Thermodynamics of Polymerization

1
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 ? bonds
  into ? 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 polymerizable under all conditions.
• The well known thermodynamic expression
• ?G ?H - T?S
• yields the basis for understanding
  polymerization/depolymerization behavior.
• For polymerization to occur (i.e., to be
  thermodynamically feasible), the Gibbs free
  energy of polymerization ?Gp
• If ?Gp O, then depolymerization will be
  favored.
• Thus Any factor that lowers the enthalpy, H
  (i.e., makes ?Hp more negative), or raises the
  entropy, S (i.e., makes ?S more positive), will
  shift the equilibrium towards polymerization.
• Standard enthalpy and entropy changes, ?Hop and
  ?Sop are reported for reactants and products in
  their appropriate standard states. Generally
• Temperature 25oC 298K
• Monomer pure, bulk monomer or 1 M solution
• Polymer 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 ?Sp processes.
• Since depolymerization is almost always
  entropically favored, the ?Hp must then be
  sufficiently negative to composite for the
  unfavorable entropic term. Only then will
  polymerization be thermodynamically favored by
  the resulting negative ?Gp.

2
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 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.
• At long chain lengths, the chain propagation
  reaction
• is characterized by the following equilibrium
  expression
• The standard-state enthalpy and entropy of
  polymerization are related to the standard-state
  monomer concentration, Mo (usually neat liquid
  or 1 M solution) as follows

3
Thermodynamics of Polymerization (continued)

• At equilibrium, ?G 0, and T Tc (assuming that
  ?Hpo and ?Spo are independent of temperature).
• Or
• Or
• At Mc Mo, Tc ?Hpo/?Spo

4
Thermodynamics of Polymerization(continued)

• Other possible effects on ?Hp
• loss of resonance stabilization upon
  polymerization
• changes in bond hybridization
• changes in H-bonding between M and P states
• Notice the small changes in the ?Sp 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.
• ?Go -RT lnKeq
• As the temperature increases, the equilibrium
  constant decreases (characteristic of an
  exothermic reaction). When Tc 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 occur well below
  the predicted Tc. In fact, few polymers actually
  match their thermal stability as predicted from
  the Tc approach.

5
Experimental Determination of ?Hop and ?Sop

• ?Hop - 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
  ?Hof (enthalpy of formation) of M and P. The
  ?Hop is thus obtained by the relationship
• ?Sop - from the absolute entropies of M and P,
  such that
• The absolute entropies may be obtained from
  calorimetric measurements of heat capacities of M
  and P over a wide T range, as given by

6
Floor Temperature Behavior

• Although the vast majority of all polymerizations
  possess negative ?H and ?S, and hence display
  ceiling temperature behavior, four distinct
  possibilities exist as outlined in the table
• As stated earlier, -?S for polymerization is
  almost universal.
• Therefore, for olefins and small cyclics,
  polymerization is possible at low temperatures.
• However, many compounds are never spontaneous
  toward polymer due to ?H (e.g. cyclohexane,
  tetrasubstituted olefins)
• ?S for polymerization is rare, but known
  examples exist (see below).
• This rare behavior leads to floor temperature
  behavior or entropy-driven polymerizations.
• Floor 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.
• Examples of monomers possessing a floor
  temperature

7
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 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 constant 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 total 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 reaction.

8
Dependence of kp on Molecular Size
9
The Reactivity of Large Molecules(continued)

• Consider the following kinetic scheme
• where A is the reactive site, M is a monomer,
  (AM) represents the pair of 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.
• Assuming a steady-state concentration of the
  trapped reactants, the rate of polymer formation
  is given by
• If the diffusion is much more rapid than the
  chemical reaction, such that k-1k2, then
• 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
  out of the rate expression.

10
Kinetics of Condensation (Step-Growth)
Polymerization

• 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 not always), the reactions involve the
  elimination of a small molecule (e.g., water).
  Condensation polymerization may be represented by
  the following reactions
• Monomer Monomer Dimer H2O
• Monomer Dimer Trimer H2O
• Monomer Trimer Tetramer H2O
• Dimer Dimer Tetramer H2O
• Dimer Trimer Pentamer H2O
• Trimer 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.
• 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-growth polymerizations, this
  section will focus on the kinetics of
  polyesterification.

