1

Lecture II

Equilibrium properties of polymer adsorption

Polymer Adsorption(dilute solution)

Impenetrable surface

Solvent molecules

T>Tc: Desorbed phase

T<Tc: Adsorbed phase

Poymers adsorb spontaneously from solution onto impenetrable surfacesIf the interaction between the polymer and the surface is more favorablethan the solvent-surface one.

Ex: Polyethylene oxideadsorbs on silica or mica.Polystyrene adsorbs on gold

Ex: DNA adsorbs on mica.

D

Applications of polymer adsorption

Control the stability of dispersions,interfaces colloids:pharmaceutical preparations, pints and inks

The adsorption of DNA onto a flat mica surface is a necessary step to perform AFM studies of DNA conformation and observe DNA-proteininteractions in physiological conditions.

Rivetti et al. JMB 264, (1996)Valle et al. PRL 95 (2005)Witz et al PRL 101 (2008)

DNA microarrays

Equilibrium vs non-equilibrium

Many adsorption processes (chemisorption, physisorption) are dominated by irreversible effects (frozen surf-mon contacs, slow equilibrium of the adsorbed layer,trapping phenomena ) expecially when

TkBmonsurf >−ε

2

Quantities measurable experimentally

Total coverage

( )A

Vccr

inb

eqb −=

Volume of the solution

Area of the substrate

Initial bulk concentrationbulk concentrationat equilibrium

Obtainedby centrifugation of the solution or by infrared spettroscopy or by frequency shift of a quartz crystal microbalancein presence of an adsorbed layer.

( ) mfhAf ∆=∆ 0

Thickness D of the adsorbed layer

Static methods:

1) Neutron and X-ray Scattering,

2) Reflectometry and Ellipsometry: light intensity and polarization changes in presence of an adsorbed layer on a flat surface (Fresnel equation)

Set up for an experiment based on ellipometry.Linearly polarized light is reflected as ellipticallypolarized light.

An ellispometer measures the changes in thepolarization state of a film when its thickness is changed.

Hydrodynamic methods:

1) Diffusion (Photon Correlation Spectroscopy)2) Sedimentation (UC)3) Viscosity4) Electrokinetic Methods (zeta potential)

If the surface of the capillary is coated with an adsorbedpolymer layer of thickness D, the resulting radius R’measured by the Poiseuille flow will be R-2D whereR is the radius of the uncoated capillary

,8

4

LQ

PR

ηπ ∆=−

∆P:pressure drop;η: viscosity;Q:volumetric flow rate;L: capillary length

Viscosity method

Two main groups of results for the thickness:

I) and D is independent of the molecularweight N

II) and roughly proportional to N1/2

I) Strongly adsorbed phaseII) Weakly adsorbed (or desorbed) phase.

°≥ AD 300

°−≅ AD 10010

Possible interpretation:

3

Volume fraction profile

1) Small angle Neutron Scattering (SANS) (difference in scattering betweenhydrogen and deuterium). Used for adsorption on colloids (Cosgrove group: Bristol)

2) Spin-spin NMR spectroscopy.

3) Evanescent Wave Induced Fluorimetry: By fluorescent labeling of the polymersthe profile can be determined by evanescent wave-induced fluorimetry.

Schematic behaviour of the volume fraction profile as a function of thedistance z from the substrate .

Ψ (z)

Ψs

Ψ b

z

sΨ

bΨ

)(zΨ

z

( )zΨ Questions arising from experiments

How the thickness D of the adsorbed layer depends on:

(a) the strength of the surface attraction;(b) the quality of the solvent;(c) the molecular weight N ?

How the volume fraction profile behaves close to the surface ?

How the most probable configurations look likein the adsorbed phase ?

Is there an adsorption transition ?Which is its nature ? 1° order ? 2° order ?

For a theorethical approach based on models we need to specifiy

Surface characteristic(geometry, degree of smoothness)

Effective monomer-surface interactions

Quality of the solvent(good solvent where self-avoidance is important, or theta conditions)

Surface characteristic we focus on

1) Solid, i.e. impenetrable;2)Atomically smooth and flat (e.g. mica)3)Homogeneous.

