Polymer Kinetics

Thermodynamics of Polymerization

• Thermodynamics of polymerization determines the position of the equilibrium between polymer and monomer(s). Thus, it impacts both polymerization, depolymerization, and degradation. The thermodynamics of polymerization of most olefins is favorable due to the exothermic nature of converting π bonds into σ bonds. For cyclic compounds, the driving force for polymerization can vary over a much wider range, and one observes a variety of behaviors ranging from completely unreactive to spontaneously polymerizablevariety of behaviors ranging from completely unreactive to spontaneously polymerizable under all conditions.

• The well known thermodynamic expression:ΔG = ΔH - TΔS

yields the basis for understanding polymerization/depolymerization behavior.• For polymerization to occur (i e to be thermodynamically feasible) the Gibbs free energyFor polymerization to occur (i.e., to be thermodynamically feasible), the Gibbs free energy

of polymerization ΔGp < O. • If ΔGp > O, then depolymerization will be favored.

Thus: Any factor that lowers the enthalpy, H (i.e., makes ΔHp more negative), or raises the entropy, S (i.e., makes ΔS more positive), will shift the equilibrium towards polymerization.

• Standard enthalpy and entropy changes, ΔHop and ΔSo

p are reported for reactants and products in their appropriate standard states Generally:products in their appropriate standard states. Generally:

Temperature = 25oC = 298KMonomer – pure, bulk monomer or 1 M solutionPolymer – solid amorphous or slightly crystalline

• Polymerization is an association reaction such that many monomers associate to form the polymer

Regardless of mechanism, there is a large loss in the total number of rotational and translation degrees of freedom in the total system as the monomers associate. This occurrence thus yields a major loss in entropy upon polymerization.Thus: ΔSp < O for nearly all polymerization processes.

• Since depolymerization is almost always entropically favored, the ΔHp must then be sufficiently negative to composite for the unfavorable entropic term. Only then will polymerization be thermodynamically favored by the resulting negative ΔGp.p

• In practice:Polymerization is favored at low temperatures: TΔSp is smallDepolymerization is favored at high temperatures: TΔSp is large

• Therefore, thermal instability of polymers results when TΔSp overrides ΔHp and thus ΔGp> O thi th t t t l d l i (if ki ti th i t )

PSC 480/740 Kinetics1

> O; this causes the system to spontaneously depolymerize (if kinetic pathway exists).

Thermodynamics of Polymerization(continued)( )

• Since most polymerizations are characterized by an exothermic propagation reaction and an endothermic depropagation reaction, the activation energy for the depropagation reaction is higher, and its rate increases more with increasing temperature compared to theincreases more with increasing temperature compared to the propagation reaction. As shown below, this results in a ceiling temperature, defined as the temperature at which the propagation and depropagation reaction rates are exactly equal at a given monomer concentration.

3

4

5

6

k [M]

kdp

c-1

300 350 400 450 500 550 6000

1

2

3

Tc

kp[M] - kdp

kp[M]

k, se

c

• Each ceiling temperature, Tc, and corresponding equilibrium monomer concentration, [M]c, are related to the enthalpy and entropy of polymerization, defined for monomer at standard 1 M concentration,

300 350 400 450 500 550 600

Temperature, oK

polymerization, defined for monomer at standard 1 M concentration, as follows:

• At equilibrium, ΔG = 0, and T = Tc.

Δ Δ ΔG H T S RTo o= − + ln Keq

Δ ΔH T S RTo o l K


Δ ΔH T S RToc

oc− = − ln Keq

Thermodynamics of Polymerization(continued)( )

• The equilibrium constant is defined by kp/kdp, or typically, as 1/[M] for standard state defined as 1 M.

HoΔT HS Rln[M]c o

c

=+Δ

Δ

ln[M] HRT

SRc

o

c

o

= −Δ Δ

• At [M]c =1 M, Tc = ΔHpo/ΔSp

oc

Specific Examples of Monomer - Polymer Equilibrium

kcal/mol cal/mol-deg (ΔH/ΔS)

Monomer ΔHp ΔSp Tc(oC)

Ethylene -21.2 -24 610

Isobutylene -12.9 -28 175

Styrene -16.7 -25.0 395

methyl styrene 8 4 24 66

• Notice the large variation in the -ΔH values.th l > i b t l tt ib t d t t i hi d l th l

α-methyl styrene -8.4 -24 66

2,4,6-trimethyl styrene -16.7 --- ---

TFE -37 -26.8 1100

ethylene > isobutylene - attributed to steric hinderance along the polymer chain, which decreases the exothermicity of the polymerization reaction.ethylene > styrene > α-metylstyrene - also due to increasing steric hinderance along the polymer chain.Note, however, that 2,4,6-trimethylstyrene has the same -ΔH value as styrene Clearly the major effect occurs for substituents directly attached


styrene. Clearly, the major effect occurs for substituents directly attached to the polymer backbone.

