L10: Cross-linked Polymers - Apple

國立交通大學應用化學系陳俊太老師 1

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L10: Cross-linked Polymers

Jiun-Tai Chen (陳俊太) Web: http://www.jtchen.com

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Outline Crosslinks and Networks Historical Development of Rubber Rubber Elasticity Concept Thermodynamic Equation of State Statistical Thermodynamics Network Defect Flory-Rehner Equation Effect of Strain on Melting Temperature Elastomers in Current Use


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Elastomer (彈性體) Elastomer

Defined as a cross-linked amorphouspolymer above its glass transition temperature.

May be stretched substantially reversibly to several hundred percent.

If the cross-linked polymer is glassy, it is often called a thermoset.

The terms elastomer (彈性體) and rubber (橡膠) are often used interchangably.


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Cross-links(交聯), Networks (網狀結構) During reaction, polymers may be cross-

linked to several distinguishable levels. At the lowest level,

branched polymers are formed the polymers remain soluble sometimes know as the sol stage

As cross-links are added, clusters form, and cluster size increases.

Eventually the structure becomes infinite in size that is, the composition gels.

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Cross-links and Networks Continued cross-linking produces

compositions where, all the polymer chains are linked to other chains at multiple points

Producing one giant covalently bonded molecule.

This is commonly called a polymer network.


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The Sol-Gel Transition The gel point.

The reaction stage referred to as the sol-gel transition.

The viscosity of the system becomes infinite.

The equilibrium modulus climbs from zero to finite values.

The polymer goes from being a liquid to being a solid.

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Three Routes for Producing Cross-linked Polymers

1. Step polymerization reaction (逐步聚合) Little molecules such as epoxies react with

amines to form short, branched chains, Eventually condensing it into epoxies.


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Three Routes for Producing Cross-linked Polymers 2. Chain polymerization (連鎖聚合)

With multifunctional molecules present. An example is styrene polymerized with

divinyl benzene

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Three Routes for Producing Cross-linked Polymers 3. Postpolymerization reactions

A linear (or branched) polymer is cross-linked after synthesis is complete.

An example is vulcanization(硫化)of rubber with sulfur.

S

S


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Gel Point A square lattice example of percolation, at the

gel point.

Note structures that span the whole“sample”.

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Idealized Structure of a Cross-Linked Polymer The primary chains are cross-linked at many points

along their length. For primary chains such as rubber bands, tires, and

gaskets may have molecular weights of

the order of 1x 105 g/mol. Be cross-linked (randomly)

every 5 to 10 x 103 g/mol along the chain,

Producing 10 to 20 cross-links per primary molecule.


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Historical Development of Rubber

A simple rubber band may be stretched several hundred percent.

Rubber elasticity takes place in the third region of polymer viscoelasticity.

https://zh.wikipedia.org/


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Region 3: rubbery plateau region After the sharp drop that the modulus takes in

the glass transition region, it becomes almost consistent again in the rubbery plateau region.

Polymers exhibit long-range rubber elasticity the elastomer(彈性體) can

be stretched, perhaps several hundred percent, and snap back to substantially its original length on being released.

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Christopher Columbus In 1493, on his second trip to America,

he found the American Indians playing a game with rubber balls

The rubber balls were made of natural rubber. These crude materials were un-

cross-linked but of high molecular weight

Able to hold their shape for significant periods of time.

wikipedia.org

www.taringa.net/.

www.kentelastomer.com/


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Charles Goodyear In 1844, Goodyear vulcanized(琉化)

rubber by heating it with sulfur.

In modern terminology, he cross-linkedthe rubber.

Vulcanization Introduced dimensional stability, Reduced creep (潛變) and flow. Permitted the manufacture of a

wide range of rubber articles (物品).

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S

wikipedia.org

www.goodyear.com.tw/

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Modern Development 1914, a polymer of 2,3-dimethylbutadiene know as

methyl rubber was made in Germany. 1920, Staudinger developed his theory of the long-

chain structure of polymers. 1939 the US government started a Synthetic Rubber

Program.

www.mindfully.org/.