Surface-monomer potential

1) Short ranged;

2) Long range effects(van der Waals or electrostatic interactions (DNA))

4

Schematic (short-ranged) surface potential

0

x

3

2

20

1

015

-1

105

0U−

)(zU

z

Uδ

>−<<−

<∞=

−U

U

zforcz

zforU

zfor

zU

δδ

η

0

0

)( 0

Long-rangeattractive tail

Impenetrability

Short-rangeattraction

Ex: unscreened van-der-Waals interactions (η=3)

Depending on TkU B0 one can distinguish between

Strongly attractive potential

Weakly attractive potential

We will see that the strength and range of the potential haveimportant consequences on the validity of the mean fieldapproach to the problem.(Lipowsky and Baumgartner, PRA (1989) JCP, R. Netz and D. Andelman, Phys. Rep. 380, (2003))

10 >>TkU B

10 <<TkU B

In equilibrium at temperature T the competition between the energy gain due to mon.-surf. contaccts and the entropy loss due to polymer localization at the surface is described by the ratio

TkU B0

Entropy loss due to confinement (adsorbed state)

N.B. Unlike single molecules adsorbed on a surface, the adsorption of a macromolecule implies a strong reduction in conformational entropy due tosurface-confinement. One can estimate this entropy loss by using the blob theory:(De Gennes 1979).

Blob picture: A polymer adsorbed on a surface with thickness layer D can be seen as a as sequence of N/g independentblobs of average size D

In the adsorbed state the total extropy loss is proportional to the number of blobs formed by confinement since the entropy within each blob is unchanged.

ν/1

)(

=∝D

bN

g

NDSN

Total entropy loss due toconfinement

Within each blob the subchain of g monomers behavesas if it were in the bulk

ν/1

≈b

Dg

5

Energy gain due to surf-mon interaction

Delicate problem since it depends on the number of monomers Ns withinthe potential range δU that in general is a strongly fluctuating variable.If we neglect the fluctuations we can use a mean field argument and seefor which potentials it is a good approximation (self consistent check)

Mean field approximationEach monomer feels an average surface potential U(z) whose value can, for example, be replaced by U evaluated at the averagedistance from the surface <z> = D/2

( )2/DUA

NE ≈

Soppose A=1 (unit area)

The total energy gain is then

Total free energyIs given by the sum of the entropy loss due to confinement (times kBT) and the surface interaction energy term E:

ν/1

)2/()(

+=D

bTNkDNUDF BN

Strongly adsorbed regime ( ) 02/ UDUD U −=⇒< δ

rU ~ D

ν/1

0)(

+−=D

bTNkNUDF BN

The two terms are of equal strength for a well depth

ν/1

*

≈D

bTkU B

The strongly adsorbed regime is realized for |U0| >|U*|In this situation the MF approx. is a good one.

DrU

Weakly adsorbed regime

The condition |U0| < |U*|, is easily satisfied for sufficiently high T

For |U0| < |U*|, D >> δU and the system is in the

Since D>> δU , the mean field approximation will be problematic sincebeing the polymer weakly adsorbed, fluctuations in monomer concentrationsare important.

There are however cases in which mean field is still a good approximation.First notice that for D>>δU, D/2 >>δU. Hence

η−−= Dc

DU2

)2/(

νη

/1

2)(

+−= −

D

bTNkD

cNDF BN

The minimization of F gives νην

ν −

≈

1/1

c

TkbD B

If η η η η > 1/νννν, D -> 0 as c decreases:non physical result

νη 1< Validity condition for the MF approach

6

For ideal chains ν = ½ and the validity condition becomes η < 2For self-avoiding chains ν=3/5 and η < 5/3: the condition ishere more stringent since configurations are more extended on average

N.B. For most interactions (including van der Walls: η = 3)the condition on η is not satisfied and the mean field approx. is not valid.