Thermodynamics of Polymerization(continued)

• Other possible effects on ΔHploss of resonance stabilization upon polymerizationchanges in bond hybridizationg ychanges in H-bonding between M and P statesNotice the small changes in the ΔSp values. This small variation is attributed to the loss of translational entropy which is about constant from system to system.

• For the systems in the table above, the equilibrium at 25 oC (i.e., at the standard state condition) favors the formation of polymer. This may be verified using the equation we examined previously.

ΔGo = -RT lnKeq

• As the temperature increases, the equilibrium constant decreases (characteristic of an exothermic reaction). When Tcis exceeded, Keq becomes less than 1, and thus,is exceeded, Keq becomes less than 1, and thus, depolymerization becomes the dominant process.

• It is very important to note that the Tc concept applies only to closed systems at equilibrium. For open systems, monomer may volatilize away and thus depolymerization may occurmay volatilize away, and thus, depolymerization may occur well below the predicted Tc. In fact, few polymers actually match their thermal stability as predicted from the Tc approach.


Experimental Determination of ΔHo

p and ΔSop

• ΔHop - by direct calorimetric measurement of amount of heat evolved

when known amount of the monomer is converted to a known amount of polymer.

or

• by heats of combustion of M and P which yields ΔHof (enthalpy of

formation) of M and P. The ΔHop is thus obtained by the relationship:

)()(1 MHPHn

H ofn

of

op Δ−Δ=Δ

• ΔSop - from the absolute entropies of M and P, such that:

)()(1 MSPSS ooo ΔΔ=Δ

• The absolute entropies may be obtained from calorimetric measurements of heat capacities of M and P over a wide T range, as

)()( MSPSn

S np Δ−Δ=Δ

p ggiven by:

dTTC

TST

po ∫=0

)(


0

Floor Temperature Behavior

• Although the vast majority of all polymerizations possess negative ΔH and ΔS, and hence display ceiling temperature behavior, four distinct possibilities exist as outlined in the table:

• As stated earlier, -ΔS for polymerization is almost universal.Therefore for olefins and small cyclics polymerization is possible at lowTherefore, for olefins and small cyclics, polymerization is possible at low temperatures.However, many compounds are never spontaneous toward polymer due to +ΔH (e.g. cyclohexane, tetrasubstituted olefins)

• +ΔS for polymerization is rare, but known examples exist (see below).This rare behavior leads to floor temperature behavior or entropy-drivenpolymerizations.p yFloor temperature monomers are invariably large cyclics containing large atoms from the third row and below of the periodic table, that yield polymers with highly flexible chains.

ΔH ΔS Behavior

Ceiling temperature- - Ceiling temperature

- + Always spontaneous toward polymer

+ + Floor temperature

+ - Never spontaneous toward polymer

Examples of monomers possessing a floor temperature:

SiO Si

O

H3CCH3

CH

H3C

H CS

S SS

O SiOSi

CH3

CH3

H3C

H3C CH3

octamethylcyclotetrasiloxane (OMCTS)ΔHp ~ 0ΔS 6 7 J/moleoK

S SSS

elemental sulfurΔHp = 13.5 kJ/moleΔS 31 J/ l oK


ΔSp = 6.7 J/mole K ΔSp = 31 J/moleoK

The Reactivity of Large Molecules

• In general, when considering growing polymer chains (i.e., regardless of the type of polymerization mechanism), the reactivity of the chain ends will be the primary focus in studying the kinetics of the polymerization reactionpolymerization reaction.

Thus, investigations of the kinetics of polymerization may be simplified by assuming that the rate constant of the chain growth reaction is independent of the size of the molecule to which the reactive functional group is attached.The validity of the assumption that the rate of polymerization is independent of changes in molecular size of the reactants may be rationalized by observing the behavior of several small molecule reactions.

• For reactions involving homologous series of reactants, the rate g gconstant levels off and becomes independent of molecular size when n > 2.

