Continuous Addition Polymerization

http://www.e-polymers.org e-Polymers 2005, no. E_003. ISSN 1618-7229

Uwe Beginn

DWI/ITMC-TexMC, RWTH-Aachen, Germany [email protected]

∆F = F (f ) - F (p )) 2.0 1 1 1,0 1 max -1.000 f = 0.5 -0.9000 1.8 1,0 -0.8000 -0.7000 -0.6000 p = 0.9 -0.5000 max -0.4000 1.6 -0.3000 -0.2000 -0.1000 0 1.4 0.1000 0.2000 0.3000 0.4000 0.5000 1.2 0.6000 0.7000 0.8000 0.9000 1.0 1.000

2 1.010 r 0.8 0.6 azeotropic line 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 r 1

(Received: July 1, 2005; published: July 6, 2005)

PD Dr. Uwe Beginn RWTH-Aachen, ITMC / TexMC Worringerweg 1 D-52056 Aachen Germany

E-mail: [email protected]

2 CONTENT

1. Introduction 5

2. Free Radical Homopolymerization 8 2.1. Reaction Mechanism of Free Radical Polymerization 8 2.2. Reaction Kinetics of Free Radical Polymerization 9 2.3. Continuous Addition Batch Homopolymerization 14 2.3.1 Continuous Addition at Constant Monomer Concentration 17 2.3.2 Continuous Addition at Constant Rate of 20 Polymerization 2.3.3 Continuous Addition at Negligible Dilution 23 2.4 Effect of an Erroneous Volume Addition Rate 25 2.5 Continuous Addition of Initiator 29 2.5.1 At Constant Specific Rate of Initiation 29 2.5.2 At Constant Total Rate of Initiation 30

3. Radical Copolymerization 32 3.1. Copolymer composition 32 3.2. Rate of Copolymerization 37 3.3. Continuous Addition Batch Copolymerization 42 3.3.1 Necessity of Continuous Batch Copolymerization 42 3.3.2 Applicability of Homopolymerization Continuous 50 Addition Rate Equations to Copolymerizations 3.3.3 Addition of One Mixed Monomer Feed solution 52 3.3.3.1 At Constant Specific Rate of Copolymerization 52 3.3.3.2 At Constant Total Rate of Copolymerization 54 3.3.4 Addition of Two Separate monomer solutions 56 3.3.4.1 At Constant Specific Rate of Copolymerization 56 3.3.4.2 At Constant Total Rate of Copolymerization 59 3.3.5 Constant Addition at Negligible Dilution 60 3.3.6. n-Monomer Feed with a Mixed Monomer Solution 61

3.3.7 Effect of Erroneous Monomer Addition Rates 62 3.3.7.1 Sensitivity of the Copolymer Composition 62 3.3.7.2 Changes in Monomer Mixtures with Small Addition 66 Errors 3.3.7.3 Changes in Monomer Mixtures with Any Addition Error 72 3.3.7.3.1 Poly(dimethylsiloxane)mono methacrylate / Maleic 73 Anhydride 3.3.7.3.2 Maleic Anhydride / Perfluorooctyl methacrylate 77 3.3.7.3.9 Styrene / Methyl acrylate 78

4. Empirical Rate Expressions of Comonomer Pairs 81 4.1. The System PDMS1k-MA / MSA 81 4.2. The System MSA / EA 83 4.3. The System MSA / AN 85 4.4. The System MSA / F8H2MA 87 4.5. The System Styrene / MA 89 4.6. The System Styrene / MMA 91

5. The Program COPODOS.BAS 93 5.1. System Description 93 5.2 Using COPODOS 96

6. COPODOS Example Recipes 98 6.1. COPODOS Recipes for P[PDMS1k-MA-co-MSA] 98 6.1.1 P[PDMS1k-MA0.95-co-MSA0.05] 98 6.1.2 P[PDMS1k-MA0.90-co-MSA0.10] 99 6.1.3 P[PDMS1k-MA0.80-co-MSA0.20] 100 6.1.4 P[PDMS1k-MA0.70-co-MSA0.30] 101 6.1.5 P[PDMS1k-MA0.60-co-MSA0.40] 102 6.1.6 P[PDMS1k-MA0.55-co-MSA0.45] 103 6.2. COPODOS Recipes for P[MSA-co-EA] 104 6.2.1 P[MSA0.05-co-EA0.95] 104 6.2.2 P[MSA0.10-co-EA0.90] 105 6.2.3 P[MSA0.20-co-EA0.80] 106 6.2.4 P[MSA0.30-co-EA0.70] 107 6.2.5 P[MSA0.35-co-EA0.65] 108

7 References 109

4 1. Introduction

Free- and controlled radical polymerizations have become standard reactions in the research laboratory to prepare a large variety of homo- and copolymers. Hence, the synthetic polymer chemist is frequently asked to supply a colleague or a project partner with 10 - 20 g of some (co)polymer. The usual mode of action is the "batch polymerization": Solvent, initiator and monomers are mixed and heat under an inert atmosphere until (almost) full conversion of the monomer has been achieved. Sometimes the researcher is not aware the fact, that this simple procedure is ineffective when used to prepared larger quantities of homopolymers and it will produce a messy product in many copolymerization reactions. The reason for the first statement is simple: The rate of polymerization is proportional to the monomer concentration. In closed reaction systems the monomer concentration will continuously decrease,- and so does the rate of polymerization. If it takes a certain time t(p=0.25) to achieve 25% monomer conversion, it will take more than the double of this time to go for 50%. Aiming for 99% conversion will cost 16 - fold the time t(p=0.25) (cf. Table 1). In particular for controlled radical polymerizations this argument can be of importance, because the yet known systems are not completely free of side reactions. The relative importance of side reactions, like termination or transfer, becomes more pronounced with decreasing rate of polymerization.

Table 1: Time to reach a certain monomer conversion, related to 25% conversion in radical homo- and copolymerizations.

monomer conversion p t(p) / t(p=0.25) 0.25 1.00 0.50 2.41 0.75 4.81 0.90 8.00 0.95 10.41 0.99 16.00

With copolymerizations the second statement becomes important, whenever the copolymerization behaviour is not perfectly random. Since the more reactive monomer becomes faster incorporated in the copolymer, the remaining monomer mixtures en- riches in the less reactive monomer. The instantaneous copolymer composition only depends on the relative ratio of the monomers in the reaction mixture. Hence, the copolymers composition will permanently change during the reaction time. The result is a mixture of macromolecules with different monomer compositions. To illustrate this effect, Figure 1 depicts the compositional distribution that arises from batch copolymeri-

5 zation of the monomer pair maleic anhydride / methyl acrylate, characterized by the copolymerization parameters r1 = 0, r2 = 4.5. It is clearly seen that the composition average - as obtained from elemental analysis or NMR spectroscopy - does not really tell the full truth. While e.g. the analysis states the product to contain 12 mol% monomer 1 (Figure 1: green curve) the copolymer consists of a mixture of macromolecules containing all molar fractions of monomer 1 between 4 and 50 mol%.

Figure 1: Compositional distribution of copolymers produced by maximum conversion batch copolymerization of two monomers characterized (0) by the copolymerization parameters r1 = 0, r2 = 4.5 (f 1 = initial concentration of monomer 1 in the reaction mixture, = average content of monomer 1 in the copolymer as obtained from elemental analysis of NMR spectroscopy).

The way out of these problems is "continuous addition polymerization". This term indicates that the converted monomers must continuously be replaced by adding a feed solution during the polymerization reaction. The technique is well established in the industrial production of copolymers, but is merely unknown to the laboratory researcher. This is partially, because the relevant information is "hidden" in journals and text books on technical chemistry or reaction engineering,- a literature not frequently consulted by the typical preparative working organic polymer chemist. Moreover the theory of technical continuous addition copolymerization is adapted to technical processes: Assuming large qu- antity vessels, complicated heat and mass transfer problems,

6 the presence of on-line analytics and a high degree of automa- tion. Things can be made easier in the laboratory, since it is frequently the aim to prepare 10 - 100 g of a copolymer of uniform composition. This task can be managed with the typical organic synthetic equipment together with a pump that is able to add a volume of 50 - 400 mL liquid over a period of some hours. An expensive on-line analytic is not required, because even a continuous addition batch polymerization with a slightly wrong addition rate gives by far better copolymers than the simple batch polymerization. This text will give a short introduction to the reaction mechanism and reaction kinetics of homo- and copolymerization reactions. On this basis it will analyse the theoretical basis for small - scale "continuous addition polymerization" reactions. The aim is to develop equations and set up a methodology to perform these experiments in the laboratory with minimal efforts. All required equations will be cast in a computer program, so that even a complicated formula cannot terrify the bold reader of this text.

Aachen, June 2nd, 2004

7 2. Free Radical Homopolymerization

2.1. Reaction Mechanism of Free Radical Homopolymerization

To perform a free radical batch homopolymerization a mixture of solvent, monomer and initiator is heated under inert atmosphere to a temperature, were the initiator decomposes in- to radicals according to Scheme 1 - 1.1 During the whole reaction time the initiator decomposition keeps on, continuously supplying the reaction mixture with new radicals. The radicals will start kinetic chain reactions that finally produce polymer chains: When an initiator radical R• meets a monomer molecule it adds to the monomers double bond, creating the chain link of a growing chain radical. This is the "chain initiation" step (cf. Scheme 1 - 2). In subsequent "propagation" steps (Scheme 1 - 3) the growing chain radical adds further monomer units, hence the resulting chain of covalently attached monomers - the macromolecule - becomes longer and longer. When two chain radicals meet (Scheme 1 - 4) they will immediately react with each other, creating non-reactive macromolecules that cease growing. The relative frequency of this "termination step" mainly determines the number of monomer units that can make up the macromolecule. k 1) Initiator decomposition: I →d 2 ⋅ f ⋅ R •

k • i • 2) Initiation: R + M → P1

k • p • 3) Propagation: Pi + M → Pi+1

k • • t 4) Termination: Pi + Pj → dead Polymer

k • tr • 5a) Transfer: Pi + T → dead Polymer + T

k • re-ini • 5b) Re- initiation: T + M → P1

Scheme 1: Generalized reaction mechanism of a free radical homopolymerization reaction

It is also possible for a growing chain radical to react with any other species T present in the reaction mixture (e.g. solvent, monomer, initiator, and other polymer chains). By ab- stracting an atom and leaving a radical function at the reaction partner T the radical function is transferred from the

8 polymer chain P• to T (Scheme 1 - 5a). Hence, this class of reactions are subsumed under the term "transfer reactions".2 The saturated polymer chain can no longer grow, but the kinetic reaction chain may proceed, if the radical T• is able to re- start a new growing chain by adding to a monomer molecule (Scheme 1 - 5b). If the re-initiation is fast compared to the propagation, the rate of polymerization remains unaltered, but the average length of polymer chain decreases. In this case T is called as transfer agent. Note that transfer reactions from a growing polymer chain to another macromolecule will cause branching, e.g. the architecture of the final product will deviate from the strictly linear arrangements of the monomer units in the polymer. With T• radicals that cannot re-initiate a new polymer chain, the radical function will be lost and the polymerization will be stopped until T is completely consumed. Such special transfer agents are named "inhibitors". In between transfer agents and inhibitors one finds "retarders",- compounds that form T• radicals by transfer reactions with growing chain radicals that slowly re-initiate and hence slow down the whole polymerization reaction.

2.2. Reaction Kinetics of Free Radical Homopolymerization

The subsequent treatment of the reaction kinetics of free radical homopolymerization reactions is limited to polymerization in diluted monomer solutions (i.e. monomer concentrations below 1 - 3 mol/L) and low monomer conversions (monomer conversion below 30 %). At larger conversions the polymer content of the reaction mixture will significantly alter the solution viscosity and modify the reaction kinetics, by strongly redu- cing the rate of termination. Furthermore a large polymer content can cause the vitrification of the reaction mixture or cause phase separation phenomena, e.g. liquid / liquid demixing or precipitation. Such circumstances should be avoided in polymer synthesis and will not be discussed here. Note that emulsion polymerizations and some precipitation polymerizations follow special kinetic laws that also are not covered by the present description. Experimentally it is easy to observe the rate of monomer consumption dc/dt (c = monomer concentration, t = reaction time). Referring to the reactions in Scheme 1, the kinetic expression for dc/dt is:3

(c = monomer concentration, [R•] = radical concentration from decom- • • posed initiator, [Pi ] = concentration of growing chain radicals, [T ] = transfer agent radicals)

9 If the degree of polymerization is sufficiently large (Xn >> 10) the monomer consumption caused by initiation and re- initiation steps is negligible, hence the monomer consumption becomes identical to the rate of polymerization:

dc = R = k ⋅[P• ]⋅ c Eq. 2) dt P P i

It is well established experimentally that the concentra- • tion of growing chain radicals [Pi ] becomes constant a few se- conds after the start of the reaction and the reaction runs under "steady state conditions". In the steady state the rate of radical formation is identical to radical consumption due to termination reaction. Hence:

• 2 k t ⋅[Pi ] = 2 ⋅ f ⋅ k d ⋅ c I Eq. 3)

• 2 ⋅ f ⋅ k d ⋅ c I [Pi ] = Eq. 4) k t

• (cI = initiator concentration, [Pi ] = concentration of growing chain radicals, f = radical efficiency factor, kd = initiator decomposition constant, kt = termination constant)

Inserting Eq. 4) in Eq.2) yields the experimentally observed rate of polymerization:3

dc k P − = R P = ⋅ 2 ⋅ f ⋅ k d ⋅ c ⋅ c I Eq. 5) dt k t

In batch polymerizations the steady state conditions hold up to a monomer conversion of 10 - 30 %. At larger conversions the polymer content of the reaction mixture will significantly alter the solution viscosity and modify the reaction kinetics considerably. Since Eq. 5) holds only for low conversions, it is common to speak of the initial rate of polymerization, Rp0. From a practical point of view Eq. 5) is simplified by packing together some of the multiple constants. According to Eq. 6 it is easy to see that the rate of polymerization RP0 is direct proportional to the monomer concentration c and proportional to the square root of the initiator concentration cI. Since the rate of polymerization depends on the square root of the initiator concentration, small changes in the initiator concentration hardly affect the reaction rate (E.g. a change in initiator concentration for ± 10% will alter the reaction rate only for about ± 5%).

10 R P0 = k app ⋅ c = q pt ⋅ R I ⋅ c Eq. 6)

R I = 2 ⋅ f ⋅ k d ⋅ c I Eq. 7)

k P q pt = Eq. 8) k t

(cI = initiator concentration, c = monomer concentration, f = radical efficiency factor, kd = initiator decomposition constant, RI = rate of initiation, kapp = qpt⋅RI = apparent rate constant of polymerization)

As long as the initiator concentration cI is constant - and in connection to the "square root law" (Rp ∝ √cI) this is a fairly good assumption up to a monomer conversion of 10 - 30 % - the rate of polymerization obeys a kinetic law of "pseudo first order" according to Eq. 7). Hence, setting cI = cI,0 inte- grating Eq. 7) let one obtain the time dependencies of the monomer concentration c(t) (Eq.9) and the polymer concentration cP(t) (Eq. 10).

-kapp⋅t c(t) = c0 ⋅ e Eq. 9)

-kapp⋅t c P (t) = c0 ⋅ (1− e ) Eq. 10)

(c(t) = monomer concentration at time t in the reaction mixture, cP(t) = polymer concentration at time t in the reaction mixture, c0 = initial monomer concentration, kapp = apparent rate constant of polymerization (cf. Eq. 6-7)).