Exceptions:

1) Charged polymer with oppositely charged surface in absenceof salt ions η = -1 (the attraction between an infinite planar, charged,

surface and the polymer is linear in z)

2) Adsorption of polyampholytes on charged surfaces(local spontaneous dipole moments η=2)

(R. Netz and D. Andelman Phys. Rep. 380, (003)

In summary

η < 1/ ν: the mean field free energy is a valid approximation(Weak fluctuations regime)

η> 1/ ν: fluctuations in the local monomer concentrationare too important to be neglected. Mean field theory is not valid any more (Strong fluctuations regime).

In the strongly adsorbed regime the mean field freeenergy is a valid approximation

For weakly adsorbed polymers if

Adsorption of an ideal chain(polymer adsorption at the Θ point)

In presence of an external potential the diffusion equation for G(R,N) becomes

( ) )0,|,(1

)0,|,(6

)0,|,( 002

2

0 RNRGRUTk

RNRGb

RNRGN B

rrrrrrr−∇=

∂∂

With initial condition ( )00 )0,|0,( RRRRGrrrr

−= δ

Since in d-1 dimensions the chain behaves as a free Gaussian chain the interestingcase is along the direction z:

( ) )0,|,(1

)0,|,(6

)0,|,( 002

22

0 zNzGzUTk

zNzGz

bzNzG

N zB

zz −∂∂=

∂∂

( )00 )0,|0,( zzzzGz −= δ

(1)

Eq. (1) is similar to a Schrodinger equation for which we know that a generalSolution is given by the expansion of eigenfunction

pN

pp

pz ezuzubzNzGε−

∑= )()()0,|,( 0*

0

where )(zup are solutions of the eigenvalues problem

)()(1

)(6

)(2

22

zuzUTk

zudz

dbzu p

Bppp +−=ε

Normalization condition1)(

0

2=∫

∞

dzzu p

The solution depend on the form of the potential U(z).

><<−

<∞=

U

U

zfor

zforU

zfor

zU

δδ

0

0

0

)( 0Let us consider

7

Tk

zUzV

B

)()( =

If 10 <<TkU B(entropy wins over energy). The eigenvalues are positive andThere are only un-bound solutions (desorbed phase)

)()()(6

)(2

22

zuzVzudz

dbzu pppp +−=ε

If 10 >>TkU B(low energy). There exist negative eigenvaluesCorresponding to bound solutions (adsorbed phase)

Ur δ> 0)()( 22

2

=− zuzudz

dpp κ where 0

62

2 >−= pbεκ

With solution zp Cezu κ−=)(

Bound state solution (at least ε0 negative)

bU ≈δIf we assume The effect to the potential is to impose the boundary conditions at z=0 .

These are the Continuity ofdz

zduzu p

p

)()( and

atFrom zp Cezu κ−=)(

κ−==0

)(

)(

1

z

p

p dz

zdu

zu

0== Uz δ

That is satisfied by the exponential

( )( ) )(

;0)(

TTTTT

TT

cc

c

−≈→

==− κ

κκκ

and

Note that κ describes the relative strength of entropy/ mon-surf interaction

Ground state dominance

If there is a finite gap between the smallest eigenvalue and the first excited level it is possible to use the ground state dominance approximation. Indeed

pN

pp

pz ezuzubzNzGε−

∑= )()()0,|,( 0*

0

can be written as

( )

+= −−

=

− ∑ 00 )()()()()0,|,(1

0*

00*00

εεε pNp

pp

Nz ezuzuzuzubezNzG

if 00

≠− εεp the second term can be neglected giving

0)()()0,|,( 00*00

εNz ezuzbuzNzG −∝

and N>>1

zzo e

ACezu κκ κ −−

==2/1

2)(

and the expression for C comes from the normalization.

( ) 002

)0,|,( 0εκκ Nzzgsd

z eeA

bzNzG +−

∝

The volume fraction profile at one point z is obtained by looking at the statistical weight for a chain, starting from an arbitrary point s, reaching z and then extending to an arbitrarry point t

zN

M z

zze

A

NzuN

sNtGdsdt

MzNtGsMzGdsdtz κκ 22

00

2)()0,|,(

),|,()0,|,()( −

=

===Ψ ∑∫

∫

1)(0

2

0 =∫∞

dzzuA

A: area of the substrate where

8

From the exponential decay of Ψ(z) one gets the effective thickness 1/(2κ).