• Note that this behavior is quite analogous to step-wise polymerization.• Further physical rationalizations for the underlying assumption

include:1. The larger and heavier the molecule, the greater the distance between the

center of mass of the molecule and the reactive chain end. Thus, the mobility of the reactive end group in solution is much greater than the mobility of the molecular center of mass (i.e., the average mobility of the y ( g ytotal chain). This enhanced mobility of the reactive sites yields an "encounter rate" which is much greater than that predicted by the total molecular mass and is approximately independent of the molecular size.

2. In most polymerization reactions, the diffusion rate of reactants (i.e., the reactive chain ends and monomers) is much more rapid than the chemical


) preaction.

Dependence of kp on Molecular Size


The Reactivity of Large Molecules(continued)

• Consider the following kinetic scheme:

kk-1~~A + M ~~(A+M)

where A is the reactive site, M is a monomer, (A+M) represents the pair

( )

k2~~(A+M) P

( ) p pof reactants trapped in the "liquid cage", and P is the product polymer.

• The rate constants k1 and k-1 represent diffusion rate constants into and out of the liquid cage, while k2 is the rate constant for the chemical reaction. A i d i f h d h• Assuming a steady-state concentration of the trapped reactants, the rate of polymer formation is given by:

]][[][

21

21 MAkk

kkdtPd

+=

• If the diffusion is much more rapid than the chemical reaction, such that k-1>>k2, then:

21−

]][[][ 1 MAkkPd

• Since diffusion into the cage is affected by molecular size in the same way as diffusion out of the cage, the effect of molecular size cancels

t f th t i

]][[21

1 MAkkdt −

=


out of the rate expression.

Kinetics of Condensation (Step-Growth) Polymerization) y

• Step-Growth polymerization occurs by consecutive reactions in which the degree of polymerization and average molecular weight of the polymer increase as the reaction proceeds Usually (although notpolymer increase as the reaction proceeds. Usually (although not always), the reactions involve the elimination of a small molecule (e.g., water). Condensation polymerization may be represented by the following reactions:

iMonomer + Monomer Dimer + H2OMonomer + Dimer Trimer + H2O

Monomer + Trimer Tetramer + H2ODimer + Dimer Tetramer + H2ODi + T i P t + H2ODimer + Trimer Pentamer + H2OTrimer + Trimer Hexamer + H2O

• Generally, the reactions are reversible, thus the eliminated water must be removed if a high molecular weight polymer is to be formed.be removed if a high molecular weight polymer is to be formed.