11
Kinetics of Condensation (Step-Growth)
Polymerization

• Polyesterification reactions are catalyzed by
  acid and the mechanism is given by
• Step 1 Fast Equilibrium
• Step 2 Nucleophilic attack slow, rate
  determining step
• Step 3 Loss of water
• Step 4 Regeneration of catalyst
• In this mechanism, step 1 is a fast equilibrium
  and step 2 is the slow, rate-determining step,
  which follows the rate law
• By applying the fast equilibrium assumption, the
  rate law becomes

12
Polyesterification Without Acidic Catalyst

• In this case, the carboxylic acid groups must
  themselves function as the catalyst such that
  H ? COOH and thus,
• 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 becomes
• which upon integration yields

RCOOHo R'OHo 2HOOC-R-COOHo 2HOR'OHo
13
Polyesterification Without Acidic Catalyst
(continued)

• Consider the fractional conversion of the
  polymerization reaction, P, expressed in terms of
  the fraction of COOH groups (or OH groups) that
  have reacted
• Substitution into the integrated rate expression
  yields
• Note that experimental data for esterification
  reactions show that plots of 1/(1-p)2 vs. time
  are linear only after ca. 80 conversion.
• This behavior has been attributed to the reaction
  medium changing from one of pure reactants to one
  in which the ester product is the solvent.
• Thus, the true rate constants for condensation
  polymerizations should 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 adipic acid and
  1,10-decamethylene glycol show that at 202oC, the
  third-order rate constant for the uncatalyzed
  polyesterification is k 1.75 x 10-2 (kg/equiv)2
  min-1.

14
Uncatalyzed Polyesterification
15
Acid-Catalyzed Polyesterification

• Recall that the rate law from the acid catalyzed
  polyesterification is given by
• If acid is added to the system, then by
  definition of a catalyst, the acid concentraion
  remains constant.
• Furthermore, at the stoichiometric feed, RCOOH
  OH, the rate expression becomes
• and in terms of their fractional conversion of
  the reactive groups,
• 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. 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.

16
Catalyzed Polyesterification
17
Time Dependence of the Degree of Polymerization

• Consider a polyesterification of bifunctional
  monomers, at a stoichiometric feed ratio.
• In general, a polymer of (AB)n may be formed in
  the reaction
• HO-(CO)-R-(CO)-OH HO-R'-OH ?
  HO-(CO)-R-(CO)-O-R'-OH H2O
• or
• HO-A-OH H-B-H ? HO-A-B-H H2O
• where A and B are the structural units
  -(CO)-R-(CO)- and -O-R-O-, respectively.
• If water is efficiently removed during the
  reaction (which must be done to obtain high
  polymer), then the number of COOH groups present
  is equal to the number of molecules present, at
  all times.
• 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 the reaction, the total number of
  structural units present at all times is constant
  and equal to the number of initial molecules.

18
The Number Average Molecular Weight in
Polycondensation

• By defining the average degree of polymerization
  of the system, Xn, as the average number of
  structural units per molecule, the relationship
  becomes
• This relationship is a special case of the
  Carother's Equation.
• 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, then the number average
  molecular weight, Mn may be defined as
• 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.
• 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 useful polymers of molecular
  weights greater than 10,000.

19
Mn as a Function of Conversion
20
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
• For large reactions times (i.e., for conversions
  greater than 80) the following approximations
  are reasonable.

(uncatalyzed)
(catalyzed)
(uncatalyzed)
(catalyzed)
21
Molecular Weight Distributions of Linear
Condensation Polymers

• 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 time.
• Given a reacting system composed of an A-B type
  monomer, we wish to define the number fraction of
  molecules, at a given conversion, p, which
  contain exactly x structural units. A key
  question becomes
• What is the probability that a molecule selected
  randomly from the polymerization mixture will
  contain exactly x structural units?
• p conversion fraction of COOH groups that
  have reacted at time t, and
• (1-p) fraction of COOH groups remaining at time
  t
• Thus, the probability of obtaining the molecule
  shown above is given by
• Prob(x) px-1(1-p)
• 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

(1)
(2)
22
Molecular Weight Distributions of (continued)

• where Nx is the number of x-mers in a system of N
  molecules. Thus, the relationship becomes
• Therefore, we can see that Prob(x) is the mole
  fraction of molecules containing x structural
  units
• If the evolved water is completely removed during
  the polymerization, then
• NCOOH N No(1-p)
• where No is the initial number of molecules.
  Combining eqs. (3) and (4) yields
• 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 molecular weight increases as the
  conversion increases.