Equations 9) and 10) have been used to calculate the data from the Table 1. It follows from Eq. 9) that the rate of polymerization exponentially decreases after the start of the reaction:

dc(t) -kapp⋅t -kapp⋅t = −k ⋅ c ⋅ e = R ⋅ e Eq. 11) dt app 0 p0

Note that these equations do not take into account the consumption of the initiator due to its decomposition. With increasing time the initiator concentration decreases and consequently the rate of initiation, RI. Taking into account the time dependence of the initiator concentration cI (Eq. 12),

-k ⋅t d c I (t) = c I,0 ⋅ e Eq. 12) the equations describing the rate of polymerization become (Eq. 13 and Eq. 14):

11 k t d k −k t − dc p d 2 = - ⋅ 2fk d ⋅ c I,0 ⋅ e = q pt ⋅ R I,0 ⋅ e Eq. 13) dt k t k ⋅ t k d app − − 2⋅ ⋅(1−e 2 ) k d c(t) = c0 ⋅ e Eq. 14)

(c(t) = monomer concentration at time t in the reaction mixture, c0 = initial monomer concentration, cI,0 = initial initiator concentration, kapp = apparent rate constant of polymerization, kd = decomposition rate constant of the initiator).

From Eq. 14 it follows that the monomer conversion, and consequently the polymer yield is limited, even for infinite long reaction times. The maximum polymer yield that can be achieved with "low concentration" batch polymerization reactions is given by Eq. 15). k app − 2⋅ k lim Yield = 1− e d Eq. 15) t→∞

Standard recipes of polymerization are designed such, that the initial initiator concentration cI,0 and the rate of initiator decomposition kd are sufficient to produce large values of kapp/ kd. Hence, within one half live time of the initiator τI = ln(2)/kd the polymerization can be run to full monomer conversion. But with polymerization times considerably exceeding τI one must keep in mind the change in reaction kinetics due to inconstant initiator concentrations (Eq. 12) ("dead end polymerization"). To ensure a constant rate of initiation RI = RI,0, cI must be kept constant by feeding of initiator to the polymerizing mixture. The number average degree of polymerization, Xn, is obtained from the ratio of the rate of polymerization Rp to the sum of the rates of reactions that kill growing chain radicals, i.e. termination (Rt) and all present transfer reactions (Rtrans- 4 fer). Hence,

R P X n = Eq. 16) R t + ∑ R transfer j

In the absence of any transfer reactions one obtains for Xn:

q c pt c X n0 = q pt ⋅ = ⋅ Eq. 17) R I 2fk d c I

(Xn0 = number average degree of polymerization without transfer reactions, c = monomer concentration cI = initiator concentration, kd = decomposition rate constant of the initiator, qpt: refer to Eq. 8). 12 Note that the degree of polymerization depends on the ratio c / √cI. With "dead end polymerization" conditions (reaction time ≈ τI) the polydispersity index XW/Xn of the resulting molecular weight distribution will become large, because with vanishing initiator the actual degree of polymerization will rapidly grow. With transfer reactions present the degree of polymerization is lowered and obeys the "Mayo-Equation", Eq. 18).5

1 1 cT , j = + ∑CT , j ⋅ Eq. 18) X n X n0 j c

(Xn = number average degree of polymerization with transfer reactions, c = monomer concentration, cT,j = concentration of transfer agent j, CT,j = transfer constant of transfer reagent j)

This short discussion of the reaction kinetics of free radical homopolymerization reactions unequivocally states the necessity to keep constant the monomer concentration c, as well as the initiator concentration cI. In any batch polymerization process both the concentrations will inevitably decrease, causing (i) a continuous decrease of the over-all rate of polymerization and (ii) a steady increase of the degree of polymerization with time. In cases that the reaction time is small compared to the half live time of the initiator, cI is approximately constant and initiator feed may be neglected. However, in any experiment where the reaction time approaches or even exceeds the initiators half live time, one definitely has to replace the decomposed initiator.

13 2.3. - Continuous addition batch homopolymerization

Figure 2: Set-up for continuous addition homopolymerization on adding a monomer feed solution

Prior to the mathematical treatment to obtain the continuous addition conditions we have to define the process, its variables and their symbols. The experimental situation is depicted in Figure 2 and is briefly described as follows:

A "stock solution" of volume V0 contains the monomer. At t = 0 the initial monomer concentration in the stock solution is t c0. A total rate of polymerization R P0 is observed. As a conse- quence a polymer is formed and monomer is consumed. To this reaction mixture a "monomer feed solution" and a "initiator feed solution" are added with the volume rates of addition dV*/dt and dV(I)/dt, respectively over a period of time τ. The monomer feed solution contain a concentration c* of the monomer, while the initiator feed solution contains the initi- (I) ator concentration cI .

The monomer concentration c is defined as the quotient of the number of moles monomer n and the total solution volume V:

n c = Eq. 19) V

Taking the time derivative of Eq. 19) yields Eq. 20):

dc d  n  1 dn n 1 dV =   = ⋅ − ⋅ ⋅ Eq. 20) dt dt  V  V dt V V dt

Hence, the total change of the concentration is composed of the change in number of monomer moles dn/dt and the change in

14 solution volume dV/dt. Let us identify the contributors to each of the differential quotients.

a) Change of the solution volume The change of the solution volume is caused by three processes, namely (i) addition of monomer solution, (ii) addition of initiator volume and (iii) volume contraction due to polymerization. The last process occurs, since the density of the polymer is larger than the density of the monomer. Equation 21 gives the differential volume change due to monomer addition:

dV* dV = dt Eq. 21) dt

Eq. 22) states the effect of initiator solution addition:

(I)  dV  dV =   dt Eq. 22)  dt 

Eq. 23) describes the volume change due to polymerization:

 1 1  dV = M ⋅ −  ⋅ R t ⋅ dt M   P Eq. 23)  ρ p ρ M 

(V = solution volume, t = time, ρP = density of the polymer, ρM = t density of the monomer, MM = monomer molecular weight, R P = total rate of polymerization).

Hence, the total change of the reaction volume with time is the sum of Eq. 21) - Eq. 23):

(I) dV dV*  dV   1 1  = + + M ⋅ −  ⋅ R t   M   P Eq. 24) dt dt  dt   ρ p ρ M 

b) Change of the monomers number of moles n in time: Monomer is consumed by polymerization and replaced by monomer addition. The two contributions to the total change in monomer moles are given by the following equations:

(reaction)  dn  t Polymerization:   = − k app ⋅ n = R p Eq. 25)  dt 

15 t Note that R P here denotes the total rate of polymerizati- t on observed with the whole reaction mixture. The unit of R P is simply mol / time, and it is related to the monomer concentration according to Eq. 26). The expression Rp = kapp ⋅ c, termed "rate of polymerization" should better be named "specific rate of polymerization", since it gives the moles of monomer converted per time and volume.

n R t = k ⋅ n = V ⋅ k ⋅ = V ⋅ k ⋅ c = V ⋅ R Eq. 26) p app app V app p

(c = monomer concentration in the reaction mixture, kapp = apparent ⋅ polymerization constant = kp/√kt √(kd⋅cI))

The second process that affects the number of moles monomer in the reaction system is the addition of a monomer feed solution with the monomer concentration c*:

(addition)  dn  dV* Monomer addition:   = + c* ⋅ Eq. 27)  dt  dt

The total change of the monomers mole number is the sum of Eq. 26) and Eq. 27): dn dV* = −R t + c* ⋅ Eq. 28) dt p dt

t (n = number of moles of monomer in the reaction mixture, R P = total rate of polymerization of the whole reaction mixture, c* = monomer concentration in the feed solution, dV*/dt = volume addition rate of the monomer feed solution).

Now the expressions for dV/dt (Eq. 24) and dn/dt (Eq.28) are inserted in Eq. 20), and n/V is identified with the monomer concentration c in the reaction solution to yield the concentration balance equation 29:

 *   * (I)    dc 1 t * dV 1 dV  dV   1 1  t = − R p + c ⋅  − c ⋅ ⋅  +   + M M ⋅ − ⋅ R p  dt V dt V dt dt  ρ ρ  Eq. 29)       p M  

(c = monomer concentration in the reaction mixture, V = volume of t the reaction mixture, R P = V0⋅kapp⋅c0 = total rate of polymerization, cf. eq.26), dV*/dt = volume addition rate of the monomer feed soluti- * (I) on with c > c0, (dV/dt) = volume addition rate of the initiator feed solution, MM = monomer molecular weight, ρP = Density of the polymer, ρM = density of the monomer).

16 At this point a decision must be made what to keep constant by the continuous addition of a monomer feed solution. Two options are possible:

(1) - Keep the specific rate of polymerization Rp constant.

t (2) - Keep the total rate of polymerization R P constant.

The first option implies to fix the monomer concentration c at a constant initial value c0, while the second option allows a certain change in c as long as the total rate of polymerization, i.e. the product V(t)⋅kapp⋅c(t), remains unaltered. In text- books and in the literature on continuous addition copoly- merizations6,7,8 frequently the first option is mentioned. Hen- ce, let us start with option 1:

2.3.1 Continuous addition at constant monomer concentration

The first possible condition for continuous monomer addition is that the monomer concentration c must not change in time. Hence, dc/dt must be zero and consequently the monomer concentration c remains constant at its initial value c0. As a result the specific rate of polymerization RP also remains constant at its initial value RP0, but as the volume of the reac- t tion mixture changes, the total rate of polymerization R P = V ⋅ RP0 cannot remain constant. To find an expression for the required monomer volume addition rate dV*/dt in equation 29) dc/ t dt is equalled to zero and R P = V⋅kapp⋅c (Eq. 26) is inserted. After re-arranging the variables one obtains Eq. 30):

 1 dV* dV (I)   1 1  1 dV* − k + ⋅ +  ⋅ c - M ⋅ −  ⋅ k ⋅ c 2 ⋅ +c* ⋅ ⋅ = 0  app  0 M   app 0 Eq. 30)  V dt dt   ρ p ρ M  V dt

Equation 30) can considerably be simplified by taking into account that the amount of initiator solution added to the reaction mixture is negligible against the added monomer volume (dV(I)/dt 〈〈 dV*/dt). In case of diluted reaction solutions (i.e. low c0) - or when working with macromonomers - the volume contraction due to monomer conversion can also be neglected. Hen- ce, the total volume change dV/dt becomes identical to the volume addition rate dV*/dt and equation 31 is a simple differential equation, relating dV*/dt, V and the reaction time:

 1 dV*  1 dV*   * − k app + ⋅  ⋅ c0 + c ⋅ ⋅ = 0 Eq. 31)  V dt  V dt

17 Equation 31 can easily be solved by means of variable separation and subsequent integration in the experiment limits (time: 0 → t, volume: V0 → V). One obtains for the volume addition rate dV*/dt:

dV* R t  c  = p0 ⋅ exp 0 ⋅ k ⋅ t *  * app  Eq. 32) dt c − c0  c − c0 

(dV*/dt = volume addition rate of a monomer feed solution with c* > t c0, R p0 = initial total rate of polymerization in the stock solution

= V0⋅kapp⋅c0, cf. eq.26), V0 = stock solution volume).

Note that dV*/dt is time dependent and increases exponentially in time. This detail will be discussed later, but at this point we will keep in mind that a time dependent addition rate is inconvenient from the experimentalist's point of view. Furthermore, exponential growth can become disastrous large and it must be checked if the planned experiment is possible with terrestrial recourses.

The added feed solution volume is obtained by integration of dV*/dt over the addition time:

t dV*  c  V(t) = ⋅ dt = V ⋅ exp 0 ⋅ k ⋅ t ∫ 0  * app  Eq. 33) 0 dt  c − c0 

 R t  V(t) = V ⋅ exp P0 ⋅ t  0  *  Eq. 34)  c − c0 

* (c = monomer concentration in the monomer feed solution, c0 = stock t solution monomer concentration, V0 = stock solution volume, R P0 = initial total rate of polymerization in the stock solution).

The addition process is continued until α times the starting amount of monomer is added. The ratio α = ∆n+ / n0 ≥ 1 will be called "addition factor". This preposition leads to the following expression:

τ τ dV* ∆n = dn = c* ⋅ dt + ∫ + ∫ Eq. 35) 0 0 dt

  R t   ∆n = c* ⋅ V ⋅ exp P0 ⋅ t −1 + 0   *   Eq. 36)   c − c0  

  R t   ∆n = α ⋅ n = c* ⋅ V ⋅ exp P0 ⋅ τ −1 + 0 0   *   Eq. 37)   c − c0  

18 Solving Eq. 37) for τ yields the required formula to calculate the length of the addition period:

c* −c 1  c  τ = 0 ⋅ ⋅ln 1+α⋅ 0 *  *  Eq. 38) c kapp  c 

(τ = addition time, c* = monomer concentration in continuously added solution, c0 = stock solution monomer concentration, kapp = apparent rate constant of polymerization, α = ratio of added amount of monomer ∆n+ to presented amount of monomer in the stock solution, n0).

Inserting τ for t in V(t) yields the total volume of the reaction mixture at the end of the addition process, V(τ), as well as the total added solution volume ∆VND(α):

 c  V(τ) = V ⋅ 1+ 0 ⋅α 0  *  Eq. 39)  c  c ∆V (α) = 0 ⋅ V ⋅ α Eq. 40) ND c* 0

* The relative volume increase ∆VND(α)/V0 is given by α⋅c0/c . At this point it is easy to see that the dilution effect cannot be neglected as long as the monomer feed solution is not much more concentrated than the stock solution: The dilution effect only becomes negligible as long as:

* α⋅c0/c 〈〈 1. Eq. 41)

This condition is fulfilled only with low values of α and c0 or * with very large feed concentrations c .

The amount of polymer formed during the addition period is identical to the number of added monomer moles (cf. Eq. 36). The polymer concentration cp is obtained inserting τ (Eq. 38) in Eq. 36) and dividing the result by the final volume V(α). One obtains:

α ⋅ n c ⋅α c = 0 = 0 Eq. 42) P V c ⋅α 1+ 0 c*

(α = addition factor = ∆n/n0, c0 = monomer concentration in the stock * solution, c = monomer concentration in the monomer feed solution, n0 = initial total number of moles of monomer in the stock solution, ∆n = total number of moles of monomer fed to the reaction during the addition time τ).

Even at very large α the polymer concentration cp cannot exceed * (max) the value c . If a maximum polymer concentration cp must not be exceeded, this information can be used to fix the monomer * (max) concentration of the monomer feed solution: c = cp . To summarize, continuously feeding a monomer solution to the the reaction mixture with keeping the specific rate of polymerization Rp constant (apply Eq. 32) has the following effects: - The monomer concentration remains constant, - The volume exponentially increases with time,

- The specific rate of polymerization kapp⋅c0 remains constant, but the total rate of polymerization (dn/dt)(reaction) = V⋅Rp exponentially increases with time. - The polymer concentration is kept below c*.

The continuous addition fixes the polymerization conditions in its initial state, i.e. the steady state conditions are con- served over the full reaction time.

The polymerization process can become very fast, because of the exponential growth in reaction volume and parallel to this, the total rate of polymerization. However, with batch processes - performed in a fixed volume vessel - a rapidly growing reaction volume is of disadvantage, since any reaction with exponentially growing rate is simply dangerous. The "constant concentration condition" may be suitable for continuous polymer production plants, but not for a laboratory batch polymerization.