• Based on the assumption that the polymerization kinetics are independent of molecular size, the condensation reactions may all be simplified to:

~~~~COOH + HO~~~~ → ~~~~COO~~~~ + H2O

• Note that there are many types of step-growth polymerization reactions which yield a wide variety of polymers including proteins, nylons, and polyesters. Although similar treatments apply to all step-


nylons, and polyesters. Although similar treatments apply to all stepgrowth polymerizations, this section will focus on the kinetics of polyesterification.

Kinetics of Condensation (Step-Growth) Polymerization) y

• Polyesterification reactions are catalyzed by acid and the mechanism is given by:

• Step 1: Fast Equilibriumk

• Step 2: Nucleophilic attack – slow, rate determining step

R-(C=O)OH + H+k1

k-1R-(C=OH⊕)OH

R' OH + R (C OH⊕)OH R C(OH) (OH ⊕) R'k2

• Step 3: Loss of water

R'-OH + R-(C=OH⊕)OH R-C(OH)2-(OH ⊕)-R'

R-C(OH)2-(OH ⊕)-R' R-(C=OH ⊕)-O-R' + H2Ok3

• Step 4: Regeneration of catalyst

• In this mechanism step 1 is a fast equilibrium and step 2 is the slow

R-(C=OH ⊕)-O-R' + H2O R-(C=O)-O-R' + H+k4

• In this mechanism, step 1 is a fast equilibrium and step 2 is the slow, rate-determining step, which follows the rate law:

-d [COOH]dt

= k2[R(C=OH ⊕)OH][R'-OH]

By applying the fast equilibrium assumption, the rate law becomes:

-d[COOH]= k K [R-(C=O)OH][R'-OH][H+]


dt= k2Keq1[R-(C=O)OH][R -OH][H ]

Polyesterification Without Acidic Catalysty

• In this case, the carboxylic acid groups must themselves function as the catalyst such that [H+] ≈ [COOH] and thus,

h k i l d k K d h f h id b

-d[COOH]dt

= kexp[COOH]2[OH]

where kexp includes k2, Keq1, and other constants of the acid-base equilibrium of the carboxylic acid.

• For a stoichiometric feed ratio of the reactants at the beginning of the reaction (t = 0),

such that [COOH] = [OH] at all times, and the rate equation

[RCOOH]o = [R'OH]o = 2[HOOC-R-COOH]o = 2[HOR'OH]o

such that [COOH] [OH] at all times, and the rate equation becomes

-d[COOH]dt

= kexp[COOH]3

which upon integration yields:

1 = 1 + 2k t


[COOH]2 =[ ]COOH 2

o+ 2kexpt

Polyesterification Without Acidic Catalyst (continued)y ( )

• Consider the fractional conversion of the polymerizationreaction, P, expressed in terms of the fraction of COOH groups(or OH groups) that have reacted:

Substitution into the integrated rate expression yields:

XCOOH = p = 1 -[COOH][COOH]o

or [COOH] = [COOH]o(1-p)

Substitution into the integrated rate expression yields:

1(1-p)2 = 1 + 2[ ] tkCOOH exp

2o

Note that experimental data for esterification reactions show thatplots of 1/(1-p)2 vs. time are linear only after ca. 80% conversion.This behavior has been attributed to the reaction mediumchanging from one of pure reactants to one in which the ester

d i h lproduct is the solvent.Thus, the true rate constants for condensation polymerizationsshould only be obtained from the linear portions of the plots (i.e.,the latter stages of polymerization).

• For example, the kinetic plots for the polymerization of adipicacid and 1,10-decamethylene glycol show that at 202oC, thethird-order rate constant for the uncatalyzed polyesterificationis k 1 75 10-2 (kg/eq i )2 min-1


is k = 1.75 x 10-2 (kg/equiv)2 min-1.

Uncatalyzed Polyesterification


Acid-Catalyzed Polyesterification

• Recall that the rate law from the acid catalyzed polyesterification is given by:

-d[COOH]

• If acid is added to the system, then by definition of a catalyst, the acid concentraion remains constant.

-d[COOH]

dt= k2Keq1[R-(C=O)OH][R'-OH][H+]

• Furthermore, at the stoichiometric feed, [RCOOH] = [OH], the rate expression becomes:

-d[COOH]

dt= kexp[COOH]2

and in terms of their fractional conversion of the reactive groups,

dt

11 p = 1 + kexp[COOH]ot

Thus a second-order plot of 1/(1-p) vs. time yields a linear relationship. Note that experimental data are usually linear only beyond ca.

i

1-p exp o

80% conversion. The polyesterification of adipic acid catalyzed by p-toluene sulfonic acid shows the the rate constant for reaction with 1,10-decamethylene glycol at 161 oC and 0.4% p-toluene sulfonic acid is k = 9.7 x 10-2 (kg/equiv) min-1.


Note that this rate constant is significantly larger than the noncatalyzed rate constant.

Catalyzed Polyesterification


Time Dependence of the Degree of Polymerizationy

• Consider a polyesterification of bifunctional monomers, at a stoichiometric feed ratio.

• In general, a polymer of (AB)n may be formed in the reaction:g , p y ( ) y

HO-(C=O)-R-(C=O)-OH + HO-R'-OH → HO-(C=O)-R-(C=O)-O-R'-OH + H2O

orHO-A-OH + H-B-H → HO-A-B-H + H2OHO A OH H B H → HO A B H H2O

where A and B are the structural units -(C=O)-R-(C=O)- and -O-R-O-, respectively.

• If water is efficiently removed during the reaction (which must wate s e c e t y e oved du g t e eact o (w c ustbe done to obtain high polymer), then the number of COOH groups present is equal to the number of molecules present, at all times.

( )pCOOHCOOHNo −== 1][][

where N is the total number of molecules in the system and V is the volume.

• Since the structural units A and B are never removed during

( )pV o][][