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Rubber Elasticity Mark and Kuhn proposed the model of a random coil

polymer chain Forms an active network chain segment in the

cross-linked polymer. When the sample was stretched, the chain had

extended in proportion, now called an affine(仿射) deformation.

When the chain is relaxed, The chain has an average end-to-end distance r0.

When the chain is stretched, The average end-to end distance increase to r.


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Retractive Stress The basic equation relating

the retractive stress, σ, of an elastomer in simple extensionto its extension ratio,

The Retractive Stress, σ, n: number of active network chain segments per unit

volume. R: the gas constant T: the absolute temperature.

The extension ratio, : length after extension. 0: original length

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Retractive Stress

n: Number of active network chain segments per unit volume.

: density : the molecular weight between cross-links


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Retractive Stress

The equation is non linear. The Hookenan simple

proportionality between stress and strain doe not hold.

Young’s modulus is often close to 2 x 106

Pa.

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Lightly Cross-linked Natural Rubber at 50oC Curve(a):

The sharp upturn of the experimental data above α = 7 is due to the limited extensibility of the chains themselves.

Part of the effect is usually strain-induced crystallinity especially for natural rubber and cis-polybutadiene.

Curve (c) Illustrates the reversible nature of the

extension up to 5.5.

Curve (b) At higher elongation, hysteresis effects

become important.


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Thermodynamic Equation of State The Helmholtz free energy, F,

U: the internal energy S: the entropy

The retractive force, f, exerted by the elastomer depends on the change in free energy with length.

For an ideal elastomer, The changes in numbers of chain conformations can

be expressed as an entropic effect.

energetic entropic


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Thermodynamic Equation of State

A relationship between the entropy and the retractive force.

The thermodynamic equation of state for rubber elasticity.

energetic entropic

energetic entropic

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General Experimental Curve

For an ideal elastomer, the quantity ,⁄ is zero.

The experimental line is straight line in the ideal case. The slope being proportional to ,⁄ or

,⁄

energetic entropic

entropic

energetic

(energetic)


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f :the retractive force fe : the energetic portion fs : the entropic portion

Retractive Force

energetic entropic

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Retractive Force

The quantity fs accounts for more than 90% of the stress, Where as fe hovers near zero.

The turndown of fe above 300% elongation may be due to incipient crystallization.

energetic entropic


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Statistical Thermodynamics of Rubber Elasticity

The statistical theory of rubber elasticity is based on

the concepts of random chain motion

the restrainingpower of cross-links.

It is a moleculartheory.


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Equation of State for Rubber Elasticity

Front factor: : the average of the squares of the relaxed end-to-end

distances. : The isotropic, unstrained end-to-end distance in the

network. Under many circumstances the quantity,

approximately equals unity. n :

the number of active network chains per unit volume Sometimes called the network or cross-link density.

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Definition of Young’s Modulus Young’s Modulus

: tensile stress Force per unit area.

: tensile strain If the sample’s initial length is L0 and its final

length is L, then the strain is /

Young’s modulus is a fundamental measure of the stiffness of the material. The higher its value, the more resistant the material

is to be stretched.


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Young’s Modulus of Cross-linked Materials Young’s modulus for small strains.

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Definition of Shear Modulus In stead of elongating (or compressing!) a

sample, it may be subjected to various shearingor twisting motion.

Shear modulus

: Shear Stress : Shear Strain


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Definition of Shear Modulus

wikipedia.org

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Shear Modulus of Cross-linked Materials Shear modulus

: poisson’s ratio : 0.5 for rubber (assume incompressibility)


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Example Calculations An elastomer of 0.1 cm x 0.1 cm x 10 cm is stretched to 25 cm

length at 35oC, a stress of 2x 107 dynes /cm2 being required. What is the concentration of active network chain segments.

Assume that

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Network Defects Two major types of network defects

(1) The formation of inactive rings or loops, where the two ends of the chain segment are connected to the samecross-link junction

(2) Loose, dangling chain ends, attached to the network by only one end.

(a) Elastically active chain

(b) Loop (c) Dangling chain end

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Network Defects Both these defects tend to decrease the

retractive stress Because they are not part of the network.

The equation in use to correct for dangling ends

Mc: the molecular weight between cross-links M: the primary chain molecular weight.

