metric Nanowire Chemical Sensor. Nano Letters, 4, 671-675.

[10] Ma, Y. & Trivedi, K.S. (1999). An algorithm for reliability analysis of phased-mission

systems. Reliability Engineering and System Safety, 66, 157-170.

25

Page 25 of 29



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

[11] Meyer, John F. (1992). Performability: a retrospective and some pointers to the future.

Performance Evaluation, 14, 139-156.

[12] Michelena, Nestor F. & Papalambros, Panos Y. (1994). A Network Reliability Ap-

proach to Optimal Decomposition of Design Problems. Advances in Design Automation,

2, 195-204.

[13] Muppala, J.K., Ciardo, G, & Trivedi, K.S. (1994). Stochastic Reward Nets for Relia-

bility Prediction. Communications in Reliability Maintainability and Serviceability, 1,

9-20.

[14] Rue, Harvard & Held, Leonhard. (2005). Gaussian Markov Random Fields Theory

and Applications. Boca Raton, Fl: Chapman & Hall/CRC.

[15] Wainwright, M.J. & Jordan, M.I. (2008). Graphical models, exponential families, and

variational inference. Foundations and Trends in Machine Learning, Vol. 1, Numbers

1–2, 1-305.

[16] Yang, Fan, Taggart, David K., & Penner, Reginald M. (2009). Fast, Sensitive Hy-

drogen Gas Detection Using Single Palladium Nanowires That Resist Fracture. Nano

Letters, 9, 2177-2182.

[17] Zeng, X.Q., Latimer, M.L., Xiao, Z.L., Panuganti, S., Welp, U., Kwok, W.K., & Xu,

T. (2011) Hydrogen Gas Sensing with Networks of Ultrasmall Palladium Nanowires

Formed on Filtration Membranes. Nano Letters, 11, 262.

[18] Zeng, X.Q., Wang, Y.L., Deng, H., Latimer, M.L., Xiao, Z.L., Pearson, J., Xu, T.,

Wang, H.H., Welp, U., Crabtree, G.W., & Kwok, W.K. (2011). Networks of Ultrasmall

Pd/Cr Nanowires as High Performance Hydrogen Sensors. ACS Nano. Manuscript sub-

mitted for publication.

26

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123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

[19] Zuo, M.J., Tian, Z., & Huang, H.Z. (2007). An efficient method for reliability eval-

uation of multistate networks given all minimal path vectors. IIE Transactions, 39,

811-817.

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Appendix

� Proof of Theorem 1

We use induction. Let m=2. We know that

P (X2×2 = x) = p2x−2q2 + 2p2x−4qpq + 2p2x−6qppq + · · ·+ 2px−2qpq

= p2x−2q2+2px−1q2[1+p+ · · ·+px−2] = p2x−2q2+2px−1q2[1−px−1

1−p ] = p2x−2q2+2px−1q[1−px−1] = p2x−2q2 + 2px−1q − 2p2(x−1)q = 2px−1q − (1− p2)p2x−2.

Now, P (X2×2 ≤ x) =∑x

k=1[2pk−1q − (1 − p2)p2k−2] = 2q[1−px

q ] − (1 − p2)[1−p2x

1−p2]

= 2− 2px − 1 + p2x = (1− px)2.

Now assume it is true for m− 1, then we will prove it for m. It is clear that

P (X2×m = x) = P (X2×(m−1) ≤ x)P (X2×1 = x) + P (X2×1 ≤ x)P (X2×(m−1) = x)

= (1− px)m−1px−1q+ (1− px−1)[(1− px)m−1 − (1− px−1)m−1] = (1− px)m − (1− px−1)m.

Therefore, P (X2×m ≤ x) = (1− px)m, and this completes the proof.


(a) E(X2×m) =∑∞

x=1 xP (X2×m = x)

=∑∞

x=1 x[(1−px)m−(1−px−1)m] =∑∞

x=1 x[∑m

k=0(−1)k(mk

)pkx−∑m

k=0(−1)k(mk

)pk(x−1)]

=∑∞

x=1 x[∑m

k=1(−1)k(mk

)pkx−k(pk − 1)] =

∑mk=1(−1)k(pk − 1)

(mk

)[∑∞

x=1 xpk(x−1)] =

∑mk=1(−1)k+1(1− pk)

(mk

)[ 1(1−pk)2

] =∑m

k=1(−1)k+1(mk

)1

1−pk.

(b) V (X2×m) = E(X22×m) − E2(X2×m) = E(X2×m(X2×m − 1)) + E(X2×m) − E2(X2×m)

= 2∑m

k=1(−1)k+1(mk

) pk

(1−pk)2+

∑mk=1(−1)k+1

(mk

)1

1−pk− [

∑mk=1(−1)k+1

(mk

)1

1−pk]2.

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(a) It is clear that P (Xn×1 = x) = (pn−1)x−1[(n−11

)pn−2q +

(n−12

)pn−3q2 + · · · + qn−1]

= (pn−1)x−1∑n−1

k=1

(n−1k

)pn−1−kqk (pn−1)x−1(1− pn−1) = p(n−1)(x−1) − p(n−1)x.

Now P (Xn×1 ≤ x) =∑x

k=1 P (Xn×1 = k) =∑x

k=1 p(n−1)(x−1) − p(n−1)x = 1−p(n−1)x

1−pn−1 −pn−1(1−p(n−1)x)

1−pn−1 = (1−pn−1)(1−p(n−1)x)1−pn−1 1− p(n−1)x.

(b) Using part (a), it is clear that Xn×1 has the geometric distribution with the parameter

1− pn−1. Thus, E(Xn×1) =1

1−pn−1 .

(c) This proof again comes from the fact that Xn×1 has the geometric distribution with

the parameter 1− pn−1.

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