2.3.2 Continuous addition at constant rate of polymerization

The second option for continuous addition polymerization reaction mentioned on page 18 was to keep constant the total t rate of polymerization R P. Hence, one takes Eq. 28) describing the total change in the number of moles of monomer and equals this expression to zero:

* dn * dV = −k app ⋅ n + c ⋅ = 0 Eq. 43) dt dt

(n = number of moles of monomer in the reaction mixture, c* = monomer concentration in the feed solution, kapp = apparent rate constant of polymerization, dV*/dt = volume addition rate of the monomer feed solution).

Note that the dilution due to monomer solution addition does not affect the monomer molar number balance equation: Al-

20 though the reaction mixture will be diluted we do not explicitly have to deal with this complication, as long as we focus on the total number of monomer molecules ! If dn/dt = 0, the number of moles monomer does not change, but remains unaltered at its initial value n0. Resolve Eq. 44) for dV*/dt, expand with the factor V0 / V0 and identify n0 / V0 t = c0 and V0⋅kapp⋅c0 = R P0. Equation 44 yields a simple expression for the required volume addition rate dV*/dt:

dV* R t = P0 Eq. 44) dt c*

(dV*/dt = volume addition rate of the monomer feed solution, c* = mo- t nomer concentration in the feed solution, R P0 = initial total rate of polymerization in the stock solution).

* t Note that dV /dt is direct proportional to R p0 and vice versa. As long as the feed volume addition rate is kept constant, the total rate of polymerization will also remain unaltered. For practical purposes it is of large importance that the addition rate needs no adjustments during the course of the polymerization reaction.

Since dV*/dt is constant, all further calculations become very simple and only the resulting equations will be given here to calculate the reaction volume, the monomer concentration, the specific rate of polymerization RP, the addition time, and the polymer concentration cP at an arbitrary reaction time t:

c Reaction volume: V(t) = V ⋅ (1+ 0 ⋅ k ⋅ t) Eq. 45) 0 c* app

c Monomer concentration: c = 0 Eq. 46) c 1+ 0 ⋅ k ⋅ t c* app

R Specific rate of polymn.: R = k ⋅c = P0 Eq. 47) p app c 1+ 0 ⋅ k ⋅ t c* app α Addition time: τ = Eq. 48) k app

c0 ⋅ k app ⋅ t Polymer concentration: c = Eq. 49) P c 1+ 0 ⋅ k ⋅ t c* app

* (c = monomer concentration in the feed solution, c0 = monomer concentration in the stock solution, c = monomer concentration in the

21 reaction mixture, V = volume of the reaction mixture, V0 = volume of t the stock solution, R P = V0⋅kapp⋅c0 = total rate of polymerization, cf. eq.26), dV*/dt = volume addition rate of the monomer feed solution * with c > c0, kapp = apparent rate constant of polymerization, α = ratio of added amount of monomer ∆n+ to presented amount of monomer in the stock solution, n0).

To summarize, continuously feeding a monomer solution to the stock reaction mixture with keeping the total rate of po- t lymerization R p constant (apply Eq. 44) has the following effects: - The monomer concentration decreases,

- The specific rate of polymerization RP decreases with c0, - The volume linearly increases with time, - The polymer concentration is kept below c*.

The total rate of polymerization over the whole reaction mixture remains constant, because the decrease in monomer concentration is compensated by the increased reaction volume:

c c R t (t) = V ⋅ k ⋅ c = V ⋅ (1+ 0 ⋅ k ⋅ t) ⋅ k ⋅ 0 = V ⋅ k ⋅ c = R t P app 0 c* app app c 0 app 0 P0 1+ 0 ⋅ k ⋅ t c* app

This continuous addition technique keeps the polymerization reaction in the steady state conditions, because the over-all reaction rates are not affected. Furthermore the constantly decreasing monomer concentrations and the limited polymer concentration prevent the system from running into regimes of non - ideal reaction kinetic.

At the end of the addition period (t = τ, ∆n+/n0 = α) the volumes, concentrations, and reaction rates are: c c V(α ) = V ⋅ (1+ 0 ⋅ α) ≈ V ⋅ 0 ⋅ α Eq. 50) 0 c* 0 c* c c* c(α ) = 0 ≈ Eq. 51) c α 1+ 0 ⋅ α c* R c* R (α ) = P0 ≈ k ⋅ Eq. 52) p c app α 1+ 0 ⋅ α c* α ⋅ c c (α ) = 0 ≈ c* Eq. 53) P c 1+ 0 ⋅ α c* For large values of α the short approximate formulae on the right side of each equation become applicable.

22 2.3.3. Continuous Addition Homopolymerization at Negligible Dilution

With low molecular weight monomers it will sometimes be possible to feed pure monomer to the polymerization reaction mixtures. Since the concentration of pure monomers like MA, MMA or styrene is in the order of 10 - 15 mol/L, while the reaction mixture contains 5 - 10 wt% monomers (c0 = 0.5 - 1 * mol/L) the relation α⋅c0/c 〈〈 1 (Eq. 41) is valid up to addition factors of α ≈ 2 - 5. Hence, the dilution effect due to the continuous addition can be negligible. In particular if the monomer concentration c0 in the stock mixture is low (< 0.5 mol/L) Eq. 29) transforms into Eq. 54):

* dc 1  * dV  = − V0 ⋅ R p,0 + c ⋅  = 0 Eq. 54) dt V  dt 

If α was selected small, so that the total added monomer volume ∆VND(α) is small compared to the stock volume V0, the relation V ≈ V0 holds throughout the reaction. Hence, the continuous addition condition simplifies considerably:

dc c* dV* = −R p,0 + ⋅ = 0 Eq. 55) dt V0 dt

Eq. 55) can directly be resolved for dV/dt to yield:

* dV V0 ⋅ R p,0 = Eq. 56) dt c*

Equation 56) is identical to Eq. 44): In the limit of vanishing volume changes the two different continuous addition approaches (constant total rate of polymerization and constant specific rate of polymerization) are characterized by identical experimental conditions. Consequently the addition time τ is now to be calculated by Eq. 48 (τ = α / kapp). At the end of the addition period the reaction volume is identical to the stock volume (V ≈ V0), the monomer concentration is identical to the stock solution monomer concentration (c ≈ c0), both the t t rates of polymerization remain unaltered (R P ≈ R P0, RP ≈ RP0) and the final polymer concentration becomes:

α ⋅ n 0 c P = ≈ c0 ⋅ α Eq. 57) V0

* (c = monomer concentration in the feed solution, c0 = monomer concentration in the stock solution, V0 = volume of the stock solution, t * R P = V0⋅kapp⋅c0 = total rate of polymerization, cf. Eq.26), dV /dt =

23 * volume addition rate of the monomer feed solution with c > c0, kapp = apparent rate constant of polymerization, α = ratio of added amount of monomer ∆n+ to presented amount of monomer in the stock solution, n0).

Problems that limit the applicability of equations 54 - 57) mainly arise from the steadily increasing polymer concentration. Since no solvent is added, the amount of formed polymer grows linear in time and so does the polymer concentration (cf. Eq. 57). Hence, the viscosity of the reaction mixture will strongly increase and the Trommsdorff - effect (= gel effect) is likely to occur. It is also possible that the reaction temperature falls short of the glass transition temperature of the polymer / monomer / solvent mixture: The system will vitrify and - on further addition of monomer - simply become heterogeneous. At latest with the occurrence of heterogenei- ties - e.g. demixing processes or polymer precipitation - the reaction must be stopped.

24 2.4. - Effect of an erroneous volume addition rate

If a continuous addition homopolymerization is performed without in-situ control of the actual monomer concentration c, a deviation of the actual rate of polymerization Rp from the rate of monomer addition c*⋅dV*/dt cannot strictly be excluded. In this section the effect of a possible error in addition rate on the resulting monomer concentration will be discussed. The method to deal with erroneous addition rates is to el- aborate the complete time dependent solution for the monomer concentration c(t) as well as the number of moles monomer n(t) in dependence of the monomer volume addition rate dV*/dt. In- serting the "correct" addition rates according to Eq. 34) and Eq. 44) should result in dc/dt = 0 or dn/dt = 0. Allowing for deviations from the correct addition rates will subsequently reveal the effect of mis-dosage on the reaction conditions. As a measure for the error in volume addition rate we introduce the "relative addition error" ϕ, defined as the ratio of the absolute error in addition rate, ε• and the correct volu- * me addition rate (dV /dt)0:

• ε ϕ = • Eq. 58) * V0

The effect of the addition error will be a deviation from the targeted monomer concentration or monomer molar number.

Hence, we define the "relative monomer deviations" δc and δn:

c(t) − c0 n(t) − n 0 δc = , δ n = Eq. 59) c0 n 0

2.4.1 Constant specific rate of polymerization

Taking Eq. 29), neglecting the volume shrinkage upon monomer conversion as well as the initiator addition and re-arranging the symbols yields a differential equation describing the change in monomer concentration upon monomer addition during polymerization:

dc  1 dV*  1 dV*   * = −k app + ⋅  ⋅ c + c ⋅ ⋅ Eq. 60) dt  V dt  V dt

The homogeneous part of the equation can be solved by the variable separation technique followed by the method of variation of the constants to cope with the inhomogeneous part. The final result is an expression for the monomer concentration c(t) at any time t:

25 •  •  V* c*  c* ⋅ V*  V k c(t) c 0 exp[ app (V V(t)] = ⋅ +  0 −  ⋅ ⋅ − • ⋅ 0 − Eq. 61) k app V(t)  V0 ⋅ k app  V(t) *   V

(c(t) = monomer concentration at time t in the reaction mixture, c0 = stock solution monomer concentration, V0 = stock solution volume, kapp = apparent rate constant of polymerization, V•* = dV*/dt = actual volume addition rate, V(t) = actual volume of the reaction mixture).

This grim looking equation can be made friendlier by con- sidering only long reaction times, when the reaction volume is large compared to the stock volume. Then the whole exponential term vanishes and one receives: • c* ⋅ V* c∞ → Eq. 62) k app ⋅ V(t) c∞ denotes the limiting value of c(t), achieved at very long reaction times (formally: t → ∞). The relative monomer concentration deviation δc is calculated by dividing Eq. 62) by (t→∞) * * c0 = c ⋅(dV /dt)0/kapp⋅V(t) and subtracting 1:

• • c V* + ε limδ = ∞ −1 = 0 −1 = ϕ Eq. 63) t→∞ • c0 * V0

(δc = relative deviation of the monomer concentration from the monomer stock concentration, c = monomer concentration at time t in the reaction mixture, c0 = initial monomer concentration).

Equation 63 simply states that the relative monomer concentration deviation δ caused by monomer addition be identical to the relative addition error ϕ.

2.4.2 Constant total rate of polymerization

In analogy to the previous section the differential equation 43 is not set to zero but solved. Equation 64) describes the change of moles monomer due to polymerization and addition: dn dV* = −k ⋅ n + c* ⋅ Eq. 64) dt app dt

(n = number of moles monomer in the reaction mixture, c* = monomer concentration in the feed solution, kapp = apparent rate constant of polymerization, dV*/dt = volume addition rate of the monomer feed solution).

26 The solution of this simple differential equation is:

c* dV* n(t) = ⋅ ⋅[]1− exp(−k app ⋅ t) + n 0 ⋅ exp(−k app ⋅ t) Eq. 65) k app dt

At sufficiently large reaction times (kapp⋅t 〉〉 1) both the exponential terms vanish. Replacing the erroneous volume addition * * • rate dV /dt by the sum (dV /dt)0 + ε allows to calculate the maximum value of the relative monomer mole number deviation δmax: * • n ∞ c limδ = δ max = −1 → ⋅ ε = ϕ Eq. 66) t→∞ t n 0 R P0

Hence, in this continuous addition strategy the relative monomer deviation will always be smaller than the relative error in volume addition rate. If e.g. the applied volume addition rate is wrong for +10%, the monomer concentration in the reaction mixture will not exceed the intended stock concentration c0 for more than 10%.

To evaluate the effect of small errors in feed addition, let one assume that the volume addition rate dV*/dt is composed * of the "ideal" addition rate (dV /dt)0=V0⋅kapp⋅c0 and a small er- • ror ε : * • dV V0 ⋅ kapp ⋅ c0 = + ε Eq. 67) dt c*

Than it can be assumed that the resulting monomer mole number n will be given by Eq. 68), were λ is small compared to n0:

n = n 0 + λ Eq. 68)

Insert Eq. 67) and Eq. 68) in the differential equation Eq. * * 64) and remember that the term -kapp⋅n0 + c ⋅(dV /dt)0 = 0 (second continuous addition condition, Eq. 43). One obtains a new differential equation that directly describes the development of the concentration deviation λ in time:

dλ • = −k ⋅ λ + c* ⋅ ε Eq. 69) dt app

• (λ = absolute deviation of the monomer mol number, ε = absolute error in volume addition rate).

* • The solution of this equation is λ = c ⋅ε ⋅[1-exp(-kappt)]/kapp. * • Dividing this by n0 and identifying c ⋅ε /kapp with the relative volume addition error ϕ yields:

27 δ = δ max ⋅[1− exp(−k app ⋅ t)],

• • Eq. 70) * δ max = ϕ = ε / V0

This treatment essentially yields the same result than obtained from the complete solution of Eq. 64), but it is more convenient since it allows directly to reveal the time dependence of λ. Furthermore this procedure is applicable with more complicated differential equations, because the introduction of "negligible" small deviations λ and ε• allows to simplify the differential equations.

Replacing t in Eq. 70) by τ = α/kapp yields the maximum deviation of the monomer mol number that can be achieved within the addition period, expressed in terms of the "relative molar number deviation ratio" β = δn / δmax: β(α ) = 1− exp(−α) ≈ 1 Eq. 71) With practically feasible addition ratios α (α ≈ 5 - 10) the full possible monomer deviation will be realized.

How long does it take to reach a certain deviation δn? Re- solve Eq. 71) for tβ and refer it to the total addition time τ(α) to obtain an expression of more general validity:

t β ln[1/(1− β)] = Eq. 72) τ α 1.0 0.9 0.8 0.7 0.6 max

δ 0.5 / δ 0.4 =

β 0.3 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 t / τ β Figure 3: Plot of the "relative molar number deviation ratio" β versus the reduced reaction time tβ/τ for different addition factors α (:α = 5, :α = 10, :α = 20)

Formally it takes infinite time to reach the maximum deviation δmax, (cf Figure 3) but 90% (β = 0.9) of this deviation are reached within ≈ 25% of the total addition period (for α ≈ 10). In other words: for ca. 75% of the total addition time the monomer molar number can be regarded as constant, with a deviation of approximately δmax from the targeted value.

28 2.5. - Continuous addition of initiator

In section 2.2 (page 12) the importance of a constant rate of chain initiation RI was mentioned, since the total rate of polymerization and the degree of polymerization depend on the square root of RI. In any case where the duration of a continuous addition experiment exceeds the half live time of the applied initiator the initiator concentration is no longer constant, and so does the rate of initiation. As with monomers the initiator must be continuously added to the reaction mixture and the mode of addition, i.e. constant specific rate of initiation or constant total rate of initiation. The initiator addition strategy must be the same as selected for the monomer.