Since κ is a linear function of ( T – Tc) this gives

( ) 1−−≈ TTD c

The fraction of monomers at the surface is then given by

( )

( )( )TTcbe

dzz

dzz

f cb

b

s −∝=−≈Ψ

Ψ∝ −

∞

∫

∫κκ 21 2

0

0

Loops and tails:

Typical adsorbed configurations are actually characterized by loops,and tailswhose different statistical weights must be taken into account in a more detailed theory of adsorption transition. This has been done recently with a theory that goes beyond the ground-state dominance assumption (N infinity) and takes into account the separate contributions of loops and tails. (Semenov et al EPL 29 (1995), Macromol. 29 (1996))

Scaling theory of polymer adsorption

( )TTc −∝δ Excess surface energy: positive for attractive and negative for depletion

Consider a contact potential :only monomers within a distance b from the surface will feel the interaction

If for each monomer in contact with the surface there is a gain δ in energy, the average energy (or internal energy) due to the interaction will be

ss fNN δδ =

The entropy loss due to confinement is

( )ν/1

∝D

bNDSN

sBN fND

bTNkTDF δ

ν

+

∝/1

),(The total free energy is

In order to go on we need to estimate fs

Simplest approximation: Uniform distribution of contacts within D(good approx. deep in the adsorbed phase but not close to the transition)

( )D

b

N

DbN

N

Nf s

s ∝=≡ / Note that in this assumption< Ns > is proportional to N

D

bN

D

bTNkTDF BN δ

ν

+

∝/1

),(

(1)

11 −−

−∝

∝νν

νν

δT

TT

TbD c

eq

Minimizing the total free energy with respect to D gives

For Gaussian chain ν=1/2 and we get back the (Tc-T)-1 behaviour

νν−

−∝≡1

T

TT

N

Nf cs

s

For self-avoiding chains ν=3/5 giving

(2)

Inserting (2) in (1) we have

2/3−

−∝T

TTD csaw

eq

9

Crossover exponent φThe uniform distribution hypothesis implicitely assumes < Ns > prop. to N.

More generally one can assume that in proximity of the transition

φNN s ∝

νφν

δ1

/1

),(

−

+

∝b

DN

D

bTNkTDF BN

1−∝ φNf s

To get fs as a function of D let us consider again the blob picture: among the g monomers in a blob, gs will be on average at the surface and

1−∝ φgg s

νφ 1−

∝==∝b

D

g

g

gg

N

gg

N

N

Nf s

ss

s(3)

10 ≤≤ φ

Minimizing with respect to D gives

( ) φν−

−∝T

TTbD c

Inserting (4) in (3) gives1

1 −

−∝φ

T

TTf c

s

(4)

(5)Since 0=< φ <=1 fs goes to 0 as T approaches Tc, as expected.

1−∝ φNf s

φ−≈

−N

NT

NTT

c

cc

)(

)(

Eq(5) is satisfied in the thermodynamic limit, where the adsorption transition occurs at a given temperature Tc. For finite chains this is not true in general. Since fs=fs(N,T) we can say that for any finite N there exists an effective Tc(N) such that T>Tc(N) the system is desorbedT<Tc(N) the system is adsorbed.

(6)

We then have a crossover region whose width|Tc-Tc(N)| goes to zero, as N goes to infinity, as a power law whose exponent f is called the crossover exponent

On the other hand at T=Tc and comparing (5) with (6) we have

Scaling of some observable

We may assume that as N increases many macroscopic variables are functionof the ratio

( )φφ

τ−− ≡−=

−−

NN

TT

NTT

TT c

cc

c

)(

In particular for a fixed T and N, if:

desorbed phaseφτ −< Nφτ −> N

Consider for example the average extension of the polymer along the surface.We should expect a scaling behaviour of the form