When M >> Mc, the correction becomes negligible.


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The Flory-Rehner Equation The equilibrium swelling theory of Flory and

Rehner treats simple polymer networks in the presence of

small molecules.

v2 : the volume fraction of polymer in the swollen mass V1 : the molar volume of the solvent X1 : the Flory-Huggins polymer-solvent dimensionless

interaction term.


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The Flory-Rehner Equation Consider forces from three sources (1) The entropy change caused by mixing polymer and

solvent. positive and favors swelling.

(2) The entropy change caused by reduction in numbers of possible chain conformations on swelling. negative and opposes swelling.

(3) The heat of mixing of polymer and solvent, which may be positive, negative, or zero. Usually it is slightly positive, opposing mixing.

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Example Calculation Poly(butadiene-stat-styrene) swelled 4.8 times its volume in

toluene at 25oC. What is Young’s modulus at 25oC? X1 for the polymer : 0.39 The molar volume (V1) of toluene can be calculated from its

density, 0.8669 g/cm3. A molecular weight of 92 g/mol for toluene yields a molar

volume of 106 cm3/mol. The quantity v2 = 1/4.8 = 0.208.


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Example Calculation Young’s modulus

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Effect of Strain on the Melting Temperature Some elastomers crystallize

during strains such as extension.

The most important of these are cis-polybutadiene, cis-polyisoprene, cis-polychloroprene.

Crystallization on extension can be responsible for a rapid upturn in the stress-

strain curves at high elongation.

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Crystallization Such crystallization is good for engineering.

For the wear mechanisms in automotive tires.

Abrasion is the most important mode of loss of tread.

If the rubber crystallizes during extension It becomes self-reinforcing when it is needed most, thus

slowing the failure process. When the strain is released, the crystallites melt,

returning the rubber to its amorphous state reversibly.

In contact with the road, these shreds are strained at each revolution of the tire, gradually tearing off more rubber.

www.fourwheeler.com


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Diene Types Polymers prepared from butadiene, isoprene, their

derivatives and copolymers. Such as natural rubber (polyisoprene).

The general polymerization scheme

Cis products have lower glass transition temperatures reduce crystallinity make superior elastomers.

Trans products highly crystalline make excellent materials such as golf ball covers.


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Saturated Elastomers The polyacrylates exemplify these materials

Where X- maybe CH3-, CH3CH2-, and so on.

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Thermoplastic Elastomers These new materials contain physical cross-links

rather than chemical cross-links.

A physical cross-link can be defined as a non-covalent bond that is stable under one condition but

no under another.

Thermal stability is the most important case.

These materials behave like cross-linked elastomers at ambient temperatures but as linear polymers at elevated temperatures, having reversible properties as the temperature is raised

or lowered.


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Inorganic Elastomers The major commercial inorganic

elastomer Poly(dimethyl siloxane)

Know widely as silicone rubber. This specialty elastomer has the lowest

known glass transition temperature, Tg= -130oC.

It also serves as a high-temperature elastomer.

A common application of this elastomer is as a caulking (堵塞) material.

It cross-links on exposure to air.

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Inorganic Elastomers Another covalently bonded inorganic elastomer

class is the polyphosphazenes.

In elastomeric compositions R and R’ are mixed substituent fluoroalkoxy groups.

The current technological applications depend on the oil resistance and nonflammability of these elastomers

Low Tg’s are also important.


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Reinforcing Fillers and Other Additives Natural rubber has a certain degree of self-reinforcement

becasue it crystallizes on elongation.

The thermoplastic elastomers also gain by the presence of hard blocks.

However, nearly all elastomeric materials have some type of reinforcing filler, usually finely divided carbon black or silicas.

These reinforcing fillers, with dimensions of the order of 100 to 200 Å, Form physical and chemical bonds with the polymer chains. Tensile and tear strength are increased, and the modulus is

raised.

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Reinforcement The reinforcement can be

understood through chain slippage mechanisms.

The filler permits local chain segment motion but restrictsactual flow.

The major application of carbon-black-reinforced elastomers is in the manufacture of automotive tires.

like-img.com/.


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A modern Tire Tread Recipe

L10: Cross-linked Polymers - Apple

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