2.5.1. Constant specific rate of initiation

As with the monomers, also a balance equation must be set up and solved for the initiator. Since only small amounts of initiator solution are added, the dilution effect caused by the initiator addition can be neglected. Merely the effect of dilution upon monomer addition must be taken into account:

* (I) dcI d  N I  1 t 1 dV * 1 dV =   = − ⋅ R I0 − cI,0 ⋅ ⋅ + c I ⋅ ⋅ = 0 Eq. 73) dt dt  V  V V dt V dt

t (cI,0 = initial initiator concentration in stock solution, R I0 = ini- * tiator decomposition rate in stock solution, c I = initiator concentration in added solution, dV(I)/dt = volume addition rate of initiator solution, kd = decomposition rate constant of the initiator)

Solving Eq. 73) for dV(I)/dt yields:

(I) t t dV R I0 R P0 c I,0 t = * ⋅ exp[ * ⋅ t] + * ⋅ R P0 Eq. 74) dt cI − c I,0 V0 (c I − cI,0 ) c

Since the initiator concentration cI,0 is negligible small against the monomer feed concentration c*, the second term can be ignored. Like the monomer addition rate also the initiator addition rate must exponentially grow in time to compensate the dilution effect caused by the monomer solution. With neglecting of the second term in equation 74) the total number of added initiator moles is calculated by inserting Eq. 38) in Eq. 74):

c* − c  c  ∆n (τ ) = V ⋅ c* ⋅ 0 ⋅ 1+ α ⋅ 0 I 0 I *  *  Eq. 75) c I − cI,0  c 

(∆nI(τ) = during the whole addition time τ added moles of initiator, * cI,0 = initial initiator concentration in stock solution, c I = initiator concentration in added solution, V0 = volume of the stock solu- 29 * tion, c = monomer concentration in the monomer feed solution, c0 = monomer concentration in the stock solution, α = addition factor =

∆n/n0 )

2.5.2. Constant total rate of initiation

To keep the total number of moles initiator in the reaction mixture constant, the balanced state expressed in Eq.76) must be fulfilled:

dn dV (I) I = −k ⋅ n + c* ⋅ = 0 Eq. 76) dt d I I dt

With dnI/dt = 0, the number of moles initiator does not change, but remains unaltered at its initial value nI0. Resolve (I) Eq. 76) for dV /dt, expand with the factor V0 / V0 and identi- t fy nI0/V0 = cI0 and V0⋅kd⋅cI0 = R I0. Equation 77) yields the expression for the required initiator volume addition rate dV(I)/dt:

(I) t dV R I0 V0 ⋅ k d ⋅ cI0 = * = * Eq. 77) dt c I c I

* (cI0 = initial initiator concentration in stock solution, c I = initiator concentration in added solution, V0 = volume of the stock solution, kd = initiator decomposition constant)

In total the following number of moles of initiator has to be added during the addition period τ:

dV* ∆n = c* ⋅ ⋅ τ Eq. 78) I I dt

α t k d ∆n I = ⋅ R I0 = ⋅ V0 ⋅ α ⋅ c I0 Eq. 79) k app k app

This corresponds to the required mass of initiator mI:

α t k d m I = M I ⋅ ⋅ R I0 = ⋅ V0 ⋅ α ⋅ M I ⋅ c I0 Eq. 80) k app k app

(mI = mass of initiator added during the addition period τ, MI = molecular weight of initiator, α = ratio of added amount of monomer

∆n+ to presented amount of monomer in the stock solution, n0, cI0 = * initiator concentration in stock solution, c I = initiator concentration in feed solution, V0 = volume of stock solution, kapp = apparent

30 rate constant of polymerization, kd = initiator decomposition rate constant). The initiator solution should be of high concentration, hence one may dissolve the initiator in a threefold excess of solvent:

V ≈ 3⋅ m Solvent I Eq. 81)

In Eq. 81) we arbitrarily set the mass of initiator in g equal to mL and assume a density of 1.3 g/mL for the initiator. Note that this equation is not required when working with a initiator solution instead of a solid initiator. In this case one can directly feed this initiator solution and apply Eq. 77). m I ∆V (Initiator) ≈ V + ND Solvent 1.3 Eq. 82)

* m I 1 c I ≈ ⋅ Eq. 83) M I ∆VND (Initiator)

(∆VND(Initiator) = added initiator solution volume, mI = total mass of initiator to feed during the addition period, Vsolvent = solvent volume used to dissolve mI, MI = molecular weight of the initiator, * c I = initiator concentration in the initiator feed solution).

31 3. Copolymerization

3.1. - Copolymer composition9,10,11

In homopolymerization experiments the laboratory continuous feed technique increases the speed of a polymer prepara- tion, but has little effect on the properties of the macromo- lecular product. With copolymerizations the properties of the product strongly depend on its monomer composition,- and this is what continuous addition of monomers has to guarantee for. "Copolymerization" is termed any polymerization of mixtures of monomers. The reaction mechanism is very similar to that of free radical homopolymerizations, i.e. initiation, chain growth, termination and transfer reaction steps are in- volved. In contrast to the homopolymerization, with binary copolymerization four different chain growth steps can be disti- nguished:

k • 11 • ~~~ M1 + M1 → ~~~ M1 P-1) k • 12 • ~~~ M1 + M 2 → ~~~ M 2 P-2) k • 21 • ~~~ M 2 + M1 → ~~~ M1 P-3) k • 22 • ~~~ M 2 + M 2 → ~~~ M 2 P-4)

Scheme 2: Chain growth steps in a binary copolymerization

The composition of the copolymer depends on the relative rate of these steps, since these rates determine the extend of incorporation of each monomer in the resulting macromolecule. According to Scheme 2 the consumption of monomer 1 and monomer 2 obeys the following equation:

dc 1 = -k ⋅[~~~ M • ]⋅ c - k ⋅[~~~ M • ]⋅ c dt 11 1 1 21 2 1 Eq. 84) dc 2 = -k ⋅[~~~ M • ]⋅ c - k ⋅[~~~ M • ]⋅ c dt 22 2 2 12 1 2

• (ci = concentration of monomer i, [~~~M i] = concentration of the growing chain radicals, terminated with a unit of monomer i, kii = rate constant of homopolymerization of monomer i, kij = rate constant of cross addition).

In the steady state it is assumed that the concentration of all growing chain radicals is constant, i.e. their respective rates of formation by initiation and destruction by terminati- • • on are identical. Moreover, to keep [~~~M1 ] and [~~~M1 ] constant, the rate of the reactions P-2 and P-3 (see Scheme 1)

32 must be identical. This assumptions lead to a relation between the radical concentrations:9

• • k12 ⋅[~~~ M1 ]⋅ c2 = k 21 ⋅[~~~ M 2 ]⋅ c1 Eq. 85)

To find the copolymer composition one divides the individual rate of monomer consumptions by each other, identifies the term (dc1/dt) / (dc2/dt) with the quotient of the molar fractions of the monomers incorporated in the polymer (F1 / F2) and obtains:

k dc1 11 ⋅ c1 + c 2 dt dc1 F1 c1 k12 = = = ⋅ Eq. 86) dc 2 dc 2 F2 c 2 k 22 ⋅ c 2 + c 2 dt k 21

The ratios of homo- to cross addition rate constants kii/kij are called "copolymerization parameter" or "reactivity 9 ratios", denominated as r1 and r2:

k11 k 22 r1 = (Eq. 87), r2 = (Eq. 88) k12 k 21

The copolymerization parameters r1 and r2 measure the intrinsic ratio of homopolymerization- to copolymerization ability of the monomers within a monomer pair: if ri = 0, the monomer i is either unable to homopolymerize (kii=0) or its tendency to co- polymerize with monomer j by far exceeds its capability to form a homopolymer. On the opposite, large values of ri (ri 〉〉 1) indicate the preference of homopolymerization over copolymerization. Note that the copolymerization parameters are always valid for one pair of monomers. It is not possible to at- tribute one parameter to one monomer, and make use of this value to predict the copolymerization behaviour of this monomer with another comonomer. The pair of copolymerization parameter must be determined independently for each monomer pair in question. With the definitions of r1 and r2 Eq. 86) becomes:

dc F c r ⋅ c + c 1 = 1 = 1 ⋅ 1 1 2 Eq. 89) dc 2 F2 c 2 r2 ⋅ c 2 + c 2

(ci = monomer concentration, Fi = molar fraction of monomer i in the copolymer, ri = copolymerization parameter).

This is the Lewis - Mayo equation, relating the instantaneous composition of the copolymer to the actual composition of the monomer mixture. One can get rid of the absolute monomer concentrations ci by multiplying out with the factor c1/c2 and using the identity ci/cj = fi / fj. Hence, with a given monomer pair the initial 33 composition of the copolymer only depends on the relative composition of the monomer mixture (cf. Eq. 90) ! Once again note that this equation is valid only for negligible small monomer conversions (see below).

F r ⋅ (f f ) +1 1 = 1 1 2 Eq. 90) F2 r2 ⋅ (f 2 f1 ) +1

(fi = monomer molar fractions in the reaction mixture, Fi = molar fraction of monomer i in the copolymer, ri = copolymerization parameter).

1.0 1.0 0.9 0.9

0.8 r = 1, r = 1 0.8 1 2 0.7 0.7 0.6 0.6 r = 0, r =0 0.5 1 2 0.5 1 F 0.4 r < 1, r < 1 0.4 1 2 A 0.3 0.3 r < 1, r > 1 0.2 1 2 0.2 0.1 0.1 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f 1 Figure 4: Different types of copolymerization diagrams (black : alternating copolymerization, red: ideal statistic copolymerization, magenta: r1 = 0.15, r2 = 0.35, blue: r1 = 0.01, r2 = 4.5, A = azeotropic point)

If one remembers that the ratio of monomer concentrations is identical to the ratio of molar fractions of the monomers in the reaction mixtures (c1 / c2 = f1 / f2), Eq. 90) can be rear- ranged into:

2 r1 ⋅ f1 + f1 ⋅ f 2 F1 = 2 2 Eq. 91) r1 ⋅ f1 + 2f1 ⋅ f 2 + r2 ⋅ f 2

(fi = molar fraction of monomer i in the reaction mixture, Fi = molar fraction of monomer i in the copolymer, ri = copolymerization parameter).

34 With help of Eq. 91) one can draw "copolymerization diagrams" by plotting the copolymer composition F1 against the composition of the monomer mixture, f1. The dependence of F1 on f1 is determined by the copolymerization parameters r1 and r2:

With r1 = r2 = 1 the copolymer composition is always identical to the composition of the monomer mixture, hence this behaviour is called "(ideal) statistical" or "random" copolymerization (red line in Figure 4). If r1 = r2 = 0, the copolymer composition is completely independent of the monomer mixture composition and in any case the polymer will contain the monomer is the ratio 1 : 1. Moreover, this behaviour is only possible because monomer i can never add to a growing chain radical that is terminated • with a unit of monomer i, ~~~Mi . Consequently the two monomers become incorporated in a strictly alternating sequence (~~~M1- M2-M1-M2-M1-M2~~~) and this type of copolymerization is named "alternating copolymerization" (black line in Figure 4). In many cases one finds the combination r1 < 1 and r2 < 1 (also: r1 > 1 and r2 > 1) (e.g.: styrene / methyl methacrylate: r1 = 0.52, r1 = 0.46, styrene / methylacrylate: r1 = 0.75, r1 = 0.18). In this case the copolymerization line intersects the line of ideal statistic copolymerization in one single point (A in Figure 4). Since in this point the composition of the polymer equals the composition of the monomer mixture, it is called an "azeotropic" point and the copolymerization is en- titled "azeotropic copolymerization". The composition of the azeotropic monomer mixture is calculated according:

azeotropic r1 −1 f1 = Eq. 92) r1 + r2 - 2

azeotropic (fi = molar fraction of monomer i in the reaction mixture at the azeotropic point, ri = copolymerization parameter).

If r1 < 1 and r2 > 1 (also: r1 > 1 and r2 < 1) (e.g. maleic anhydride / methyl methacrylate: r1 = 0, r1 = 4.5) the copolymerization line forms a curve lying below the ideal statistic line. Since r1 < 1 indicates preference of copolymerization over homopolymerization for monomer 1, while r2 > 1 point towards the preference of homopolymerization of monomer 2, long blocks made of monomer 2, interrupted from more or less single units of monomer 2 will make up the monomer distribution along the polymer chain (e.g. ~~~ M1-M2-M2-M2-M2-M2- M2-M2-M1-M2-M2-M2-M2-M2-M2-M2-M2-M2-M2-M2-M2-M1~~~). Hence this copolymerization characteristic is sometimes called "blocky copolymerization". Note that this term must not be mixed up with block copolymers arising from living- or controlled polymerization techniques.

35 r1 = r2 = 1: (ideal) statistic / random copolymerization r1 = r2 = 0: alternating copolymerization r1 < 1, r2 > 1: "Blocky Copolymerization" r1 < 1, r2 < 1: Azeotropic Copolymerization

With respect to the discussion of copolymerization behaviour two items should be remembered:

(1) - Except from random copolymerizations and from azeotropic points the composition of the copolymer deviates from the composition of the monomer reaction mixture.

(2) - The Lewis Mayo equation gives the copolymer composition only for differential conversions.

From these terms follows that the composition of a copolymer will change during the run of the copolymerization: The more reactive monomer is incorporated faster than its comonomer and the monomer mixture becomes enriched in the less-reactive monomer. This law determines the direction of the compositional drift: In the "Blocky Copolymerization" depicted in Figure 5 the monomer composition changes towards f1 = 1 (blue arrows), while in an azeotropic copolymerization the monomer composition will always drift away from the azeotropic point. 1.0 1.0 0.9 0.9

0.8 r = 1, r = 1 0.8 1 2 0.7 0.7 0.6 0.6 r = 0, r =0 0.5 1 2 0.5 1 F 0.4 r < 1, r < 1 0.4 1 2 A 0.3 0.3 r < 1, r > 1 0.2 1 2 0.2 0.1 0.1 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f 1 Figure 5: Direction of the compositional drift in azeotropic (magenta) and "Block Copolymerization" reactions (blue line).

36 If one has to prepare a copolymer of uniform monomer composition, one of the following four possibilities must be chosen:

(a) Select a monomer pair that exhibits an ideal random copolymerization. In this case no compositional drift will occur. (b) Select a monomer pair that exhibits an azeotropic copolymerization and polymerize the azeotropic mixture. One is restricted to one composition of copolymer and care must be taken not to deviate from the azeotropic point, since any divergence will be amplified during the reaction. (c) Polymerize only to small monomer conversions (< 10 - 15 %), because in this case the compositional drift is negligible. This technique can be applied to any monomer pair, but 85 - 90% of the monomer is wasted. (d) Perform a continuous addition batch copolymerization.

