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Chapter 14. Steric and Fluctuation Forces. Department of Chemistry 김 도 환 조 남 성 임 도 경. Diffuse Interface. Diffuse surface = Dynamically rough (thermally mobile surface groups). Surface. Liq. 1. Solvent. Liq. 2. Chain molecules attached to a surface, dangled out into the solution

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chapter 14 steric and fluctuation forces
Chapter 14. Steric and Fluctuation Forces

Department of Chemistry

김 도 환 조 남 성 임 도 경

diffuse interface
Diffuse Interface

Diffuse surface = Dynamically rough (thermally mobile surface groups)


Liq. 1


Liq. 2

Chain molecules

attached to a surface,

dangled out into the solution

 polymer covered surface

Inherently mobile or fluid-like


what is a polymer
What is a Polymer ?
  • Polymer :
  • Macromolecule composed of many monomers

Homopolymer : ~A-A-A-A~

Copolymer : ~A-B-A-B~

volume of polymers
Volume of Polymers





  • Unperturbed Random Coil
    • Unperturbed radius of gyration, Rg

n = # of segments

l= effective segment length,

( Only in ideal solvent ! )

  • Flory radius, RF = α Rg ≒ ln 3/5(in real solvents)
    • (α: Intermolecualr expansion factor)

Good solvent : repulsion btw. segments (α<1)

Poor solvent : attraction btw. segments (α>1)

Theta temperature, Tθ : ideal solvent (α =1)

polymers at surfaces
Polymers at Surfaces

In solution Chemisorbed Physisorbed

Low coverage high coverage Bridging


Repulsive Steric Forces between Polymer-Coated Surfaces

Force development process

Outer segments overlap ⇒ Repulsive steric force

▶ Coverage of polymer on each surface▶ Reversible adsorption or irreversible grafting onto the surface▶ Quality of the solvent

Factors affecting steric force


Theories of Steric Interaction I

per molecule

Consider repulsive steric interaction between surfaces containing an adsorbed polymer layer where each molecule is grafted at one end to the surface but is otherwise inert1) At low surface coverage▶ No overlap of neighboring chains ▶ Each chain interact with the opposite surface independent of other chains Repulsive energy per unit area for two surfaces in a theta solvent over the distance from D=8Rgdown to D=2Rg

G : number of grafted chains per unit areaG = 1/s2 (s: mean distance between attachment point)

Valid for low coverage s > Rg

when layer thickness (L) is equivalent to Rg

- In theta solvent : vary as M 0.5- In good solvent : vary as M 0.6

▶ Layer thickness


Theories of Steric Interaction II

2) At high surface coverage (Brush) ▶ Adsorbed or grafted chains close to each other⇒ Chains are forced to extend away from the surface much farther than Rg ▶ Brush Layer thickness - End-grafted chain : L ∝ M - General equation for brush in a theta solvent L ∝ Mu ∝ nuu : 0.5 ~ 1 (from low to high coverage) - For a brush in a good solvent

  • = 1/s2

RF = ln 3/5


Theories of Steric Interaction III

Alexander-de Gennes theory

▶ Repulsive pressure between two brush-bearing surfaces closer than 2L

for D < 2L

Elastic energy of the chain

⇒ Oppose stretching

⇒ Decrease D

Osmotic repulsion between the coils

⇒ favor stretching

⇒ increase D

▶ For 0.2 < D/2L < 0.9,

W = Fs


Steric forces between surfaces withend-grafted chains

▶ End-grafted polymers : well understood

- Each molecule is attached to the surface at one end

-Coverage is fixed

-Molecules do not interact either with each other or with the two surfaces

- ex) Di-block copolymer

One block for anchoring, the other protruding into the

solvent to form polymer layer

Fig 14.3 Forces between two polystyrene brush layers end-grafted onto mica surfaces in toluene

▶ Measured forces agree with theoretical fits of Alexander - de Gennes eqn.

