Dilatometry is used to determine the rate of styrene polymerization and the effects of initiator concentration and chain transfer agent on the rate.

Radical chain polymerization is a chain reaction consisting of a sequence of three steps: initiation, propagation, and termination. The initiation step is considered to involve two reactions. The first is the production of free radicals by any one of a number of reactions. The usual case is the homolytic dissociation of an initiator species I to give a pair of radicals R^.

where k_d is the rate constant for the initiator dissociation. The second step of initiation involves the addition of the initiator radical to the first monomer molecule to produce the "real" chain initiating species M₁^.

As propagation continues and each monomer unit is added, the radical has the same identity as the radical before except that it is larger by one unit. Therefore, equation (2) becomes

Propagation with growth of the chain to higher molecular weight polymer takes place very rapidly. But at some point the propagating radical at the end of the polymer chain stops growing and terminates. Termination of the radical centers occurs by bimolecular reaction between two radicals. They react with each other by coupling or by disproportionation. The two different modes of termination can be expressed by

and

where k_tc and k_td are the rate constants for termination by coupling and disproportionation, respectively.

In order to derive kinetic equations for these reactions certain approximations are made. The first is that the rate of initiation, R_i, is the same as the rate of initiator decomposition, R_d, and from equations (1) and (2)

where f is the mole fraction of initiator radicals formed which actually add to monomer and initiate polymerization. This is called the initiator efficiency, and the factor 2 takes into account that 2 I^. are derived from each initiator molecule.

The rate of termination can be expressed kinetically by:

and here, the 2 denotes that two radicals combine in the termination step.

The next assumption implemented in this derivation assumes that the concentration of free radicals quickly reaches a value which does not change substantially. This assumption is known as the steady state assumption, and can be expressed as

or

The radical concentration can be expressed in terms of measurable quantities:

The overall rate of polymerization (R_p) can be considered as the rate of disappearance of monomer with respect to time, -d[M]/dt. This depletion is due to both the initiator - monomer reaction and the propagation reaction:

Replacing the radical concentration term with Eq. (10)

Therefore, the rate of polymerization is dependent on the monomer concentration and the square root of the initiator concentration. If one were to double [M], the rate would double, but doubling [I] would only increase the rate by a factor of 2^1/2=1.414.

There is an effect found in many polymerizations called the gel effect or Trommsdorff effect which can cause an autoacceleration of the reaction and complicate the kinetic study of polymer formation. This involves an increase in viscosity which leads to a decrease in the rate of termination, since the bulky growing polymer radical cannot diffuse easily through the medium. Thus, the possibility for two polymer radicals to approach each other and participate in a termination process becomes lessened. However, the smaller monomer molecules can diffuse much more readily to the polymer radical and the propagation reaction can continue. In the polystyrene system, autoacceleration is usually not important below 10% conversion, and this system will be done below this conversion. Another reason for measuring the polymerization rate at low conversions is that during the early stages of the reaction, the initiator and monomer concentrations remain relatively constant, and you don't have to worry about measuring changes in these values.

Molecular weight can be moderated through the addition of chain transfer agents such as an alkylmercaptan. One which is widely used is dodecyl mercaptan. This molecule readily transfers a hydrogen from the sulfur to a carbon radical:

The various relations of the rate constants k_tr, k_i, and k_p and how they affect the rate of polymerization can be summarized as:

1.	kp >> ktr	and	ki` ~ kp	no change in Rp
2.	kp >> ktr	and	ki` < kp	decrease in Rp
3.	kp << ktr	and	ki` < kp	large decrease in Rp
4.	kp << ktr	and	ki` ~ kp	no change in Rp
5.	kp >> ktr	and	ki` > kp	no change in Rp
6.	kp << ktr	and	ki` > kp	no change in Rp

There is a characteristic chain transfer constant (C_tr) for each chain transfer agent - monomer system defined as:

In this experiment the rate of polymerization will be measured by the use of dilatometry. Dilatometry utilizes the volume change that occurs upon polymerization to follow conversion versus time. The conversion is conveniently followed in a dilitometer whose volume includes a capillary region. The dilatometer is placed in a constant temperature bath and the volume change of the polymerizing system, which is quantitatively related to the percent conversion, is followed with time. Dilatometry is not useful for most step polymerizations where there is a small molecule by-product that results in no appreciable volume change upon polymerization.

As the dilatometer is placed into the constant temperature bath, initial meniscus movement is due to two factors:

1. thermal expansion of the styrene monomer

2. contraction due to polymerization.

After approximately 5-10 minutes the effects due to thermal expansion become negligible. If the capillary cross-section area is determined, usually in terms of volume as a function of length, the change in height of monomer in the capillary may be expressed as a volume change. Thus the slope of a plot of meniscus height versus time would give us DH/Dt, which can easily be converted to a volume to give DV/Dt. Some dilatometers have the capillary calibrated in volume increments, thus DV/Dt is directly accessible.

The total fractional change in volume corresponds to complete conversion to polymer of density d₂ from W₁ grams of monomer of density d₁. The weight of the polymer would also be W₁, since in an addition polymerization, no weight is gained or lost. For the total fractional change in volume, this gives:

which simplifies to:

The degree of monomer conversion would then be:

where D[M] is the incremental change in monomer concentration [M] and DV is the change in volume from the initial volume V_o. This is true, since the term (d₂-d₁)/d₂ is the fractional volume change which would occur at 100% conversion and DV/V_o is the fractional volume change at any time Dt. The ratio of these two quantities should give the fraction of conversion. Note that the quantity is dimensionless, so that D[M] and [M] could be in units of grams, moles, or molar quantities since the units cancel out. Now if both sides of Equation 17 are divided by Dt, incremental time, and rearranged, then

and

The value -d[M]/dt has been previously defined as the overall rate of polymerization, R_p. Equation 13 can be tested by varying the initiator concentration. Additionally, this experiment will test the effect of a chain transfer agent, dodecyl mercaptan, on the rate of the polymerization.

Experiment

Calculations

1. Prepare a table listing time (t) and a measure of capillary volume, either direct volume (V) or liquid height in the capillary (H) for each run.

2. Make a graph of either V vs t or H vs t, depending on which quantity was measured, and take the slope of the straight portion of the line. If the graph is V vs t, the slope will be DV/Dt. If the graph is of DH/Dt, this quantity can be changed to DV/Dt by the use of the calibration constant in ml/cm.

3. Using Equation A we can determine 3. D[M]/ Dt since we have DV/Dt from the previous step. d1 and d2 are the densities of styrene and polystyrene respectively (0.860 and 1.046 g/m; respectively at 70oC), and the term [M]/Vo can be calculated where Vo is the original volume (bulb + capillary) and [M] is the molar concentration of styrene. If we assume that the amount of DDM does not affect the concentration of styrene in bulk styrene, we can use the density (0.860 g/ml) and the to get the concentration in mol/l .

4. The overall rate of polymerization should be calculated for all three samples so that the effect of initiator concentration of the rate may be verified (samples 1 and 2) and also the effect of a chain transfer agent on the rate may be investigated (samples 1 and 3).