3.2.- Rate of copolymerization

The application of the continuous addition process requires the knowledge of the rate of polymerization, Rp0,i(f1) for each monomer i. In a "scout" experiment the reaction is first performed at the desired monomer composition f1 and the time / monomer - conversion curve is measured in the regime of low monomer conversions (< 10%). The individual monomer rates of polymerization, RP0,i(f1) are obtained from the initial slope of the individual monomer time / conversion curves. In case the individual monomer conversions cannot be measured the required information can be extracted from the initial specific rate of copolymerization, RP0 combined to the initial composition of the copolymer F2/F1. The copolymers compositional quotient F2/F1 is either obtained experimentally from the isolated copolymer or, if the parameters r1 and r2 of the monomer pair are known, calculated from Eq. 90). The value of RP0 is easily obtained from measurements of the time / polymer yield curve, since the polymer yield is identical to the total amount of converted monomers. Care must be taken during the polymer isolation not to loose considerab- t le amounts of the polymer, since otherwise the measured R P0 will be systematically to low. In a gravimetric determination experiment of the polymer yield one measures the weight conversion of the monomers instead of the converted number of moles. The initial slope of a polymer yield / conversion curve is termed RPµ0 and is frequently expressed in units of wt%/time. The initial rate of polymerization of monomer 1 (RP0,1) and 2 (RP0,2) can be obtained from this simply obtainable value with knowledge of the total weight concentration of monomer c0µ:

37 c0µ R Pµµ,/100 c0µ R Pµµ,/100 R = ⋅ , R = ⋅ Eq. 93) P0,1 M M F P0,2 M M F 1 1+ 2 ⋅ 2 2 1+ 1 ⋅ 1 M1 F1 M 2 F2

(RP0,i = specific rate of polymerization of monomer i, RPµ0 = weight rate of copolymerization in wt%/time, c0µ = total weight concentration of monomers = (m10+m20)/V0, mi0 = initial weight of monomer I, V0 = reaction volume, Mi = molecular weight of monomer i, Fi = molar fraction of monomer I incorporated in the copolymer)

The total rate is the sum of the individual rates of polymerization of the comonomers:

R P0 = R P0,1 + R P0,2 Eq. 94)

Note that the specific rate of copolymerization RP0 is related to the individual rates of polymerization: In equation t t 94) RP0,1 can be factored out and the quotient R P0,2 / R P0,1 be identified with the quotient of the copolymer composition, t F2/F1. Subsequently one resolves for R P0,1 and obtains:

R P0 R P0,1 = Eq. 95) 1+ F2 /F1

t (RP0,1 = specific rate of polymerization of monomer 1, R P0 = specific rate of copolymerization, Fi = molar fraction of monomer I incorporated in the copolymer)

If RP0,i(f1) was measured at the desired monomer composition f1 all information's are available and one can proceed to perform the continuous monomer addition copolymerization (next chapter). Unfortunately, in many cases RP0,i(f1) is known from earlier experiments at different monomer compositions than ac- tually needed. Since the performance of additional "scout experiments" is inconvenient, an analytic expression to interpo- late the rates of polymerization between present data points is required.

2 2 r1 ⋅ f1 + 2 ⋅ f1 ⋅ f 2 + r2 ⋅ f 2 R p0 (f1 ) = ⋅ R I ⋅ c0 Eq. 96) 2 2 2 2 2 2 r1 ⋅ δ1 ⋅ f1 + 2 ⋅ Φ ⋅ r1 ⋅ r2 ⋅ δ1 ⋅ δ 2 ⋅ f1 ⋅ f 2 + r2 ⋅ δ 2 ⋅ f 2

k pii 2 ⋅ k tii k t12 ri = ,δi = ,Φ = Eq. 97) k pij k pii 2 ⋅ k t11 ⋅ k t22

(p = polymerization, t = termination, fi = molar fraction of monomer i in the monomer mixture, RI = rate of initiation = kd ⋅ cI, c0 = total monomer concentration)

Two theoretical expressions have been derived to calculate the rate polymerisation with copolymerizations from the reac-

38 tion mechanisms. The first equation (Eq. 96) was developed by Walling under the assumption that the termination reactions are governed by "chemical control", i.e. the chemical nature of the radical chain ends.12 Walling's equation has been criti- cised because it became known that termination reactions are diffusion controlled, while the chemical nature of the radical chain end is of minor importance to the rate of termination.

As a matter of fact, Φ frequently depends of f1 and one single value of Φ cannot describe the full Rp0(f1) / f1 curve. Hence it is not allowed to interpret Φ primarily in terms of the chemical effects of the radical chain ends. Furthermore Eq. 96) is not satisfactory if one of the copolymerisation parameters equals zero. In this case Eq. 96) predicts a linear dependence of the rate of polymerization with the molar fraction of the monomers,- a behaviour that is not observed experimentally (cf. PDMS-MA / MSA, PS/MSA, F8H2-MA / MSA etc.). An alternative theory taking into account diffusion control of the termination reaction came from Atherton and North and yielded Eq. 98).13,14 The diffusion control is hidden in the expression for the single termination rate constant kt(12). The expectation that kt(12) can be obtained from a composition weig- hted average of the individual monomer termination constants kt(12), Eq. 99) was disappointed.

2 2 r1 ⋅ f1 + 2 ⋅ f1 ⋅ f 2 + r2 ⋅ f 2 R P0 (f1 ) = ⋅ R I ⋅ c0 Eq. 98) r1 ⋅ f1 r2 ⋅ f 2 k t (12) ⋅ + k11 k 22

k t(12) = F1 (f1 ) ⋅ k t11 + [1− F1 (f1 )]⋅ k t22 Eq. 99)

Up until today no valid theoretical expression to calculate kt(12) has been proposed, consequently empirical relations must be used to describe the function kt(12)(f1). Frequently the denominator of Eq. 99) is replaced by an empiric expression containing two constants that are simply treated as fit-variables:

2 2 r1 ⋅ f1 + 2 ⋅ f1 ⋅ f 2 + r2 ⋅ f 2 R P0 (f1 ) = ⋅ R I ⋅ c0 Eq. 100) d1 ⋅ f1 + d 2 ⋅ f 2

(RP0 = rate of polymerization, fi = molar fraction of monomer i in the monomer mixture, RI = rate of initiation = kd ⋅ cI, c0 = total monomer concentration, ri = copolymerization parameter, di = fit constants)

Because of the unsatisfactory theoretical description of the rate equations no valid analytically integrated rate equation is available, i.e. it is impossible to express c1 and c2 as a function of the reaction time. Only the time dependence of the ratio f1 / f2 = c1 / c2 has been calculated by DeButts 39 based on Eq. 96),15 but this equations cannot be resolved for f1/f2. If the aim of a work is not to elucidate the copolymerization kinetics from a mechanistic point of view but to prepare copolymers of constant composition, the knowledge of the dependence of the rate of copolymerization on the monomer composition in a single monomer / comonomer system is sufficient. Since empirical relations are anyhow required with Eq. 96) (to describe Φ(f1)) and Eq. 98), it is more convenient directly to use empiric relations for Rp,0. First, an apparent rate constant kapp(f1) is introduced, that explicitly depends on the composition of the monomer mixture:

R P0 (f1 ) = k app (f1 ) ⋅ R I ⋅ c0 Eq. 101)

(RP0 = rate of polymerization, fi = molar fraction of monomer i in the monomer mixture, RI = rate of initiation = kd ⋅ cI, c0 = total monomer concentration, kapp(f1) = monomer composition dependent apparent rate constant)

Note that Eq. 101) seems to imply the prediction of RP0 for taking other initiators than used in the experiments to measure kapp(f1). This is in fact not true, since the initiator efficiency f, that is enclosed in the expression of RI = 1/2 (2⋅f⋅kd⋅cI) , is valid for a specific monomer / initiator system only.

n 2 3 i k app (f1 ) = a 0 + a1 ⋅ f1 + a 2 ⋅ f1 + a 3 ⋅ f1 + ⋅⋅⋅ = ∑a i ⋅ f1 Eq. 102) i=0

(f1 = molar fraction of monomer 1 in the monomer mixture, ai = fit parameters)

At one single polymerisation temperature a set of kapp(f1) values is easily obtained from experimental initial rates of polymerization, on dividing the respective Rp0(f1) by the total monomer concentration c0 and the square root of the rate of initiation. In a subsequent step a polynomial of sufficient order in f1 is fit to the data set and kapp(f1) becomes represented by Eq. 102). The knowledge of the empirical relation between f1 and kapp(f1) allows to calculate the total rate of polymerization from Eq. 102) for virtually all combinations of monomer- and initiator concentrations. The individual rates of polymerisation Rp0,i of each monomer i can subsequently be obtained by means of Eq. 93) and Eq. 95) from RP0 and the copolymerization parameters. On principle the individual apparent rate constants kapp,i could be obtained for each monomer. However, practical work revealed that inevitable small errors in representing the reaction rate curves cause intolerable large errors in the calculated copolymer compositions. Hence, the approach "describe

40 the total rate of polymerization empirically and split it in the individual rate of polymerizations according to the copolymer composition from the Lewis-Mayo equation" is superior to the approach "describe the individual rates of polymerization empirically and calculate the copolymers composition from RP0,1/RP0,2 = F1/F2".

41 3.3. - Continuous addition batch copolymerization

3.3.1. Necessity of continuous addition copolymerization procedures

Before the equations of continuous addition copolymerization procedures are developed it is necessary to find criteria when the method must be applied. Continuous addition has to make sure that (i) the rate of polymerization and (ii) the comonomer composition remains constant. Since the rates of homo- and copolymerization exponentially decrease with time, the application of monomer feed is mandatory whenever the rates need to be kept constant. It is well known that random copolymerizations and copolymerization of azeotropic mixtures will automatically meet the second condition. Hence, if one does not care about reaction rates, continuous monomer feed is of course not necessary in such cases. But frequently one has to apply a comonomer mixture different from the azeotropic composition or worse - no azeotropic behaviour is presented by the comonomer pair of in- terest. Do monomer compositions exist under such circumstances that will allow for a batch copolymerization with only minor compositional drift ? The answer to this question is obtained by calculation of the compositional drift of the copolymer composition connected to a finite monomer conversion.

1 1.0 r = 0.15 1 r = 0.35 2 F = 0.83 0.8 1,e

∆F = 0.37 0.6 1

F = 0.46 1,0 0.4 F = 0.34 1,0 copolymer composition F copolymer ∆F = -0.2 1 f = 0.5 0.2 1,0 F = 0.14 1,e

0.0 0.00.20.40.60.81.0 comonomer mixture composition f 1 Figure 6: Schematic sketch of a copolymerization diagram showing the initial composition of monomer and copolymer at two staring compositions as well as the finally obtained compositions after a certain monomer conversion pmax.

42 Figure 6 depicts a copolymerization diagram showing the initial composition of monomer mixture and copolymer as well as the finally obtained compositions after a certain monomer conversion pmax. It is obvious that the sign and the magnitude of the copolymers compositional drift ∆F1 will depend on the initial monomer composition (cf. blue and red "drift path" in Figure 6). Hence, predicting the necessity of continuous addition procedure becomes a four parameter problem, since ∆F1 will depend on the two copolymerization parameter, the initial monomer composition and the monomer conversion. The change of the monomer composition f1 with monomer conversion p is given by Eq. 103:16

df 1 1 = ⋅ (f − F ) Eq. 103) dp 1− p 1 1

(f1 = molar fraction of monomer 1 in the reaction mixture, F1 = molar fraction of monomer 1 incorporated in the copolymer, p = monomer conversion)

The connection between the monomer conversion p and the monomer composition at this conversion, f1,e, is made by integration of this equation Eq. 103) (see Eq. 104).16 Practically one first calculates the expression ln(1-p) for a given value of p and subsequently works out the value of the integral from f1,0 to x = f1,e. One has to increase x, starting from x = f1,0, until the integral becomes equal to ln(1-p): The finally chosen value of x is identical to the final monomer composition f1,e that can be obtained with the given value of the monomer conversion p.

f 1,e df ln(1− p) = ∫ 1 Eq. 104) F1 − f1 f1,0

(f1 = molar fraction of monomer 1 in the reaction mixture, F1 = molar fraction of monomer 1 incorporated in the copolymer, p = monomer conversion)

From the known final monomer composition f1,e the copolymer composition F1,e = F1(f1,e) can be calculated. With this fact in mind one can compute the resulting drift in copolymer composition ∆F1 = F1,e - F1,0 for all possible comonomer pairs by sy- stematic variation of the copolymerization parameter r1 and r2 from 0 to some final value. Now one constructs a three dimen- sional chart, where ∆F1 is plot versus r1 and r2. In the resulting chart one marks the regions of tolerable compositional drifts, e.g. |∆F1| < 0.1. With known r - parameters of a certain comonomer pair it is possible to decide at a glance whether or not continuous addition is required. Note, that such a chart will be valid for one specific monomer conversion pmax and one initial mono- 43 mer composition only. The first point is met by setting pmax to an arbitrary value, e.g. pmax = 0.9: The chart will tell then if a monomer system can be converted for 90% without producing a messy copolymer. The second point is solved by creating charts for every possible initial monomer composition f1. In praxis a few charts will be sufficient to cover the compositional space. Figure 7 depicts a colour coded plot of the expectable copolymer compositional drift ∆F1 in batch copolymerizations with 90% monomer conversion at the initial monomer composition f1 = f2 = 0.5. Ideal statistic copolymerizations (∆F1 =0) can only be expected along the white "azeotropic line", with all other systems compositional deviation are unavoidable. One can limit the deviation to ± 10% only in a wedge shaped region aside the azeotropic line and in this area continuous addition polymerizations are in fact not necessary.

∆F = F (f ) - F (p )) 1 1 1,0 1 max 2.0 -1.000 -0.9000 -0.8000 1.8 -0.7000 f = 0.5 -0.6000 1,0 -0.5000 -0.4000 -0.3000 1.6 p = 0.9 -0.2000 max -0.1000 0 0.1000 1.4 0.2000 0.3000 0.4000 0.5000 1.2 0.6000 0.7000 0.8000 0.9000 1.0 azeotropic line 1.000

2 1.010 r 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 r 1 Figure 7: Color coded plot depicting the expectable copolymer compositional drift ∆F1 in batch copolymerizations with 90% monomer conversion at the initial monomer composition f1 = f2 = 0.5.

Figure 7 contains more information than needed to make a decision on the necessity of continuous monomer addition. In the subsequent charts (Figure 8) only the ± 10% regions are colour encoded, while systems exhibiting larger drifts became

44 grey. All charts base on the assumption of 90% monomer conversion. a) ∆F = F (f ) - F (p )) 1 1 1,0 1 max 2.0 f = 0.1 -0.1000 1.8 1,0 p = 0.9 -0.05000 max 1.6 0

1.4 0.05000

1.2 0.1000 1.0 0.1100 2 r 0.8 azeotropic line 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 r 1 b) ∆F = F (f ) - F (p )) 1 1 1,0 1 max 2.0 f = 0.25 -0.1000 1.8 1,0 p = 0.9 -0.05000 1.6 max 0

1.4 0.05000

1.2 0.1000 1.0 0.1100 2 r 0.8 azeotropic line 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 r 1 Figure 8a,b: "Decision charts" depicting the expectable copolymer compositional drift ∆F1 in batch copolymerizations with 90% monomer conversion at the initial monomer composition f1 = 0.1 (a) and f1 = 0.25 (b). 45 c) ∆F = F (f ) - F (p )) 1 1 1,0 1 max 2.0 f = 0.333 -0.1000 1.8 1,0 p = 0.9 -0.05000 max 1.6 0

1.4 0.05000

1.2 0.1000 1.0 0.1100 2 r 0.8 azeotropic line 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 r 1 d) ∆F = F (f ) - F (p )) 2.0 1 1 1,0 1 max

-0.1000 1.8 f = 0.5 1,0 -0.05000 1.6 p = 0.9 max 0

1.4 0.05000

1.2 0.1000 1.0 0.1100 2 r 0.8 azeotropic line 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 r 1 Figure 8c,d: "Decision charts" depicting the expectable copolymer compositional drift ∆F1 in batch copolymerizations with 90% monomer conversion at the initial monomer composition f1 = 0.333 (c) and f1 = 0.5 (d)

46 e) ∆F = F (f ) - F (p )) 1 1 1,0 1 max 2.0

f = 0.666 -0.1000 1.8 1,0 p = 0.9 1.6 max -0.05000 0 1.4 0.05000 1.2 0.1000 1.0 2 r 0.8 azeotropic line 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 r 1 f) ∆F = F (f ) - F (p )) 1 1 1,0 1 max 2.0 f = 0.75 -0.1000 1.8 1,0 p = 0.9 -0.05000 max 1.6 0

1.4 0.05000

1.2 0.1000 1.0 0.1100 2 r 0.8 azeotropic line 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 r 1 Figure 8e,f: "Decision charts" depicting the expectable copolymer compositional drift ∆F1 in batch copolymerizations with 90% monomer conversion at the initial monomer composition f1 = 0.666 (e) and f1 = 0.75 (f)

47 g) ∆F = F (f ) - F (p )) 1 1 1,0 1 max 2.0 f = 0.9 -0.1000 1.8 1,0 p = 0.9 -0.05000 max 1.6 0

1.4 0.05000

1.2 0.1000 1.0 0.1100 2 r 0.8 azeotropic line 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 r 1 Figure 8g: "Decision charts" depicting the expectable copolymer compositional drift ∆F1 in batch copolymerizations with 90% monomer conversion at the initial monomer composition f1 = 0.9 (g).