∝ −φτ

NhRR g||

)(NTTTT ccc −<− )(NTT c>

)(NTTTT ccc −>− )(NTT c< adsorbed phase

This is possible if

In the desorbed phase we expectφτ −< N

In the adsorbed phase we expectφτ −> N

νNR ≈||

10/ →

→−

−φτφ

τ N

Nh

dNR 2||

ν≈

Assuming

p

NNh

≈

−− φφ

ττ

and matching the exponents givesφ

νν −= dp 2

10

For ν = 3/5 and ν2d = ¾

Hence φνν

ν τ−

∝d

dNR2

2||

φτ 20

34/3

|| NR ∝

If we consider instead the z-component of average extension of the polymerwe may again assume a scaling form

∝ −φτ

NhRR zgz

φτ −< N νNRz ≈ 10/ →

→−

−φτφ

τ Nz N

h

The system is in the adsorbed phase andφτ −> N 0NRz ∝

φν−=p φ

ν

τ−

∝||RSimilar to the layer

thickness D as expected

Volume fraction profile and proximal exponent

By solving the Gaussian chain model with ground state dominance we have seen thatthe volume fraction profile decays exponentially to zero as z increases. This solutionwas obtained in the assumption of D very small i.e. comparable to δU.

The exponential decay is indeed true in the distal region i.e. for z>>D.

For b < z < D, proximal region the volume fraction profile behaves as

( ) ( ) ms bzz −∝Ψ ϕ

To compute m let us consider the scaling behaviour of sϕ

Clearly νφϕ 22

2−≈∝ N

R

bN

g

ss

We may then assume ( )φνφ τϕ −−∝ NhNs /2

In the adsorbed phase we expectφτ −>> N0Ns ≈ϕ

φφν −= 2

p φφν

τϕ−

∝2

s

On the other hand ( )∫−≈Ψ=

gR

g

NdzzR

N

0

212

ν

And since ( ) ( )∫∫∫

−−−−

−−− ≈=

∝Ψ=ggg R

mmm

R m

s

R

NNdzzb

Ndz

b

zdzzN

0

122

00

21 ννφνφ

ν ϕ

We have

ννφ 1−+=m

Excluded volume interaction: Lattice models of polymer adsorption

11

Positive SAWs and polygons

A positive SAW (polygon) is a SAW (polygon) with one (two) vertex at the plane z=0 and with all the z-coordinates greateror equal to 0.

Def.

A positive walk A positive polygon

wN+ = # positive SAWs pN+ = # positive polygons

++= ∆−+

LN

BNeAw N

N11

1 11γσ

++= ∆

−+L

N

BNeAp

pNp

N13

1 11ασ

Positive SAWs and positive polygons have the same σ thanSAWs and polygons in the bulk

γ1 and α1 are surface entropic exponents (different from γ and α in the bulk)

A 2d –positive self avoiding walkWith N=38 steps and Ns=6 visits

SAW model of polymer adsorption

visit

( ) ∑=

−+=N

N

TkNsNN

s

BseNwTZ0

/)( ε

wN+(Ns) : # of positive SAWs with Ns visits (the first vertex is not counted)

Partition function foradsorbing SAWs

Since ε= -1 consider TkB/1=α

( ) ∑=

+=N

N

NsNN

s

seNwZ0

)( αα

Existence of the limiting free energy

Bounds on F(α):

Using concatenation argument for positive walks it is possible to prove theexistence of the limiting free energy for any finite α

( ) ( )αα FZN N

N=

∞→log

1lim

F(α) is a convexfunction of α

Clearly: ( ) NN

N

N

NsNN eNweNwZ

s

s ααα )()(0

+

=

+ ≥= ∑

Where w+N(N) :# of SAWs completely adsorbed on the plane i.e.the # of SAWs in (d-1)

( ) ( ) [ ] ασαα α +=≥= −+

∞→∞→ 1)(log1

limlog1

lim dN

NN

NN

eNwN

FZN

Lower bound

α > 0

Repellingsurface

On the other hand:

( ) ( ) +=≤ NNN wZZ 0α

From each positive walk with N-1 steps and arbitrary number of visits Ns, we can form a positive walk with no visits by lifting up the walk and adding an edge at the first vertex down to the surface. Since this is not the only way to obtain a positive walk with N steps and no visits we have:

