ChE 553 Lecture 9

Statistical Mechanics Of Adsorption

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Objective

• Start to discuss thermodynamics of surface phase transitions

• Landau analysis – what phases do we see

• Solve one dimensional surface – what is phase behavior like

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Background• So far we discussed simple adsorption

with no interactions between adsorbed species, complete mobility– Results in Langmuir behavior

• At saturation densities like liquid densities – Molecules interact

• Diffusion rates not always fast– Leads to defects in adsorbed layer

• Predicts no ordering of adsorbed layer

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Real Surface systems Are Different

• Usually a series of phases that vary with coverage and temperature

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P(2x2)

C(2x2)

(2x1)

(1x2)

i

1b

1a1a

2 2

3

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4b

4a 4a

4b

5 5

1b 22

3

4b

4a4a

4b

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i

1b

1a1a

2 2

3

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4b

4a 4a

4b

5 5

1b 22

3

4b

4a4a

4b

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i

1b

1a1a

2 2

3

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4b

4a 4a

4b

5 5

1b 22

3

4b

4a4a

4b

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i

1b

1a1a

2 2

3

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4b

4a 4a

4b

5 5

1b 22

3

4b

4a4a

4b

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Typical Surface phases

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Figure 4.22 The absorption of molecules in a P(2x2), C(2x2), (2x1) overlayer. The dark circles represent sites, the red circles represent adsorption on the sites.

Why Do These Phases Arise?

• Forces Between Adsorbates keep them apart– Direct forces (Van der waals repulsions)– Indirect forces (Adsorbate attracts local

electrons, depleting nearby regions for adsorption

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Today

• Consider the effects of pair-wise interactions on the behavior of adsorbed layer– Qualitative features: Landau analysis– Quantifying results, analytically and with

monte carlo

• Try to predict surface phase behavior

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Landau Analysis

• Attempt to predict what phases are produced as gas adsorbs on solid surfaces

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P(2x2)

C(2x2)

(2x1)

(1x2)

i

1b

1a1a

2 2

3

33

4b

4a 4a

4b

5 5

1b 22

3

4b

4a4a

4b

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i

1b

1a1a

2 2

3

33

4b

4a 4a

4b

5 5

1b 22

3

4b

4a4a

4b

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i

1b

1a1a

2 2

3

33

4b

4a 4a

4b

5 5

1b 22

3

4b

4a4a

4b

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i

1b

1a1a

2 2

3

33

4b

4a 4a

4b

5 5

1b 22

3

4b

4a4a

4b

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Example:When will Each Of The Following Phases Form?

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Figure 4.22 The absorption of molecules in a P(2x2), C(2x2), (2x1) overlayer. The dark circles represent sites, the red circles represent adsorption on the sites.

Steps in Landau Analysis

• Make a list of possible arrangements of the adsorbate molecules to consider

• Calculate the energy of each arrangement at 0 oK– Usually use pairwise additive assumption to

calculate energy

• The structure with the lowest energy fills first, the one with the second lowest energy fills second, …, until the surface fills up

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Energy Expression

-Ei=H1+nnhnn+n2nnh2nn

Ei=Energy molecule on site

nn = Number of nearest

neighbors occupied

n2nn= = Number of second

newest neighbors occupied

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Solution

Random adsorption on a (1x1) layer: For random absorption the probability of any site being filled is .

the energy of a atom absorbed at site i, becomes

4 because 4 first and second nearest neighbors

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,iE

nn2nn1)1x1( h4h4HE(4.202)

Random Absorption On A C(2x2) Layer:

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With a C(2x2) layer there are no first nearest neighbors, but there are several second nearest neighbors. If it is assumed that the C(2x2) layer is partially filled, then

)22(21)22( 4 xcnnxc hHE 3

1)22( xc

(4.203)

Random Absorption On A P(2x2) Layer:

With a P(2x2) layer there are no first or second nearest neighbors

(4.204)

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1)2x2(p HE

Random Absorption On A (2x1) Layer:

With a (2x1) layer there are two first nearest neighbors and no second nearest neighbors

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)12(1)12( 2 xnnx hHE 2

1)12( x

Summary Of The Equations

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nn2nn1i h4h4HE

nn2h2HE 1i

(1x1)

C(2x2)

1i HE

nn1i hHE

(p2x2)

(2x1)

When Does The (1x1) Have The Lowest Energy?

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E(1x1)< E(2x1)

E(1x1)< EC(2x2)

E(1x1)< EP(2x2) 11

2121

2121

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244

44

HhhH

hHhhH

hHhhH

nnnn

nnnnnn

nnnnnn

Solving hnn+h2nn>O hnn>O

Strong attractions and insufficiently weak

second nearest neighbors to prevent

1x1

When Does The P(2x2) Have The Lowest Energy?

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P(2x2)< E(2x1)

P(2x2)< EC(2x2)

P(2x2)< EP(2x2)

Solving h2nn<0 hnn<0

Occurs when First and second

neighbors are repulsive

What Happens If We Squeeze On More Molecules When P(2x2) Is Lowest Energy?

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Also Get Incommensurate Adsorption

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Figure 3.15 The domain wall structure of CO on Pt(100). (Proposed by Persson et al. [1990].)

Domain wallDomain wall

(2x2) Domain

Actual Phase Behavior More Complex

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Need Statistical Mechanics To Solve Real System

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Define occupancy number by:ξ=1 (occupied site) ξ =-1 (empty site)

Partition function definition:

Combining with expression

nEexpQS

11211

S

iiiiiSS

FhexpHexpQ

S 111 2411211

(4.78)

(4.67)

Analytical Solution Of Equations In One Dimension

• Assume a circular chain with So sites

• Calculate partition function analytically (see p 270 in Adsorption)

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Qso=exp(Hso){(1)So+(2)So}

h) (-0.5+F) ( h) (0.5+F) ( )4

h( = 2

1 expsinhexpcoshexp

h) (-0.5+F) ( h) (0.5-F) ( )4

h( = 2

2 expsinhexpcoshexp

(4.103)

(4.104)

Calculate Coverage Analytically

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)4h

exp(- + F)(sinh

F)sinh(

2

2S +

2S = > N < 00

(4.115)

Fig 4.17 A series of isotherms calculated from Equation 4.115 for βh = 3, 2, 1, 0.5, 0, -0.5, -1. The Langmuir

line (βh = 0) is shaded.

Replot vs. Pressure

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Figure 4.18 A replot of the data from Figure 4.17 versus dimensionless pressure.

Fig 4.17 A series of isotherms calculated from Equation 4.115 for βh = 3, 2, 1, 0.5, 0, -0.5, -1. The

Langmuir line (βh = 0) is shaded.

Key Features

• Characteristic S like behavior

• No first order phase transitions

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Fig 4.17 A series of isotherms calculated from Equation 4.115 for βh = 3, 2, 1, 0.5, 0, -0.5, -1. The Langmuir

line (βh = 0) is shaded.

Real Surfaces Show Phase Transitions

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Need Monte Carlo Calculation

Summary

• Can use statistical mechanics to calculate surface phase behavior

• Qualitatively – almost Langmuir behavior if only nearest neighbor interactions in 1 D

• More complex in 2 D.

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