Take the comonomer pair styrene / methyl acrylate (r1 = 0.75, r2 = 0.18) as example: On the charts in Figure 8a - 8e the point (0.75/0.18) is located in a grey area, well below the colored region indicating possible batch polymerization. But in Figure 8f (f1,0 = 0.75) and 8g (f1,0 =.9) the comonomer pair falls in the colored stripes. Hence, styrene / methyl acrylate can be batch polymerized with small copolymer compositional drifts ∆FStyrene as long as fStyrene,0 > 0.7 - 0.75. If other monomer compositions were required, the polymerization must be carried out with continuous monomer addition. Each chart in Figure 8 can be divided in four quadrants, namely:

Quadrant I: 0 < r1 < 1, r2 > 1 Quadrant II: 0 < r1 < 1, 0 < r2 < 1 Quadrant III: r1 >1, 0 < r2 < 1 Quadrant IV: r1 >1, r2 >1

Azeotropic copolymerization conditions are only met in quadrant II and IV, while the edges of the other quadrants that neighbour quadrant II and VI in some cases offer this opportu- nity. Hence, monomer pairs like maleic anhydride / ethyl acrylate (r1 = 0, r2 = 3.69) or styrene / cinnamic acid (r1 = 0, r2

48 = 1.85) can under no circumstances be run to high monomer conversions without continuous monomer addition. Figure 9 depicts the distribution of 70 common comonomer pairs over the four quadrants.17 The occupation density is lar- gest in quadrant II (42 pairs), followed by I/III (24 pairs) while only four monomer pairs can be found in quadrant IV. If one bears in mind that simple batch copolymerization is possible with monomer pairs in quadrant II only in a restricted interval of monomer compositions, it becomes obvious that the majority of copolymerization systems requires continuous addition polymerization.

2.0

1.8 I VI 1.6

1.4

1.2

2 1.0 r 0.8 II III

0.6 r r = 1 1 2 0.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 r 1

Figure 9: Distribution of 70 common comonomer pairs over the r1-r2 space (• = comonomer pair i - {r1,i, r2,i}, ο = mirror position of monomer pair i - {r2,i, r1,i})

Note, that pairs of homologous monomers, e.g. (meth)acry- late1 / (meth)acrylate2 or R1-styrene / R2-styrene, most frequently form azeotropic copolymerization systems (located in quadrant I), but this is by far no guaranty for ideal statistical copolymerization.

49 3.3.2. Applicability of homopolymerization continuous addition rate equations to copolymerizations

The equations describing the volume addition rate dV*/dt and the addition time τ in case of a homopolymerization do not hold for copolymerizations. Although it is true that adding a * * mixture of two monomers with a total concentration c = c 1 + * c 2 to a stock solution containing c0 = c10 + c20 monomer according to Eq. 44) would keep the total number of moles monomer constant at n = n0. But in this case the individual mole numbers ni of the two monomers would not remain constant, as can be seen by calculating the change in n1 and n2:

* *  dn1  c1  t c1 t   =  * −1 ⋅ R P1,0 + * ⋅ R P2,0 ≠ 0 Eq. 105)  dt t=0 c  c

Although the sum of the changes in the individual monomer molar numbers dni/dt vanishes (Eq. 106), the individual dni/dt can be large.

 dn1  * t * t   = −(1− f1 ) ⋅ R P1,0 + f1 ⋅ R P2,0  dt  dn dn1 dn 2 t=0 } = + = 0 Eq. 106)  dn 2  * t * t dt dt dt   = (1− f1 ) ⋅ R P1,0 − f1 ⋅ R P2,0  dt t=0

* (dni/dt = change in number of moles of monomer i, f i = molar fraction t of monomer i in the monomer feed solution, R Pi,0 = initial individual rate of polymerization of monomer i in the stock solution).

The equations derived for batch addition homopolymerizations can only be used to design a copolymerization experiment, * if one adjusts the ci to fulfil the condition:

t * R P1,0 F1 c1 t = = * Eq. 107) R P2,0 F2 c 2 In this section we will derive feed equations for copolymerizations that systematically take into account these facts. The basic idea of a continuous addition batch copolymerization is to replace the consumed monomer at any time of the copolymerization run to keep the ratio c1/c2 = f1/f2 constant. This requires the continuous addition of monomers in such a way, that the monomer rate of addition is identical to the individual rate of polymerization of the respective monomer. Hence, the ratio of comonomer concentrations remains constant all the time, and so does the copolymer composition. Two possible strategies can be employed: (1) - Addition of the faster polymerizing monomer only, to keep c1 / c2 = const. (2) - Feed both the monomers to the system and replace the

50 consumed monomer at any time to keep the absolute values of the monomer concentrations constant. Although the first approach will finally yield full conversion of the monomers it will not discussed here, because it requires a time dependent monomer addition program that must perfectly mirror the time dependence of the rate of polymerization. Furthermore it is a slow process compared to the second approach. In the second approach a "stock solution" of volume V0 contains the two monomers (1) and (2). At t=0 the initial monomer concentrations in the stock solutions are c0,1 and c0,2, respectively. The mixture hence contains a total monomer con- t centration c0 = c0,1+c0,2 and a total rate of polymerization R P = t t t R P1+R P2 is observed. The symbols R Pi denote the individual total rates of polymerization of the monomers. Consequently a copolymer is formed containing the molar fraction Fi of monomer i. To this reaction mixture either two separate "monomer feed solutions" are added or the two monomers are added together in one single feed mixture. In the first case the volume rate of * * addition is dV1 /dt for monomer 1 and dV2 /dt for monomer 2, * * * hence dV /dt = dV1 /dt + dV2 /dt is the total change of the volume due to monomer addition. The concentrations of the mono- * * mers in the two monomer addition solutions are c 1 and c 2. With the second method (addition of a monomer mixture) there exists only one volume addition rate dV*/dt, and the addition mixture * * * contains the over-all monomer concentration c = c 1 + c 2. Note that the monomer content of the stock solution will not be consumed during the whole addition period (any consumption is compensated by the monomer addition) and has to be se- parated from the polymer later. One can renounce this separation by polymerizing the stock monomer content subsequent to the addition procedure. Then the amount of stock monomer must be small compared to the totally added amount of monomers, otherwise the inevitable compositional drift in the resulting batch polymer may deteriorate the composition distribution of the continuous addition polymer. As with continuous addition batch homopolymerization reactions one has to decide whether to keep constant the total- or the specific rate of copolymerization. In any case balance equations must be set up for each individual monomer i. With n monomers one obtains a set of n simultaneous equations that must be solved for the respective volume addition rates. It will be shown in the next section that this task is most easily performed with the addition of one mixed monomer feed solution and keeping the total rate of copolymerization constant. To describe the required method binary copolymerizations will be discussed first.

51 3.3.3. - Addition of one mixed monomer feed solution

Figure 10: Set-up for continuous addition binary copolymerization on adding one monomer feed solution, containing a mixture of both the monomers

In the case that one feed solution containing a mixture of the two monomers is added, only one volume addition rate dV*/dt can exist. Hence, the two monomers cannot be added independently as in the previous treatment, but their delivery is coupled. Let us consider the two different addition strategies:

3.3.3.1 Constant specific rate of copolymerization

The individual concentration balance equation for each monomer I, when adding a solution containing two monomers to a stock solution reads (Eq. 108):

* n * (I) dc 1  dV  1 M p  ρ ρ  dV dV  i t *  p i  t     = − R P,i + ci ⋅  − ci ⋅ ⋅  ⋅1− ⋅ ∑  ⋅ R Pi +   +    dt V  dt  V  ρ p  M p i=1 M i   dt   dt  

The standard treatment is to neglect the volume contraction as well as the initiator addition ant to simplify the ba- t lance equations (furthermore R P,I is replaced by V⋅ki⋅ci):

* * dc1 1  * dV  c1 dV = ⋅ − V ⋅ k1 ⋅ c1 + c1 ⋅  − ⋅ = 0 Eq. 109) dt V  dt  V dt * * dc2 1  * dV  c 2 dV = ⋅ − V ⋅ k 2 ⋅ c 2 + c 2 ⋅  − ⋅ = 0 Eq. 110) dt V  dt  V dt

From each equation 109) or 110) the volume addition rate can be obtained by solving the differential equation for dV*/dt and

52 replacing ci by c0,i and RP,i by RPi.0. Remembering, that RPi,0 / RP,0 = Fi, let one find from Eq. 109):

* dV F1 ⋅ V0 ⋅ R P,0  F1 ⋅ R P,0  = * ⋅ exp * ⋅ t Eq. 111) dt c1 − c0,1 c1 − c0,1 

It is important to note that the selection of one of these equations automatically fixes the concentration of the second monomer in the addition mixture. The couplings of the delivery of the two monomers by adding the mixture is expressed form the following relation (simply obtained by equalling Eq. 109) and 110): * * (c2 − c0,2 ) ⋅ F1 = (c1 − c0,1 ) ⋅ F2 Eq. 112)

If Eq. 108) is used to determine the volume addition rate, * c2 becomes:

* F2 * c 2 = c0,2 + ⋅ (c1 − c0,1 ) Eq. 113) F1

To calculate the required addition time, the total number of moles monomer is needed that were introduced during the addition period:

t=τ dV*   F ⋅ R ⋅τ   ∆n = α ⋅ n = (c* + c* ) ⋅ ⋅dt = (c* + c* ) ⋅ V ⋅ exp 1 P,0  −1 Eq. 114) 0 ∫ 1 2 dt 1 2 0  c* − c  t=0   1 0,1  

* * * Resolving for τ and replacing c 1 + c 2 = c and n0 / V0 in Eq. 114) results in:

* c1 − c0,1 c0 τ = ⋅ ln[1+ α ⋅ * ] Eq. 115) F1 ⋅ R P,0 c

In this time period τ the total volume of the reaction mixture V becomes:

 c  V = V ⋅ 1+ α ⋅ 0 0  *  Eq. 116)  c 

The resulting polymer concentration cP is limited to the maximum value at t = τ:

α ⋅ c c = 0 P * Eq. 117) 1+ α ⋅ c0 /c

53 * * (dV /dt = volume addition rate of the mixed feed solution, ci = concentration of monomer i in the mixed feed solution, c0,i = concentration of monomer i in the stock solution, Fi = molar fraction of t monomer i incorporated in the copolymer, R P0 = initial total rate of polymerization in the stock solution = V0⋅kapp⋅c0, V0 = volume of the stock solution, α = addition factor = ∆nadded / n0).

Also with the addition of mixed monomer feed solutions the addition equations can considerably be simplified if the dilu- * tion effect is negligible. Hence, in case that c0⋅α/c 〈〈 1 the related expressions become:

* dV F1 ≈ * ⋅ V0 ⋅ R P,0 Eq. 118) dt c1

* F2 * c 2 ≈ ⋅ c1 Eq. 119) F1 α ⋅ c τ ≈ 0 Eq. 120) R P,0

 α ⋅ c0 ⋅ F1  V ≈ V0 ⋅ 1+ *  Eq. 121)  c1 

α ⋅ c0 c P ≈ Eq. 122) α ⋅ c0 ⋅ F1 1+ * c1 * * (dV /dt = volume addition rate of the mixed feed solution, ci = concentration of monomer i in the mixed feed solution, Fi = molar fraction of monomer i incorporated in the copolymer, RP0 = initial rate of polymerization in the stock solution = kapp⋅c0, V0 = volume of the stock solution, α = addition factor = ∆nadded / n0).

3.3.3.2 Constant total rate of copolymerization

The molar number balance equation reads for each monomer:

* dn i * dV = −k i ⋅ n i + ci ⋅ = 0 Eq. 123) dt dt

Like in the last section, the balance equation for monomer 1 was used to calculate dV*/dt. Inserting dV*/dt in the equation * for monomer 2 yields the required concentration c 2 in the mixed monomer feed solution. The relation coupling the monomer delivery in one mixed feed solution becomes in this case:

* * c 2 ⋅ F1 = c1 ⋅ F2 Eq. 124)

54 * * * Taking into account the relations RPi = Fi⋅Rp and c i/c = f i one obtains the following set of equations that describe the experimental conditions for continuous addition batch copolymerization at constant over-all rate of copolymerization:

* dV F1 = * ⋅ V0 ⋅ R P,0 Eq. 125) dt c1

f * α Addition time: τ = 1 ⋅ Eq. 126) F1 k app

F1 Reaction volume: V(t) = V0 ⋅ (1+ * R P0 ⋅ t) Eq. 127) c1

c Monomer concentration: c = 0 Eq. 128) F1 1+ * R P0 ⋅ t c1

R P0 Specific rate of polymn.: R p = k app ⋅ c = Eq. 129) F1 1+ * R P0 ⋅ t c1

F1 * ⋅ R P0 ⋅ t f1 Polymer concentration: c P = Eq. 130) F1 1+ * R P0 ⋅ t c1

* * (dV /dt = volume addition rate of the mixed feed solution, fi = molar fraction of monomer i in the mixed feed solution, Fi = molar fraction of monomer i incorporated in the copolymer, RP0 = initial specific rate of polymerization in the stock solution = kapp⋅c0, V0 = volume of the stock solution, α = addition factor = ∆nadded / n0).

At the end of the addition period (t = τ) the reaction mixture volume V(τ), the concentrations c(τ), cP(τ) and reaction t rates (RP, R P) become identical to the situation discussed in section 2.3.2 (continuous addition batch homopolymerization at constant total rate of polymerization) and obey to Eq. 50) - Eq. 51). Hence, at the end of the addition period the system * * * behaves as if a single monomer of concentration c (= c 1 + c 2) was fed to a stock mixture containing a monomer concentration c0 (= c0,1 + c0,2).