∑ −+

−+ =≥

sNNsNN ZNww )0()()0( 11

0≤α

( ) ( )0+≥ NN wZ α

( ) ( ) ( )001 NNN ZZZ ≤≤− αTaking the logs, divide by N and taking the limit gives for

( ) σα =F

0≤α

w+N(0) :# of SAWs with no contacts

12

( ) σα =F0≤α

0>α ( ) ασα +≥ −1dF

Existence of a phaseTransition at a critical value

0≥cα

With a lot of hard work one can show that actually 0>cα

cαα ≤

0>> cααDesorbed phase

Adsorbed phase

cαα = Critical point that governs the adsorption transition

N.B. The adsorption transition is one of the few transition for whicha rigorous proof of its existence is available

F( )α

α c α

κ

Limiting free energy for adsorbing SAWs

α = αc the free energy is not analytic (phase transition)

desorbed

adsorbed

Ν.Β. The curve of the limiting free energy is exact for α < αc but it is only asketch for α > αc since we do not know its exact functional form but only that isconvex with given asymptots and bounds

F(α)=σd

Asymptote: α+σd-1

Fraction of monomers at the surface fs

( ) ( ) ( )ααα

α∂

∂= →∞→ F

N

Nf

Ns

s

cαα ≤

0>> cαα

( ) 0=αsf

( ) 0≠αsf Adsorption phase: finite fraction of visits.

The complete adsorption regime in which the walk isa (d-1) object is reached in the limit

∞→α

Monte Carlo simulations of adsorbing polymers

1) Sampling in the space of positive walks2) Adsorption energy must be taken into account (importance sampling)3) Finite size effects in data analysis

Consider first the problem of sampling in free space,i.e. sampling un-weighted walks (or polygons)

We will focus on lattice models but some ideas and algorithms can be easiliy adapted to simulate off-lattice models.

13

Sampling SAWs in free space: one pivot-move

Use Dynamic Monte Carlo methods: Generate a trajectory in the space of configurations by using a stochastic dynamic.

Stochastic Dynamic: The configuration at time t is obtained by attempting a deformation of the configuration at time t-1. The attempted move is accepted with a given probability to satisfy detailed balance condition. This condition is enoughto establish the convergence of the dynamic to the desidered probability distribution

The attempted moves must be designed in order to have an algorithm that isErgodic in the class of the configurations to be sampled. Roughly speaking ergodicity means that each configuration is accessible by a finite sequence of proposed moves. Ergodicity assures unicity of the limiting distribution.

ErgodicityDetailed Balance

Are the main conditions to be satisfied in Designing a MC algorithm

One point-pivot algorithm

1. Choose randomly one among the N vertices of the walk Ω (pivot point)

2. Choose one of the two sub-walks (Ω−ω and ω) in which the pivot partitions the walk

3. Perform a trasformation of the chosen sub-walk, say, ω, according to one of the possible symmetry transformation of the underlined lattice (see below)

4. Accept the move if the transformed sub-walk ω’ does not intersect the sub-walk Ω−ω

5. If excluded volume condition is satisfied accept the new walk Ω’= (Ω−ω) U ω’

6. Otherwise reject the attempted move and keep the old walk as the new configuration i.e. Ω’=Ω (important for detailed balance)

Is a dynamical, fixed N, algorithm that goes as follow:

’

’

ω

Ω−ω Ω−ω

ω

ω

Ω−ω

ω

Ω−ω

ω ’

Ω−ωω

Ω−ω

Examples of One point-pivot moves

Reflection through a plane

90°rotation

Self avoidance violation

Two points-pivot algorithm(polygons)

1. Choose randomly two vertices among the N vertices of the polygon Ω (pivot points)

2. Choose one of the two sub-walks (Ω−ω and ω) in which the two pivot partitions the polygon

3. Perform a trasformation of the chosen sub-walk, say, ω, according to one of the possible symmetry transformation of the underlined lattice (see below)