▶ No hysterisis on approach and separation of polymers

steric forces between surfaces with physisorbed layers i
Steric forces between surfaces withphysisorbed layers I

▶ No anchoring group that chemisorb to the surface

▶ Binding to the surface via much weaker physical forces

▶Highly dynamic layers

: Individual segments continually attaching and detaching from the surface

: whole molecules slowly exchange with those in bulk solution

When two surface approach each other

▶ Amount of adsorbed polymer changing

▶ Number of binding sites per molecule changing

▶ Different segments from the same coil bound to both surfaces ⇒ Bridging

▶ Long time to reach equilibrium

▶ Hysteretic force profile


Steric forces between surfaces with physisorbed layers II

Fig 14.4-inset Evolution of the forces with the time allowed for high MW polymer to adsorb from solution

Fig 14.4 Forces between two polyethylene oxide layers physisorbed onto mica

Solvent : aquous 0.1M potassium nitride

▶ time ↑, adsorption ↑, brush layer ↑

▶ Gradual reduction in the attractive

bridging component

▶ Hysterisis on approach and separation

of physisorbed polymer

▶ The range of repulsive steric force may be many times Rg ( > 10 Rg )


Forces inPure Polymer Liquids(Polymer melts)

▶ Polymer molecules are surrounded by molecules of its own kind

⇒ Much the same interactions as that of a polymer in a theta solvent

▶ Terminally anchored to the surface


Expermental force behavior in Pure Polymer Liquid

▶ Forces between

mica surfaces across

pure polymer melts

▶ At small distance : Oscillations with a periodicity equal to the segment width

▶ Non-equilibrium monotonically decaying repulsion farther out upto 10Rg

: Strong binding or effective immobilization of polymers at the surface

▶ Forces between inert hydrocarbon surfaces

across chain-like hydorcarbon liquids

▶Attractive tail

⇒ Much weaker binding to the surfaces

▶ Forces between Irregularly shaped polymers

(e.g. bumpy segments or randomly branched

side group)

▶ No short range oscillation,

but smooth monotonic repulsion

⇒ Inability to order into discrete, well-defined layers

▶ Limitation of force measurement in polymer molecules concentrated within adsorbed surface layer or

confined within a thin film between two surfaces

- Molecular relaxation time higher than in the bulk

- Liquid molecules in the bulk freeze into amorphous glassy state at the surface

⇒ Measurement not at true thermodynamic equilibrium


Attractive Intersegment Forces

▶ Polymer segments

Segments attract each other in a poor solvent : van der Waals force, solvation forces

Isolated coil shrinks below Rg in solution

▶ Polymer-coated surfaces

▶ As two polymer-coated surfaces come together in poor solvent,

Attraction between the outermost segments

⇒ Initial Intersegment attraction between surfaces

▶ As two polymer-coated surfaces come closer,

Steric overlap repulsion wins out


Attractive Intersegment Forces / Bridging forces

(a) End-grafted polystyrene

brush in toluene ( = 35 ℃)

(b) Physisorbed polystyrene

in cyclohexane ( = 34.5 ℃)

Rg = 7 nm

Rg = 11 nm

Rg = 8.5 nm

Rg = 21 nm

Attraction due to intersegment force

Attraction due to Bridging force

Attraction due to intersegment force

▶ Fig 14.5 Interactions between grafted and adsorbed polystyrene

layers below the theta temperature in poor solvent


Attractive Bridging Forces


Bridge formation

► Segment – surface force : attractive

►Available binding sites for segments on the opposite surface

►Polymer coil will form bridge between two surfaces

⇒ Attractive bridging force between two surfaces

►Coverage too high (brush), few free binding sites for bridges

⇒ Brush layer thicker than RF, no bridging attraction

►Coverage too low, density of bridges will be low


on coverage

►Bridging force decays exponentially with distance

Decay length ≈ Rg of the tail and/or loops on the surfaces


on distance

► under suitable conditions, Sometimes strong far exceeding

van der Waals interaction



Effect of surface coverage and solvent quality

By polymer property

(Reactivity, M.W. …)

▶ Fig 14.6 Effect of surface coverage and solvent quality

on force profiles of adsorbed and grafted chains


Attractive Depletion Forces I

▶ Polymers repelled from surfaces

⇒ no adsorption from solution at all

⇒ But weak attractive interaction


▶ Two surfaces in a dilute solution of coils of average radius Rg

▶ Polymer coils have no interaction with the surface

When surfaces are closer than Rg, coils will be pushed out from the gap

⇒ Reduced polymer concentration between the surfaces

  • ▶ Bulk polymer concentrationr,
  • Applying contact value theorem : P(D) = kT [rs(D) – rs (∞)]
  • Attractive depletion force per unit area between the surfaces at contact
  • P (D→0) = - r kT
  • Depletion free energy per unit area
  • W (D→0) ≈ - rRgkT
  • example) r = 1024 m-3, Rg = 5 nm T=25 ℃
  • Interaction energy between two surfaces decrease by 0.2 mJ/m2due to depletion