55 3.3.4. - Addition of two separate monomer solutions

Figure 11: Set-up for continuous addition binary copolymerization on adding two separate monomer feed solutions to the stock solution

3.3.4.1 Constant specific rate of copolymerization

With binary copolymerizations we have to set up a concentration balance equations for each monomer i, describing the change in concentration of this monomer caused by altering the systems volume and changing the individual numbers of moles of monomer i. All contributions to dn/dt and dV/dt described in connection to homopolymerizations can be transferred, only the volume contraction caused by the polymerization of n monomers is more complicated to obtain. With n monomers the molecular weight of the copolymers statistical repetition unit MP depends on the copolymer composition F1, as well as the polymers density ρP (cf. Eq. 131).

n M p  ρ ρ   p i  t dV = ⋅1− ⋅ ∑  ⋅ R P ⋅ dt ρ M p M p  i=1 i  Eq. 131) n M p = ∑ Fi ⋅ M i i=1 t (ρP = density of the polymer, ρi = density of monomer i, R P = total rate of polymerization in the system, Fi = molar fraction of monomer I incorporated in the copolymer)

With this expression the full concentration balance equation for each monomer becomes (Eq. 132):

* n * (I) dc 1  dV  1 M p  ρ ρ  dV dV  i t * i  p i  t     = − R P,i + ci ⋅  − ci ⋅ ⋅  ⋅1− ⋅ ∑  ⋅ R Pi +   +    dt V  dt  V  ρ p  M p i=1 M i   dt   dt  

56 In the following argumentation the volume contraction and the contribution of initiator solution addition will be neglected. Equation 133) and 134) describing the alteration in the monomer concentrations for monomer (1) and monomer (2) in the reaction mixture on adding two separate monomer solutions are derived with this approximations:

* * * dc1 1  t * dV1  c1 dV1 dV2  = ⋅ − R P,1 + c1 ⋅  − ⋅  +  = 0 Eq. 133) dt V  dt  V  dt dt  * * * dc2 1  t * dV2  c 2 dV1 dV2  = ⋅ − R P,2 + c 2 ⋅  − ⋅  +  = 0 Eq. 134) dt V  dt  V  dt dt 

With perfect addition rates both the monomer concentrations must remain constant at their initial stock values (ci = c0,i), while the individual rates of polymerizations also must not change (Rp,i = RP0,i). With these propositions valid one obtains the equations to determine the addition rates (Eq. 135 / 136).

dV * dV * (c* − c ) ⋅ 1 − c ⋅ 2 = R ⋅ V Eq. 135) 1 0,1 dt 0,1 dt P0,1 dV * dV * (c* − c ) ⋅ 2 − c ⋅ 1 = R ⋅ V Eq. 136) 2 0,2 dt 0,2 dt P0,2

* * Resolving Eq. 135) and 136) for dV1 /dt and dV2 /dt yields:

* * dV1 (c2 − c0,2 ) ⋅ F1 + c0,1 ⋅ F2 = * * ⋅ R P0,1 ⋅ V Eq. 137) dt (c1 − c0,1 ) ⋅ (c 2 − c0,2 ) − c0,1 ⋅ c0,2

* * dV2 (c1 − c0,1 ) ⋅ F2 + c0,2 ⋅ F1 = * * ⋅ R P0,2 ⋅ V Eq. 138) dt (c1 − c0,1 ) ⋅ (c2 − c0,2 ) − c0,1 ⋅ c0,2

The total volume change is equal to the sum of the individual * * volume addition rates (dV/dt = dV 1/dt + dV 2/dt), hence adding Eq. 137) and Eq. 138) yields the differential equation that must be solved to yield the total reaction mixture volume V(t) at an arbitrary time t:

* * dV c1 ⋅ F1 + c 2 ⋅ F2 = * * ⋅ R P0 ⋅ V Eq. 139) dt (c1 − c0,1 ) ⋅ (c2 − c0,2 ) − c0,1 ⋅ c0,2

* (dVi /dt = volume addition rate of the solution containing monomer i, * ci = concentration of monomer i in the feed solution, c0,i = concentration of monomer i in the stock solution, Fi = molar fraction of t monomer i incorporated in the copolymer, R P0 = initial total rate of polymerization in the stock solution = V0⋅kapp⋅c0, V0 = volume of the stock solution).

57 Solving Eq. 139) yields the volume V(t) and the first time derivative let one find dV/dt:

 * *  c1 ⋅ F1 + c 2 ⋅ F2 V(t) = V0 ⋅ exp * * ⋅ R P0 ⋅ t Eq. 140) (c1 − c0,1 ) ⋅ (c 2 − c0,2 ) − c0,1 ⋅ c0,2 

* *  * *  dV (c1 ⋅ F1 + c 2 ⋅ F2 ) ⋅ R P0 ⋅ V0 (c1 ⋅ F1 + c 2 ⋅ F2 ) ⋅ R P0 ⋅ t = * * ⋅ exp * *  Eq. 141) dt (c1 − c0,1 ) ⋅ (c2 − c0,2 ) − c0,1 ⋅ c0,2 (c1 − c0,1 ) ⋅ (c2 − c0,2 ) − c0,1 ⋅ c0,2 

Inserting Eq. 140) in the expressions Eq. 139) and Eq. 138) allows for calculation of the individual volume addition rates of the two individual monomer solutions (1) and (2). Because of the length of the resulting expressions three abbreviations are introduced:

* [(c2 − c0,2 ) ⋅ F1 + c0,1 ⋅ F2 ] A1 = * * Eq. 142) (c1 − c0,1 ) ⋅ (c2 − c0,2 ) − c0,1 ⋅ c0,2

* [(c1 − c0,1 ) ⋅ F2 + c0,2 ⋅ F1 ] A 2 = * * Eq. 143) (c1 − c0,1 ) ⋅ (c2 − c0,2 ) − c0,1 ⋅ c0,2

* * (c1 ⋅ F1 + c 2 ⋅ F2 ) B = * * Eq. 144) (c1 − c0,1 ) ⋅ (c2 − c0,2 ) − c0,1 ⋅ c0,2

In terms of these definitions the addition rates become:

dV * 1 = A ⋅ R ⋅ V ⋅ exp[]B⋅ R ⋅ t Eq. 145) dt 1 P0,1 0 P0

* dV2 = A ⋅ R ⋅ V ⋅ exp[]B⋅ R ⋅ t Eq. 146) dt 2 P0,2 0 P0

The required time of monomer addition depends on the targeted "addition factor" α, being the ratio between added- and initially present amount of monomers:

t=τ dV* t=τ dV* ∆n = α ⋅ n = c* ⋅ 1 ⋅ dt + c* ⋅ 2 ⋅ dt 0 1 ∫ 2 ∫ Eq. 147) t=0 dt t=0 dt

* Resolving for τ and inserting the expressions for dVi /dt let one find the required expression.

1  B c0  τ = ⋅ ln1+ * * ⋅ * ⋅ α Eq. 148) R P0 ⋅ B  f1 ⋅ A1 + f 2 ⋅ A 2 c 

58 The volume of the reaction mixture at t = τ is then:

 B c0  V = V0 ⋅ 1+ * * ⋅ * ⋅ α Eq. 149)  f1 ⋅ A1 + f 2 ⋅ A 2 c 

Dividing the total amount of fed monomer (= α⋅n0) by the reaction mixture volume at the end of the addition period yields the final polymer concentration:

α ⋅ c0 c P = Eq. 150) B c0 1+ * * ⋅ * ⋅ α f1 ⋅ A1 + f 2 ⋅ A 2 c

Although this expression is very complicated one can see that the polymer concentration is limited to a value lying in bet- * * ween c1 and c2 for sufficiently large addition factors α.

3.3.4.2 Constant total rate of copolymerization

If the monomers are added in two separate solutions, the molar number balance equation reads for each monomer:

* dn i * dV = −k i ⋅ n i + ci ⋅ = 0 Eq. 151) dt dt

Since Eq. 151) is identical to Eq. 123), the "constant total rate approach" does not distinguish between addition of a mixture of monomers in one solution and adding two different monomer solutions.

All equations Eq. 124) - Eq. 130) can be applied without any variation.

59 3.3.5. - Continuous Addition Copolymerization At Negligible Dilution

If pure low molecular weight monomers are continuously fed, while the polymerization occurs in a sufficiently diluted reaction mixture (c0 = 0.5 - 1 mol/L), the dilution effect can become negligible. In this section it will be assumed that the * * concentrations c1 and c2 of the monomer addition solutions are large compared to the stock solution concentration. In parti- * * cular in all equations we assume: ci ± c0,j ≈ ci With this prepositions valid it is obtained for the monomer volume addition rates, the addition time, the system volume and the polymer concentration:

* dV1 F1 t = * ⋅ R P0 Eq. 152) dt c1

* dV2 F2 t = * ⋅ R P0 Eq. 153) dt c 2

dV  F1 F2  t =  * + *  ⋅ R P0 ≈ 0 Eq. 154) dt c1 c 2 

c ⋅ V τ = 0 0 ⋅ α Eq. 155) R P0

  F F  1 2 Eq. 156) V = V0 ⋅ 1 + α ⋅ c 0 ⋅  * + *   c1 c 2 

c ⋅α 0 Eq. 157) c P =  F F  1 + c ⋅α ⋅  1 + 2  0  * *   c1 c2 

* (dVi /dt = volume addition rate of the solution containing monomer i, * ci = concentration of monomer i in the feed solution, c0,i = concentration of monomer i in the stock solution, Fi = molar fraction t of monomer i incorporated in the copolymer, R P0 = initial total rate of polymerization in the stock solution = V0⋅kapp⋅c0, V0 = volume of the stock solution).

Note that in this case the differences between "constant total rate" and "constant specific rate" approach vanish, as well as the differences between "addition of one mixed monomer solution" and "addition of individual monomer solutions" dis- appear.

60 3.3.6. - n-monomer feed with a mixed monomer solution

The approach of feeding a mixed solution at constant total rate of copolymerization can easily be generalised to the copolymerization of n monomers. The concentration balance equation of the first monomer is used to determine the addition volume rate of the mixed monomer feed solution:

* t dV F1 ⋅ R P0 = * Eq. 158) dt c1

Subsequently the balance equations of all other monomers are used to obtain the respective mixed feed solution concen- * tration c i of the respective monomer:

* Fi * ci = ⋅ c1 Eq. 159) F1

* Summing up all individual monomer concentrations c i of the mixed feed solution yield a relation between the total feed monomer concentration c* and the feed monomer concentrati- * on of the reference monomer, c 1:

c* c* = 1 Eq. 160) F1

With this, the total amount of fed monomer is expressed as

t=τ dV* n ∆n = α ⋅ n = ⋅ c* ⋅ dt = V ⋅ R ⋅τ 0 ∫ ∑ i 0 P,0 (Eq. 161), t=0 dt i=1 and the addition time τ is calculated to be:

α ⋅ c α τ = 0 = Eq. 162) R P,0 k app

 α ⋅ c  V = V ⋅ 1+ 0 0  *  Eq. 163)  c 

* (c = total concentration of monomer in the feed solution, c0 = total concentration of monomer in the stock solution, dV*/dt = volume addi- * tion rate of the mixed feed solution, ci = concentration of monomer i in the mixed feed solution, Fi = molar fraction of monomer i incorporated in the copolymer, RP0 = initial rate of polymerization in the stock solution = kapp⋅c0, V0 = volume of the stock solution, α = addition factor = ∆nadded / n0).

61 3.3.7. - Effect of erroneous monomer addition rates

In continuous addition homopolymerization the only effect of an error in monomer addition rate was a change in the total rate of polymerization, accompanied by an alteration of the final polymer concentration. With copolymerizations we expect in addition a shift in the copolymer composition. Prior to the discussion of the effect of mis-dosage on the reaction kinetics it is convenient to look how sensitive the copolymer composition F1 reacts to changes in the monomer composition of the reaction mixture. First the relevant measures of compositional errors must be defined. Let one assume that the actual monomer mixture composition, in terms of the monomer molar fraction f1, deviates for a value δ1 from the required composition f1,0:

f1 = f1,0 + δ1 Eq. 164)

Then the composition of the copolymer will deviate for a value ∆F1 = F1 - F1,0 from the targeted value F1,0. The relative error in copolymer composition is hence defined as:

∆F1 F1 − F1,0 F(f1 ) − F(f1,0 ) δ F1 = = = Eq. 165) F1,0 F1,0 F(f1,0 )

Obviously the relative error can be calculated by inserting the Mayo Equation (Eq. 91) in Eq. 165).

3.3.7.1 - Sensitivity of the copolymer composition on changes in monomer mixture composition

Dividing δF1 by δ1 yields a function σ1(f1,0) that should be independent of δ1 for small values of δ1. Strictly speaking, in the limit of vanishing monomer composition error δ1 the function σ1(f1,0)0 is defined as the reduced derivative of F1 for δ1:

1 dF1 σ1 (f1,0 )0 = lim ⋅ Eq. 166) δ →0 1 F1,0 dδ1

(∆F1 = F1 - F1,0 = deviation of the copolymer composition F1 from the target composition F1,0, σ1(f1,0)0 = copolymer composition sensitivity,

δ1 = deviation of the monomer molar fraction in the reaction mixture from the target value f1,0, cf. Eq. 162).

The higher the value of σ1(f1,0)0 the larger the compositional error of the copolymer will be. Hence, the function σ1(f1,0)0 will be termed the "copolymer composition sensitivity" of the copolymerization system. Note that this function is re-

62 lated to the incorporation of monomer 1 ! A vanishing sensitivity value do not opine that any error in monomer composition will be possible, but one has to consider the corresponding function σ2(f1,0)0. The latter is simply done by looking at the copolymerization system obtained by exchanging r1 vs. r2. For practical purposes we state Eq. 167) that allows to calculate the function σ1(f1,0,δ1) for arbitrary values of δ1.

1 ∆F1 F(f1,0 + δ1 ) − F(f1,0 ) σ1 (f1,0 ,δ1 ) ≈ ⋅ = Eq. 167) F1,0 δ1 F(f1,0 ) ⋅ δ1

If f1,0, δ1 and the sensitivity σ1(f1,0) of a copolymerization system are known, one can calculate the resulting deviation of the copolymer composition from the targeted value:

∆F1 = F1,0 ⋅ σ1 (f1,0 ,δ1 ) ⋅ δ1 ≈ F1,0 ⋅ σ1 (f1,0 )0 ⋅ δ1 Eq. 168)

(∆F1 = F1 - F1,0 = deviation of the copolymer composition F1 from the target composition F1,0, σ1(f1,0) = copolymer composition sensitivity

(cf. Eq. 165), δ1 = deviation of the monomer molar fraction in the reaction mixture from the target value f1,0, cf. Eq. 162).

Since σ1(f1,0,δ1) can also be calculated from the Mayo equation, its value should depend on the copolymerization parameters r1, r2 and the initial monomer composition f1,0. Figure 12 depicts four nomograms of σ1(f1,0)0 for copolymerization systems arranged with increasing values of r1. All diagrams have in common that σ1(f1,0)0 diverges at low values of f1,0, indicating that the relative compositional error δF1 of the copolymer will be large. This is simply because F1,0 goes for zero at small contents of monomer 1 in the reaction mixture. Hence, continuous addition experiments need an excellent control of the volume addition rates when targeting for monomer compositions that are poor in one monomer. All copolymerization systems that are characterized by a large value of r2 exhibit high sensitivities over the whole range of monomer compositions (cf. Figure 8a, b, c, blue lines: r2 = 5). In the central region of the monomer composition diagrams (0.3 < f1,0 < 0.7) the sensitivities become σ1(f1,0)0 > 4 - 5. If one allows for a relative copolymers composition error of 10% (∆F1/F1 ≤ 0.1) this claim limits the deviation of the monomers molar fraction below 2 - 3%. On the other hand small values of the copolymerization parameter r2 tend to lower the sensitivity in general (cf. Fi- gure 12, black lines). With sensitivities around one, a monomer compositional error of δ1 ≈ 0.1 - 0.15 is possible and not to exceed ∆F1/F1 = 0.1.

63 a) 10 b) 10

8 8

6 1 1 δ δ ] / ] / 1,0 1,0 /F /F 1 1 4 4 F F ∆ ∆ [ [

2 2

0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 f f 1,0 1,0 c) 10 d) 10

8 8

6 6 1 1 δ δ ] / ] / 1,0 1,0 /F /F 1 4 1 4 F F ∆ ∆ [ [

2 2

0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 f f 1,0 1,0 Figure 12: Nomograms depicting plots of the copolymerization composition sensitivity σ1(f1,0)0 (Eq. 164) for (a) r1=0, (b) r1=0.5, (c) r1=1.0 and (d) r1=5.0 versus the initial monomer composition f10

(r2=0.1 (), r2=0.5 (), r2=1.0 (), r2=5.0 ()).