4. Accept the move if the transformed sub-walk ω’ does not intersect the sub-walk Ω−ω

5. If excluded volume condition is ok accept the new walk Ω’ = (Ω−ω) U ω’

6. Otherwise reject the attempted move and keep the old polygon as the new configuration i.e. Ω’ = Ω (important for detailed balance)

14

k1

k1

k1

k1

k2

k2

k2

k2

I

II

Two points-pivot movesω

ω

'ω

'ω

Two points-pivot moves for 3D polygons

N.B. The two-points pivot move may change the knot type of the polygon

Trefoil Unknot

31

Features of the pivot algorithms

Very efficient in sampling uncorrelated configurations

When a pivot move is accepted a big change in the conformation is likely to occur

One-pivot move: ergodic in the class of SAWs

Two-points pivot move: ergodic in whole the class of polygons.

Ergodicity

Efficiency

Sampling positive SAWs

Impenetrability condition (to be verified after step 4. of the algorithm)Each vertex of the transformed sub-walk ω’ must havenon negative z-coordinate

Ergodicity property still holds for positive SAWs and positive polygons.The pivot algorithms are very effcient also for positive SAWs and polygons

NO

15

Sampling adsorbing polymers: importance sampling technique

Aim A: to sample configurationsC that occur with probability

( ) ( ))(1 CN

N

seZ

C αα α

π =

Ns(C) : # surface contacs; α = 1/ T

( ) ∑=C

CNN

seZ )(αα Partition function for the adsorption problem

Aim B: to estimate averages of observables

( ) ( )∑=C

CN

NN

seCOZ

O )()(1 α

αα

( ) ( ))(1 CN

N

seZ

C αα α

π =

Not efficient to sample positive SAWs withuniform distribution (un-weighted positive walks)

Better to sample configurations according to (1): Importance sampling

If C(t) , t=1,M is a series of configurations sampledaccording to (1) then

( ) ( )∑=

≈M

tN

tCOM

O1

)(1α

Since (1)

( )sNep ∆∝ α,1min

Use, as underlying Markov chain, the one based on pivotmoves and accept the proposed move(given that self-avoidance and impenetrability conditions are ok)with a probability p proportional to :

How to generate C(t) , according to (1) ?

( ) ( )CNCNN sss −=∆ 'where

(2)

Condition (2): Metropolis scheme(not the only scheme available but probably the easiest to implement)

C:old conf. C’ new conf.

The Metropolis algorithm does the job in theorybut is not efficient as α increases (slow convergence)

Quasi – ergodic problem for Metropolis schemes

( ) ( )CNCNN sss −=∆ '

In the strongly adsorbed phase (high values of α ) any pivot movethat proposes the desorption of a big chunk of the walk willbe rejected because ∆Ns <<-1 and exp(α ∆Ns) very small

C:old conf. C’ new conf.

N.B. This is true for any Markov chain based on Metropolis (and related) schemes

The problem is more dramatic if only pivot moves are attempted

Reason:

16

( c )

( a )

( b )

Local moves

One bead flip

180°crankshaft

90°crankshaft

No big changes in the energy, hence more probable to be accepted.

N.B. Local moves are not ergodic the full MC algorithm needs alsopivot moves.

The quasi ergodic problem can be slightly milded byusing local moves but is still there expecially forα very big (strongly interacting system)!!

How may we improve the mobility of a Markov chainfor strongly interacting systems ?(Question more general than sampling polymer adsorption)

Umbrella sampling; Multicanonical;Flat-histogram; Simulated tempering; Wang-Landau method;Multiple Markov chain method

Different methods available

MMC scheme

( ) ( ))(1 CN

N

seZ

C αα α

π =Run in parallel M Markov chains each

with is own α and converging to

Mαααα <<< K,321

Fast converging Slow converging

Let the M Markov chains to mutually interact by swapping configurations

between contiguous (in α) chains as follows

Consider the set

4

β

β

β

β

1

2

3

MMC swapping scheme

Pick a pair of contiguous chain, αi , αi+1 and interchange the configurations Ci and Ci+1 with probability

( ) ( ) ( )( ) ( )