• Since the structural units A and B are never removed during the reaction, the total number of structural units present at all times is constant and equal to the number of initial molecules.

COOHN ][unitsstructural


oCOOHV

][unitsstructural =

The Number Average Molecular Weight in Polycondensationg y

• By defining the average degree of polymerization of the system, DP, as the average number of structural units per molecule, the relationship becomes:

• This relationship is known as the Carother's Equation.

PCOOHCOOHDP o

−==

11

][][

Note that for condensation polymers prepared from two reactants, the average number of repeating units per molecule is one-half the average degree of polymerization.

• If Mo is the average molecular weight of the structural units that make up a repeat unit (i e the average monomerthat make up a repeat unit (i.e., the average monomer molecular weight), then the number average molecular weight, Mn may be defined as:

1818 +=+⋅==∑∑ MMDP

mnM o

oxx

n

where nx is the moles of x-mer of mass mx, and 18 is added to account for the unreacted (HOH) groups at the ends of each polyester chain.

1−∑ Pn ox

n

• The following figure demonstrates the dependence of the number average molecular weight on the fractional conversion.

• Clearly, very high conversions are required in order to obtain f l l f l l i ht t th 10 000


useful polymers of molecular weights greater than 10,000.

Mn as a Function of Conversion


The Number Average Molecular Weight (continued)

• Using the kinetic relationships derived earlier, a dependence of the molecular weight on reaction time may be given by:

g ( )

( ) 18][21 212 ++= ktCOOHMM oon (uncatalyzed)

• For large reactions times (i.e., for conversions greater than

( ) 18][1 ++= ktCOOHMM oon(catalyzed)

For large reactions times (i.e., for conversions greater than 80%) the following approximations are reasonable.

ktCOOHMM oon 2][≈ (uncatalyzed)

ktCOOHMM oon ][≈ (catalyzed)


Molecular Weight Distributions of Linear Condensation Polymersy

• While the average degree of polymerization may be determined at any time t using the above relationships, it is equally important to know the distribution of molecular weights and the dependence of this distribution on reaction timethe dependence of this distribution on reaction time.

• A key question becomes:What is the probability that a molecule selected randomly from the polymerization mixture will contain exactly x structural units?

• Recall that:• Recall that:P = fraction of COOH groups that have reacted at time = t, and(1-P) = fraction of COOH groups remaining at time = t

• In other words:P = probability that at time = t a given COOH group will haveP probability that at time t a given COOH group will have reacted.(1-P) = probability that at time = t a given COOH group will nothave reacted.

• By considering the linear combinations of x structural units, the probability that the randomly selected molecule will contain exactly x structural units is given by:

Prob(x) = Px-1(1-P)

where (1-P) may be considered as the probability that the chain end will not have reacted, and P considered as the probability that the chain will have grown from x reactions (-1 for the end group).


Molecular Weight Distributions of (continued)

• The chance that a randomly selected molecule contains exactly x structural units is equal to the fraction of molecules composed of x-mers, such that

• where Nx is the number of x-mers in a system of N molecules.

Prob(x)=NNx

Thus, the relationship becomes:

• If the evolved water is completely removed during the

)1(1 PNPN xx −= −

polymerization, then

NCOOH = N = No(1-P)

• where No is the initial number of molecules. Thus,

Nx = No (1-P)2 Px-1

• As shown in the following Figure, for any given conversion P, low molecular weight polymers (i.e., the low values of x) have the highest probability of being formed in the total distribution.

• However, the distribution becomes broader and the average


, gmolecular weight increases as the conversion increases.

Effect of Conversion on the Number Distribution of Structural Units

Numerical distribution of the number of structural units in a condensation polymer for various conversions.



• Neglecting the weight of water on the terminal groups of the condensation polymer, the molecular weight of an x-mer is given by:

( )

Mx-mer = xMo

where Mo is the average molecular weight of the structural units.

• Thus, the number average molecular weight may be obtained from the probability function and the usual definition of an arithmetic average as given by:

∑∑=

−

=

−==oo N

x

xo

N

xon xPPMxxMM

1

1

1)1()(Prob



• The weight fraction of x-mers, Wx, may be defined as the total weight of molecules containing exactly x structural units divided by the total weight of polymer.

( )

• Again, the following Figure shows that this distribution of Wxfavors low molecular weight polymer at low conversions.

• In addition, the weight average molecular weight, Mw , may be defined as:

∑ ∑∑∑

= =− ===

o oN

x

N

xxo

x

xxw xWMWM

wmw

M1 1

merx

where, wx is the weight of x-mer of mass mx.

