soft matter review 10 january 2012. characteristics of soft matter (1)length scales between atomic...
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![Page 1: Soft Matter Review 10 January 2012. Characteristics of Soft Matter (1)Length scales between atomic and macroscopic (sometimes called mesoscopic) (2) The](https://reader030.vdocuments.us/reader030/viewer/2022032801/56649d545503460f94a301ea/html5/thumbnails/1.jpg)
Soft Matter Review
10 January 2012
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Characteristics of Soft Matter(1) Length scales between atomic and macroscopic
(sometimes called mesoscopic)(2) The importance of thermal fluctuations and Brownian
motion
(3) Tendency to self-assemble into hierarchical structures (i.e. ordered on multiple size scales beyond the molecular)
(4) Short-range forces and interfaces are important.
Lecture 1
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Lecture 2:
• Discussed polar molecules and dipole moments (Debye units) and described charge-dipole and dipole-dipole interactions.
• Discussed polarisability of molecules (electronic and orientational) and described charge-nonpolar and dispersive (London) interactions.
• Summarised ways to measure polarisability.• Related the interaction energy to cohesive energy and
boiling temperatures.
+ +- +
+
-
-
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SummaryType of Interaction Interaction Energy, w(r)
Charge-charge rQQ
o421 Coulombic
Nonpolar-nonpolar 62
2
443
r
hrw
o
o
)(_=)(
Dispersive
Charge-nonpolar 42
2
42 rQ
o )(_
Dipole-charge24 r
Qu
ocos_
42
22
46 kTruQ
o )(_
Dipole-dipole
62
22
21
43 kTruu
o )(_
Keesom
321
22
21
4 rfuu
o ),,(_
Dipole-nonpolar
62
2
4 ru
o )(_
Debye
62
22
4231
ru
o )()cos+(_
In vacuum: =1
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Measuring Polarisability
From Israelachvili, Intermol.& Surf. Forces
Polarisability determined from van der Waals gas (a) and u measurements.
Polarisability determined from dielectric/index measurements.
<
<
<
High f
Low f
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Lecture 3
• Lennard-Jones potential energy for pairs of atoms and for pairs within molecular crystals
• Evaluation of the Young’s (elastic) modulus for molecular crystals starting from the L-J potentials
• Response of soft matter to shear stress: Hookean (elastic) solids versus Newtonian (viscous) liquids
• Description of viscoelasticity with a transition from elastic to viscous response at a characteristic relaxation time,
• An important relationship between elastic and viscous components: = Go
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Interaction Potentials: w = -Cr -n
• If n <3, molecules interact with all others in the system of size, L. If n >3, molecules interact only with the nearer neighbours.
• Gravity: negligible at the molecular level. W(r) = -Cr -1
• Coulombic: relevant for salts, ionic liquids and charged molecules. W(r) = -Cr -1
• van der Waals’ Interaction: three types; usually quite weak; causes attraction between ANY two molecules. W(r) = -Cr -6
• Covalent bonds: usually the strongest type of bond; directional forces - not described by a simple potential.
• Hydrogen bonding: stronger than van der Waals bonds; charge attracting resulting from unshielded proton in H.
In the previous lecture:
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Comparison of Theory and Experiment
63
42
CNE A
mole ~
Evaluated at close contact where r = .
k
rwTB
23
)(=Note that o and C increase with .
Non-polar
London equation
RTbVV
aP ))(( 2
(Per mole, n = 1)
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Lecture 4• Viscosity and relaxation times increase strongly with
decreasing temperature: Arrhenius and Vogel-Fulcher equations
• First and second-order phase transitions are defined by derivatives of Gibbs’ free energy.
• The glass transition occurs at a temperature where config exp and is dependent on thermal history. In a glass,config > exp .
• Glass structure is described by a radial distribution function.
• The Kauzmann temperature could represent the temperature at which there is a first-order phase transition underlying the glass transition – possibly at a temperature of T0.
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Lecture 5• For mixing to occur, the free energy (F) of the system must
decrease; Fmix < 0.• The change in free energy upon mixing is determined by changes
in internal energy (U) and entropy (S): Fmix = U - TS.• The interaction parameter is a unitless parameter to compare
the interaction energy between dissimilar molecules and their self-interaction energy.
• The change of Fmix with (and T) leads to stable, metastable, and unstable regions of the phase diagram.
• For simple liquids, with molecules of the same size, assuming non-compressibility, the critical point occurs when = 2.
• At the critical point, interfacial energy, = 0.
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Constructing a Phase Diagram
T1
T2
T3
T4
T5
kTFmix
T1<T2<T3….