Attractive Depletion Forces II

▶Strong depletion force

- High bulk concentration of polymer molecule (high r)

- Large Rg(high MW)

⇒ Choose high polymer concentration (high r , low MW, low Rg)

▶ In the limit of very high r , low Rg

- Adhesive minimum becomes deeper

- Range of depletion force decrease (low Rg)

- By the time polymer mole fraction reach unity,

⇒Characteristic of a pure liquid or polymer melt

▶ Attractive depletion force

- Explain colloidal particle coagulation

non equilibrium aspects of polymer interactions
Non-Equilibrium Aspects of Polymer Interactions

Polymer mediated interaction is not always in equilibrium !

▶ Molecular relaxation mechanisms

  • - Solvent has to flow out through the network of entangled polymer coils
  • Coils themselves must reorder as they become compressed
  • New binding sites and bridges have to be formed
  • A certain fraction of polymer molecules may have to enter or leave the gap region altogether
  • Concerted motions of many entangled molecules
  • Require many hours or days (c.f. 10-6 sec for isolated coils)
  • Hysteresis, time-dependent effects

Thermal Fluctuation Forces between Fluid-like Surfaces

▶ Thermally mobile or fluid-like surfaces

  • - micelles, microemulsion droplets, biological membranes
  • - constantly changing shapes
  • - a number of repulsive ‘ thermal fluctuation ’
  • protrusion, undulation, peristaltic motion


Undulation forces

Peristaltic forces


Protrusion Forces I

▶ Approaching two amphiphilic surfaces (molecular - scale overlap)

  • protruding segments are forced into the surfaces
  • for graftedchain :remain between the surfaces
  • - for adsorbedchain : forced out into the bulk liquid

Protrusion Forces II

  • ▶ Approximation
  • energy increase linearlywith the distance z, that the molecules protrude into the water
    • Protrusion energy
    • = πσzy = αpz Eq. (13.1)
  • υ(zi) = αpzi
  • (αp : interaction parameter units , Jm-1)

The density of protrusion extending distance z from the surface

( the protrusion decay length )


Protrusion Forces III

▶ The protrusion force between two amphiphilic surfaces

Lateral dimension σ

Extending a distance zi into the solution

Γ protrusion sites per unit area,

by potential distribution theorem (Eq. 4.9)

The interaction free energy

Protrusion pressure

P(D) = 2.7 Г αp e-D/ λ

(λ ≈ kT/αp ) (14.12)

λ < D < 10 λ

D < λ


Protrusion Energy

▶ Eq. (14.12) corresponds to an energy per unit area

W(D) = 2.7 Гαp λ e -D/ λ ≈ 3 Г kTe -D/ λ (14.14)

▶ Compare with…

(end grafted chains)

(two brush layers)


Undulation and Peristaltic Forces I

Arise from the entropic confinement of their undulation and peristaltic waves as two membranes approach each other

⇒ Derived fromcontact value theorem (entropic force per unit area)

P(D) = kT[ρs(D) – ρs(∞)]

(ρs : Volume density of molecules in contact with the surfaces)

▶ Undulation force - Membrane’s bending modulus, kb

In thermally excited waves

The density of contacts

By the ‘chord theorem’ Eq.(9.7) x2≈2RD

The undulation pressure

Elastic bending energy, Eb of a curved membrane with local radii R1 and R2


Undulation and Peristaltic Forces II

At temperature T, suppose that each mode has area πx2 & energy ~kT

Eq. (14.16)

  • Undulation force can be drastically reduced or even eliminated !
    • - when a membrane carries a surface charge
    • - when it is in tension

▶ Peristaltic force - Area expansion modulus, ka

Mean area, a = πx2

Exceeding surface area per mode, Δa = πD2

Elastic energy Ea

The peristaltic pressure