(3)
(4)
(5)
23
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.
24
Molecular Weight Distributions of (continued)

• The number average molecular weight is obtained
  from Prob(x) and the definition of an average.
  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, we have
• Now, it can be shown that for p 1,
• Combining eqs. (8) and (9) yields

(6)
(7)
(8)
(9)
(10)
25
Molecular Weight Distributions of (continued)

• 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
• The following is true for p 1
• Combination of eqs. (11) and (12) yields the
  simplification
• Again, the following Figure shows that this
  distribution of Wx favors low molecular weight
  polymer at low conversions.
• In addition, the weight average molecular weight,
  Mw , may be defined as
• In view of eq. (11) we have

(11)
(12)
(13)
(14)
(15)
26
Effect of Conversion on the Weight Distribution
of Structural Units
27
Molecular Weight Distributions of (continued)

• Combination of eqs. (10) and (15) shows that the
  polydispersity is given by

28
Effect of Non-Stoichiometric Reactant Ratios

• The highest possible molecular weight is achieved
  in polycondensation reactions using equal
  concentrations of reacting groups.
• However, it is often desirable to produce a
  specific molecular weight in polymerization.
  This is accomplished by designing the system so
  that unreacted or unreactive end groups are
  incorporated into the polymer. Since molecular
  weight is inversely proportional to the number of
  end groups, this offers a means for molecular
  weight control. We will consider three types of
  systems.
• Type 1 A system of A-A and B-B monomers in
  which the total number of A functional groups,
  NA, is less than (or equal to) the total number
  of B functional groups, NB. We define a
  stoichiometric imbalance parameter, r, where,
• In this situation, reaction proceeds until the A
  groups are completely consumed and all the chain
  ends possess unreacted B groups. It is obvious
  that the greater the stoichiometric imbalance,
  the more leftover B groups there will be, and the
  lower the molecular weight.
• Type 2 A system of A-A and B-B monomers in
  which molecular weight control is achieved by the
  addition of small amounts of a monofunctional
  monomer containing either a single A or B group.
• Type 3 A system of A-B monomers in which
  molecular weight control is achieved by addition
  of small amounts of mono- and/or polyfunctional
  monomers containing only A or only B groups.

29
Effect of Non-Stoichiometric Reactant Ratios
(continued)

• For all types of systems, the polymerization can
  be designed to yield the desired through the use
  of the Carothers Equation. The key to this
  method is the concept of number average
  functionality, favg. To compute favg, one must
  first identify which is the minority or deficient
  type of group, A or B. Thus for the case in
  which the A groups are deficient in number,
• where,
• N(A)s the number of moles of each type of
  monomer carrying an A group
• f(A)s functionality of each type of monomer
  carrying an A group
• Ns the number of moles of each type of
  monomer present (A and B)
• The Carothers Equation is
• or
• where,
• p the fractional conversion of the deficient
  groups
• the number average degree of
  polymerization

30
Effect of Non-Stoichiometric Reactant Ratios
(continued)

• For Type 1 systems, the total number of molecules
  at any time is given by
• 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 given by
• Note that at a stoichiometric ratio r 1 the
  above relationship reduces to the previous form
• In addition, the maximum average degree of
  polymerization possible corresponds to a complete
  conversion of the A groups (i.e., p 1), such
  that

31
Effect of Non-Stoichiometric Reactant Ratios
(continued)

• For Type 2 systems (with NA NB NB)
• where, NB number of B groups contributed by a
  monofunctional monomer, and
• For Type 3 systems (with NA NB)
• where, NBf number of B groups contributed by a
  polyfunctional monomer, and f functionality of
  polyfunctional monomer, and,
• All other cases should be treated using favg and
  the general Carothers Equation.

32
Branched and Cross-Linked Condensation Polymers

• Mono and 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 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
  crosslinking).
• Since gelation is presumed to occur when the
  average degree of polymerization becomes
  infinitely large, the Carothers equation reduces
  to
• 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.