• In terms of the probability function, the weight fraction of x-In terms of the probability function, the weight fraction of xmers and the weight average molecular weight are given by:

12)1( −−= xx PPxW

and thus:

122)1( −∑−= xN

PxMPMo


1

)1(=∑=x

ow PxMPM

Effect of Conversion on the Weight Distribution of Structural Units



• Since P < 1, the summations appearing in the expressions for the number and weight average molecular weights may be evaluated as:

( )

1

)(for 1

1 2

1

1n

N

x

x

P

MP

xPo

+

⎟⎠⎞

⎜⎝⎛−

==

−∑

• Thus, the expressions for the average molecular weights become:

( ))(for

11

312

wx M

PPPx

−+

=−∑

⎟⎠⎞

⎜⎝⎛−+

=−

=PPMM

PMM ow

on 1

1 and 1

• Therefore simple combination shows that the weight average molecule weight is always greater than the number average molecular weight, such that:

• and for linear condensation polymers, the polydispersity is given by:

)1( PMM nw +=

M


g yP

MM

n

w += 1

Effect of Non-Stoichiometric Reactant Ratios

• In general, it is only possible to produce high molecular weight condensation polymers when equal concentrations of reactive functional groups are maintained throughout the course of the reaction.

The reaction must be initiated with stoichiometric ratios and the reaction mixture must be free of impurities that contain the same functional groups.

• For example, consider the A-A, B-B polymerization reaction:

A~~~~A + B~~~~B → A~~~~ab~~~~B + a’b’

where A and B are the reactive functional groups. • Furthermore, consider the feed ratio to contain an excess of

monomer B, such that:

Note that NAo = twice the number of initial AA molecules

1<=o

o

B

A

NN

r

Ao

• Since A and B must react on a 1:1 basis, and polymerization will be complete when all of A is gone, consider the conversion P = XA


P XA.

Effect of Non-Stoichiometric Reactant Ratios (continued)

• Thus, for bifunctional monomers, the total number of molecules at any time is given by:

( )

⎟⎞

⎜⎛ − rPNN 11

• With the degree of polymerization defined as the average number of structural units per molecule, the average degree of polymerization in terms of conversion and feed ratio is now

⎟⎠⎞

⎜⎝⎛ +−=

rPNN

oA 21

polymerization in terms of conversion and feed ratio is now given by:

( ) rPrrDP21

1−++

=

• Note that at a stoichiometric ratio r = 1 the above relationship reduces to the previous form:

1

• In addition, the maximum average degree of polymerization ibl d t l t i f th A

PDP

−=

11

possible corresponds to a complete conversion of the A groups (i.e., P = 1), such that:

( ) ⎟⎞

⎜⎛ +

=+

=rMMrDP 1and1


( ) ⎟⎠

⎜⎝ −− r

MMr

DP on 1 and

1 maxmax

Branched and Cross-Linked Condensation Polymersy

• Bifunctional monomers yield linear polymers; however, if one of the reactants is a tri- or multifunctional monomer, then a branched or crosslinked polymer will result.

The degree of crosslinking or branching in these systems may be controlled by the relative amounts of multifunctional and difunctional monomers.

• In a system containing equivalent numbers of A and B groups, the number of monomer units present initially is No and the corresponding number of functional groups is NoFavg,

where Favg is the average functionality of the monomers as defined by:

• where Ni is the number of molecules of monomer i with

∑∑=

i

iiavg N

FNF

where Ni is the number of molecules of monomer i with functionality Fi.

• For example, if a system contains 2 mol of a triol and 3 mol of a diacid then F = [2(3)+3(2)]/(2+3) = 12/5 or 2 4a diacid, then Favg = [2(3)+3(2)]/(2+3) = 12/5 or 2.4.

• If N is the number of molecules after the reaction has occurred, then 2(No - N) is the total number of functional groups (e.g.,


COOH's and OH's) that have reacted.

Branched and Cross-Linked Condensation Polymers (continued)

• Therefore, the extent of reaction P may also be defined as the fraction of total functional groups reacted, as given by:

y ( )

NN )(2 −

• Since the degree of polymerization is given by:

avgo

o

FNNNP )(2

=

• combination yields:

NNDP o=

combination yields:

avgPFDP

−=

22

• which may be rearranged to yield the familiar (and general) form of the Carothers equation:

P 22−=

• which relates the extent of reaction and degree of l i ti t th f ti lit f th t

avgavg FDPFP −=


polymerization to the average functionality of the system.

Branched and Cross-Linked Condensation Polymers (continued)

• This general form of the Carothers equation allows the possibility of calculating the conditions needed to avoid or ensure the reaching of the gel point (i.e., the point of extensive

y ( )

crosslinking). • Since gelation is presumed to occur when the average degree of

polymerization becomes infinitely large, the above relationship reduces to:

h P i th iti l i

avgc F

P 2=

• where Pc is the critical conversion.

• In practice, it is important to note that this approach often overestimates the reaction point at which gelation occurs.

• This overestimation is attributed to the broad molecular weight distribution in which the high molecular weight molecules reach the gelation point before those which have the average value of the molecular weight.


Polymer Kinetics

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