=

++

+

++

+ii

iiiii CC

CCCC

ii

ii

i

αα

αααα ππ

πππ

11,

1

11

1,1min,

17

MMC scheme

1) The coupling induces dependence along the chains (each single chain

is not Markov any more)

2) The whole process (MMC) is however a Markov

process

3) Since each Markov chain is ergodic the MMC is ergodic

4) The form of satisfies detailed balance condition

for the MMC process

( )1, ,1 ++ iii CC

iααπ

MMC converges to a unique invariant distribution given by the product

( ) ( ) ( ) ( )MCCCCMααα πππα L21 21

, =Π

Each singular Markov chain is asymptotically independent from the others

Features of the MMC:

( ) ( )απααα

αα

α

OtCOM

MM

t→∑

∞→

=1

)(1

The coupling increases the mobility of each markov chain

and attenuates quasi-ergodic problems

N.B.: to have a good accemptance ratios of the attempted swaps a good overlap between contiguous π(α) is necessary

Monte Carlo study of 3D SAWs interacting with a surface

Vertex model Edge model

Pivot movesLocal MovesMMC sampling scheme

TkB/1=α ( ) ∑=

+=N

N

NsNN

s

seNwZ0

)( αα Ns: # vertices (or edges) at z=0

Monte Carlo based on

N=100

Efficiency of the MMC scheme in decresing autocorrelation times

18

( ) ( )( )

αα

αα α

d

dFNeNwN

ZN N

N

N

NsNs

NNs

s

s == ∑=

+

0

)(1

Observables to estimate

( )( ) ( )2

22

22var

ααα

d

FdNNNN N

ss =−≡

“Energy”

“Specific heat”

( ) ( ) ∑=

+=N

N

NsNg

NNg

s

seNwRZ

R0

22 )(1 α

αα

( ) ( ) ∑=

+=N

N

NsNg

NNzg

s

s

zeNwR

ZR

0

22 )(1 α

αα

Mean squaredradius of gyration

z-component of theMean squaredradius of gyration

Adsorption “energy”

As α increases (going inside the adsorbed phase) the Average number of contacts increases. Which is the scaling with N ?

( )α

αd

dF

N

Nf Ns

s =≡

Fraction of verticesat the surface.

→ ∞→

constN

N Ns 0We expect

T > Tc (desorbed phase)

T < Tc (adsorbed phase)

At the transition (T=Tc) we expect 1−∝ φN

N

N s 10 << φ 4.05/2 ==φMean field

In general ( )τφφ NhNN s '=

αvsNM

N

N

Ns

Ms

loglog

( )φα ,cAll the curves should intersect at the point

At α = αc, t=0 and ( )0'hNN sφ=

0, =

= τφ

where N

M

N

N

Ns

Ms

For any pair of values M and N , at α = αc , the energy ratios are all equal to

φ

N

M

Both αc , and the crossover exponent φ may be determined by plotting (for example)

19

By extrapolating the intersection pointsAs N-> infinity one gets an estimate of αc

290.0≈cα

Near Tc and for N >> 1 we expect the scaling form

( ) [ ]L+−= − 111

φαα cs Af

( ) ( ) [ ]L+−== − 1)(var 2

1

2

2

φααα

αc

Ns Bd

Fd

N

N

( ) ( ) [ ]L+−= − 1var 2

1

φαα cs B

N

N

12

1 << φ The specific heat should diverge as a power law. if

For finite N we expect peakswhose height increases as

Moreover, since, ( ) φαα −∝− Nc

(C1)

inserting in (c1) we get ( ) [ ]L+= − 1

var 12φANN

N s

12 −φN

T > Tc

T < Tc

ν22|| NR ∝

dNR 222||

ν∝

4/32 =dν

589.0=ν

cαα <

cαα <

20

T > Tc

T < Tc

constNR Nz → ∞→ν22

589.0=νcαα <

cαα < 022 → ∞→Nz NR ν

Crossings

By extrapolating the intersection pointsAs N-> infinity one gets an estimate of αc

287.0≈cα

N=50

Volume fraction profile0=α 2.0=α

3.0=α 5.0=α

21

3.0=α cαα >= 5.0

Adsorbed phase