Co-existence where:
0=d
dF
Spinodal where:
02
2
=d
Fd
03
3
=d
Fd
G
=2
>2
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Phase Diagram for Two Liquids Described by the Regular Solution Model
G
Immiscible
Miscible
T1~
Low T
High T
Spinodal and co-existence lines meet at the critical point.
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Lecture 6• The thermodynamics of polymer phase separation is
similar to that of simple liquids, with consideration given to the number of repeat units, N.
• For polymers, the critical point occurs at N=2, with the result that most polymers are immiscible.
• As N decreases toward 2, the interfacial width of polymers becomes broader.
• The Stokes’ drag force on a colloidal particle is Fs=6av.
• Colloids undergo Brownian motion, which can be described by random walk statistics: <R2>1/2 = n1/2 , where is the step-size and n is the number of steps.
• The Stokes-Einstein diffusion coefficient of a colloidal particle is given by D = kT (6a)-1.
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Lecture 7
• The viscosity of colloidal dispersions depends on the volume fraction of the particles (Einstein equation):
• The Peclet number, Pe, describes the competition between particle disordering because of Brownian diffusion and particle ordering under a shear stress.
• At high Pe (high shear strain rate), the particles are more ordered; shear thinning behaviour occurs and decreases.
• van der Waals’ energy acting between a colloidal particle and a semi- slab (or another particle) can be calculated by summing up the intermolecular energy between the constituent molecules.
• Macroscopic interactions can be related to the molecular level.• The Hamaker constant, A, contains information about molecular
density () and the strength of intermolecular interactions (via the London constant, C): A = 22C
...)1( 22 bbo
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Israelachvili, p. 177
D
AR
R
RR
D
AW
662
1
21
If R1 > R2:
Colloidal particles
Summary of Molecular and Macroscopic Interaction Energies
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• Polymer crystals have a hierarchical structure: aligned chains, lamella, spherulites.
• Melting point is inversely related to the crystal’s lamellar thickness.
• Lamellar thickness is inversely related to the amount of undercooling.
• The maximum crystal growth rate usually occurs at temperatures between the melting temperature and the glass transition temperature.
•Tacticity and chain branching prevents or interrupts polymer crystal growth.
Lecture 8
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Lecture 9• The root-mean-squared end-to-end distance, <R2>1/2, of a freely-
jointed polymer molecule is N1/2a, when there are N repeat units, each of length a.
• Polymer coiling is favoured by entropy.• The elastic free energy of a polymer coil is given as
• Copolymers can be random, statistical, alternating or diblock.• Thinner lamellar layers in a diblock copolymer will increase the
interfacial energy and are not favourable. Thicker layers require chain stretch and likewise are not favourable! A compromise in the lamellar thickness, d, is reached as:
.++=)( constTNa
kRRF 2
2
2
3
32315
2//)(= N
kTa
d
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Lecture 9
• Elastic (entropic) effects cause a polymer molecule to coil up.• Excluded volume effects cause polymer molecules to swell (in a
self-avoiding walk).• Polymer-solvent interactions, described by the -parameter,
can favour tight polymer coiling into a globule (large ) or swelling (low ).
• Thus there is a competition between three effects!• The radius-of-gyration of a polymer, Rg, is 1/6 of its root-mean-
square end-to-end distance <R2>1/2.
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Lecture 9• When = 1/2, excluded volume effects are exactly balanced by
polymer/solvent interactions. Elastic effects (from an entropic spring) lead to a random coil: <R2>1/2 ~ aN1/2
• When < 1/2, excluded volume effects dominate over polymer/solvent interactions. In competition with elastic effects, they lead to a swollen coil: <R2>1/2 ~ aN3/5
• When > 1/2, polymer/solvent interactions are dominant over excluded volume effects. They lead to polymer coiling: a globule results.
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330 ~~~ NNNG tubeP
Lecture 10:
Applies for higher N:
N>NC
when chains are entangled.
G.Strobl, The Physics of Polymers, p. 221
Data shifted for clarity!
Viscosity is shear-strain rate dependent. Usually measure in the limit of a low shear rate: o
3.4
Reptation occurs when polymer chains are entangled (in melts or in concentrated solutions where chains overlap).
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Testing of Scaling Relation: D ~N -2
M=Nmo
-2
Experimentally, D ~ N-2.3
Data for poly(butadiene)
Jones, Soft Condensed Matter, p. 92
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Relaxation Modulus for Polymer Melts
Viscous flow
T
Gedde, Polymer Physics,
p. 103