The nomograms in Figure 12 depict the limiting case of differential changes in monomer composition δ1 (δ1→0). In Figure 9 the effect of larger values of δ1 on the composition sensitivity σ1(f1,0,δ1) is shown with the example of the comonomer pair maleic anhydride / PDMS-MA: Up to δ1 values of 0.03 - 0.05 the

64 sensitivity is very well approximated by σ1(f1,0)0, but with larger values of the monomer compositional error the sensitivity grows faster the proportional. One learns from this curves that monomer compositional errors δ1 must not exceed 0.1-0.15, since otherwise the absolute error in copolymer composition exceeds one. δ = 0.75 10 1

1 δ = 0.5 1 δ ] / 1,0 /F 1 4 F ∆ [ δ = 0.1 1 δ = 0.05 1 δ = 0.01 2 1 δ = 0.001 1

0 0.00.20.40.60.81.0 f MSA,0

Figure 13: Nomogram depicting plots of the copolymerization composition sensitivity σ1(f1,0)0 in the system MSA / PDMS1k-MA (r1 = 0, r2 =

4.5) versus the initial monomer composition fMSA,0 (: δMSA = 0.001,

: δMSA = 0.01, : δMSA = 0.05, : δMSA = 0.1, : δMSA = 0.5, : δMSA = 0.75).

65 3.3.7.2 - Changes in monomer mixture composition due to small monomer volume addition errors

For sake of simplicity we will restrict the discussion of the copolymerization kinetics in the presence of monomer addition to the "constant total rate of polymerization" approach (dn/dt = 0) and we will assume only small errors in the volume addition rate. The reaction kinetic of a binary copolymerization is described by two simultaneous differential equations (Eq. 169) that are coupled via the apparent individual rate constants ki:

* * dn1 * dV dn 2 * dV = -k1 (f1 ) ⋅ n1 + c1 ⋅ and = -k 2 (f1 ) ⋅ n 2 + c 2 ⋅ Eq. 169) dt dt dt dt

* (ni = molar number of monomer i in the reaction mixture, ci = concentration of monomer i in the feed solution, dV*/dt = monomer solution volume addition rate, ki(f1) = individual apparent rate constant of polymerization for monomer i, f1 = ni / (n1+n2)= actual molar fraction of monomer i in the reaction mixture)

The relation between the ki and the apparent rate constant of the polymerization is given by Eq. 170):

k app (f1 ) 1 k1 (f1 ) = ⋅ Eq. 170) f1 1+ F2 F1

(ki(f1) = individual apparent rate constant of polymerization for monomer i, kapp = apparent rate constant of the copolymerization monomer mixture, f1 = ni / (n1+n2)= actual molar fraction of monomer i in the reaction mixture, Fi = molar fraction of monomer i, incorporated to the copolymer).

According to Eq. 170) both the ki explicitly depend on the molar fraction f1 of monomer 1 in the reaction mixture. Since f1 = n1 / (n1+n2), in Eq. 169) the differential quotients dni/dt are correlated in a very complicated manner to both the molar numbers ni. No analytical solution is possible when using Eq. 96), Eq. 98) or Eq. 102) to describe the function kapp(f1). However, an approximate solution is possible with a list of assumptions: a) The error in volume addition rate ε• should be very small •* • compared to the "ideal" volume addition rate (V )0: ε / •* (V )0 〈〈 1. b) The actual molar number of monomer i is close to the initial stock solution molar number, ni,0: ni = ni,0 + υi. c) The individual rate constants vary linearly with the molar

fraction of monomer f1: ki(f1) ≈ ki(f1,0) + γ(f1,0)⋅(f1-f1,0).

66 The assumptions are consistent, since a small error in feed addition will cause only a small deviation of the monomers molar numbers from the initial values. Hence, the monomers molar fractions fi are only slightly altered, and in close vicinity of the starting composition the reaction rate constants can be linearized. With these assumptions one obtains for the molar fraction f1 of monomer 1 in the reaction mixture:

ν1 ν 2 f1 ≈ f1,0 + (1− f1,0 ) ⋅ − f1,0 ⋅ Eq. 171) n 0 n 0

(f1 = ni / (n1+n2)= actual molar fraction of monomer i in the reaction mixture, f1,0 = initial molar fraction of monomer i in the reaction mixture, n0 = initial number of moles monomer in the reaction mixture, υi = deviation of the mole number of monomer i from the initial value: n1 = n1,0 +υ1).

The individual rate constants ki become approximated by the relations:

γ1,0 γ1,0 k1 ≈ k1,0 + ⋅ (1− f1,0 ) ⋅ ν1 − ⋅ f1,0 ⋅ ν 2 Eq. 172) n 0 n 0

γ 2,0 γ1,0 k 2 ≈ k 2,0 − ⋅ (1− f1,0 ) ⋅ ν1 + ⋅ f1,0 ⋅ ν 2 Eq. 173) n 0 n 0

The coefficients γ1,0 describing the linear change of k1 with the molar fraction f1 is obtained by taking the first derivative of k1(f1) (Eq. 170) with respect to f1 at the value f1 = f1,0:

 dk   1  γ1,0 =   Eq. 174) df1  f1 =f1,0

    k1 (f1,0 )  dk app  k1 (f1,0 ) F2 (f1,0 )  dF1  γ1,0 = ⋅   − k1 (f1,0 ) + ⋅ 1+  ⋅  Eq. 175)   F (f )  df  k app (f1,0 )  df1   F1 (f1,0 )  1 1,0   1 f =f  f1 =f1,0    1 1,0

 dF  (2r -1) f + f (r -1) f − (r -1) f  1  = 1 1,0 2,0 − 2F (f ) ⋅ 1 1,0 2 2,0 with:   2 2 1 1,0 2 2 Eq. 176) df1 r1f1,0 + 2 f1,0f 2,0 + r2f 2,0 r1f1,0 + 2 f1,0f 2,0 + r2f 2,0  f1 =f1,0

(ki = apparent rate constant of polymerization for monomer i at an arbitrary monomer composition, ki,0 = apparent rate constant of polymerization for monomer i at the initial stock monomer composition fi,0, n0 = initial number of moles monomer in the reaction mixture, υi = deviation of the mole number of monomer i from the initial value: ni = ni,0 +υi, γi,0 = (dki/df1)f1=f1,0).

Analogous expressions can be derived for γ2,0.

67 Although Eq. 175) and Eq. 177) look very complicated, they are entirely composed of constants, they result in constant values and will hence not introduce difficulties to the integration of the rate equations. Insert the pre-requisites in the differential equation Eq. * •* 169), remember that dn1,0/dt = -k1,0⋅n1,0 + c1 ⋅(V )0 = 0 (Eq. 123) 2 and assume that υ1 ⋅γ1,0/n0 and υ1⋅υ2⋅γ1,0/n0 are negligible against all terms linear in υi. The remaining Eq. 177) gives a first order linear differential equation system in υ1, υ2 and the reaction time t with constant coefficients:

dν • 1 ≈ -(k + γ ⋅ f ⋅ f ) ⋅ ν + γ ⋅ f 2 ⋅ ν + c* ⋅ ε Eq. 177) dt 1,0 1,0 1,0 2,0 1 1,0 1,0 2 1 dν • 2 ≈ -(k + γ ⋅f ⋅f ) ⋅ ν + γ ⋅f 2 ⋅ ν + c* ⋅ε Eq. 178) dt 2,0 2,0 1,0 2,0 2 2,0 2,0 1 2

The terms ki,0+γ1,0⋅f1,0⋅f2,0 denote new rate constants that will be (eff) abbreviated as the "effective" rate constants ki of monomer i at the respective experimental conditions. For the further treatment of the differential equation system we introduce two additional abbreviations for not to overload the next equations with symbols:

dν 1 = -k(eff) ⋅ ν + g ⋅ ν + ∆ Eq. 179) dt 1,0 1 1,0 2 1 dν 2 = -k (eff) ⋅ ν + g ⋅ ν + ∆ Eq. 180) dt 2,0 2 2,0 1 2

(eff) with: k i,0 = k i,0 + γ i,0 ⋅ f1,0 ⋅ f 2,0 Eq. 181) 2 gi,0 = γi,0 ⋅ fi,0 Eq. 182) • * ∆ i = ci ⋅ ε Eq. 183)

(ki,0 = apparent rate constant of polymerization for monomer i at the initial stock monomer composition fi,0, υi = deviation of the mole number of monomer i from the initial value: ni = ni,0 +υi, n0 = initial number of moles monomer in the reaction mixture, γi,0=(dki/df1)f1=f1,0).

To solve a system of two first-order differential equations it is convenient to transform the system in one single second order differential equation by resolving Eq. 178) for υ1, calculating the derivative dυ1/dt and inserting both expressions (cf. Eq. 184) in Eq. 179.

(eff) 2 (eff) 1 dν 2 k 2,0 ∆ 2 dν1 1 d ν 2 k 2,0 dν 2 ν1 = ⋅ + ⋅ ν 2 − , = ⋅ 2 + ⋅ Eq. 184) g 2,0 dt g 2,0 g 2,0 dt g 2,0 dt g 2,0 dt

68 After rearranging the symbols in descending order of time de- rivatives of υ2 one finds: d 2 ν dν 2 + (k (eff) + k (eff) ) ⋅ 2 + (k (eff) ⋅ k (eff) − g ⋅ g ) ⋅ ν = k (eff) ⋅ ∆ − g ⋅ ∆ Eq. 185) dt 2 1,0 2,0 dt 1,0 2,0 1,0 2,0 2 1,0 2 2,0 1

d 2 ν dν 2 + a ⋅ 2 + b ⋅ ν = c Eq. 186) dt 2 dt 2

(eff) (eff) with: a = k1,0 + k 2,0 Eq. 187)

(eff) (eff) b = k1,0 ⋅ k 2,0 − g1,0 ⋅ g 2,0 Eq. 188)

(eff) c = k1,0 ⋅ ∆ 2 − g1,0 ⋅ ∆ 2 Eq. 189)

The general solution of an inhomogeneous differential equation of second order with constant coefficients and a time invari- ant perturbation function c can be obtained from the literature:18 c a ν = + []C ⋅ y (t) + C ⋅ y (t) ⋅ exp(− ⋅ t) Eq. 190) 2 b 1 1 2 2 2

2 D > 0 : y1 (t) = exp(D ⋅ t/2), y2 (t) = exp(−D ⋅ t/2) D 2 = 0 : y (t) = t, y (t) = 1 1 2 Eq. 191) 2 D < 0 : y1 (t) = sin(D'⋅t/2), y2 (t) = cos(−D'⋅t/2) D'= b − 4a • ∆ 2 ⋅ y2 (t = 0) − ν 2,∞ ⋅ y2 (t = 0) C1 = − • • Eq. 192) y2 (t = 0) ⋅ y1 (t = 0) − y1 (t = 0) ⋅ y2 (t = 0)

• ∆ 2 ⋅ y1 (t = 0) − ν 2,∞ ⋅ y1 (t = 0) C1 = • • Eq. 193) y2 (t = 0) ⋅ y1 (t = 0) − y1 (t = 0) ⋅ y2 (t = 0)

The applicable expressions for the solution functions y1(t) and 2 y2(t) depend on the sign of the square determinant D = 4a-b. By means of Eq. 187) and 188) it is simple to calculate D2 (see Eq. 192), but without specific numerical values of ki,0, γi,0 and 2 f1,0 it is impossible to assign a sign to D .

2 2 2 D = (k1,0 - k 2,0 ) + 2 ⋅ (k1,0 + k 2,0 )(γ 2,0 − γ1,0 ) ⋅ f1,0f 2,0 + [(γ 2,0 + γ1,0 ) ⋅ f1,0f 2,0 ] Eq. 194)

(ki,0 = apparent rate constant of polymerization for monomer i at the initial stock monomer composition fi,0, γi,0 = (dki/df1)f1=f1,0 = slope of the ki / f1 curve at f1 = f1,0).

69 Figure 14 depicts the three possible types of solution curves. In all cases the curves start at t = 0 with υ2 = 0 and they converge to a value υ2,∞ at long reaction times (a⋅t 〉〉 1). Hence, after long reaction times a constant deviation of the monomer mixture composition as well as the copolymer composition is achieved, because the exponential term vanishes. The limiting value υ2,∞ at infinite reaction times is given by the quotient c/b. The limiting values υ2,∞ and υ1,∞ become:

* (eff) * • g1,0 ⋅ c 2 − k 2,0 ⋅ c1 ν1,∞ = (eff) (eff) ⋅ ε Eq. 195) k1,0 ⋅ k 2,0 − g1,0 ⋅ g 2,0

(eff) * * • k1,0 ⋅ c 2 − g 2,0 ⋅ c1 ν 2,∞ = (eff) (eff) ⋅ ε Eq. 196) k1,0 ⋅ k 2,0 − g1,0 ⋅ g 2,0

On inserting Eq. 195) and 196) in Eq. 170) one obtains the limiting value of the error in monomer composition δ1,∞ = f1,∞ - f1,0. • ν1,∞ ν 2,∞ * δ1,∞ ≈ (1− f1,0 ) ⋅ − f1,0 ⋅ ∝ c1 ⋅ ε Eq. 197) n 0 n 0

Note that this final compositional error in the monomer mixture does not necessarily represent the maximum error that can occur during the reaction time (see Figure 14).

(max) ν i i ν

ν i,00

012345678910 reaction time τ = a t

Figure 14: Plots of the solution functions υi(t) (Eq. 188) for the three different values of the square determinant D2 (: D2 < 0, : D2 = 0, : D2 > 0).

70 2 According to Figure 14 only for D > 0 the deviation υ2 2 steadily grows and eventually reaches υ2,∞. With D < 0 (Figure 9, black curve) the value of υ2(t) oscillates some times around 2 υ2,∞ and with D = 0 (Figure 9, green line) one maximum deviati- (max) on υ2 can occur. The number of oscillations and the value of (max) υ2 depend on the specific numerical values of ki,0, γi,0 and (max) f1,0. Finding the value of υ2 and δ1,max requires a detailed discussion of the time dependence of υ1(t) and υ2(t). The full expression 197) may become very complicated when replacing υ1,∞ and υ2,∞ as well as the other abbreviations, but this awfully procedure is not required. Note that at any reaction time the monomer mixture compositional error δ1 is direct proportional to the error in volume addition ε• and the concentration of monomer 1 in the monomer feed solution (this is a general property of the solution equations 190) / 191). Re- garding Eq. 168) the compositional deviation of the copolymer is also proportional to this values and the compositional sensitivity:

• ∆F1 * ∝ σ1 (f1,0 )0 ⋅ c1 ⋅ ε Eq. 198) F1,0

(∆F1 = F1 - F1,0 = deviation of the copolymer composition F1 from the target composition F1,0, σ1(f1,0) = copolymer composition sensiti- * vity (cf. Eq. 165), c1 = concentration of monomer 1 in the mixed monomer feed solution, ε• = absolute error in feed volume addition rate).

Although the value of the copolymers compositional error is not calculated from Eq. 197, three conclusions can be drawn to design continuous addition copolymerizations:

(1) Keep the error in volume addition rate ε• as small as pos- • sible, since ∆F1 ∝ ε .

* (2) Since ∆F1 ∝ c1 , keep the monomer feed concentration low (a further benefit comes from the lower final polymer concentration in the reaction mixture, see Eq. 129).

(3) Avoid continuous addition experiments in any region of the copolymerization diagram exhibiting large compositional sensitivities: ∆F1 ∝ σ1(f1,0).

3.3.7.3 - Changes in monomer mixture composition due to arbi trary volume addition errors

In the previous section the monomers molar number balance differential equation system, Eq. 169), was analytically treated in the limit of small relative volume addition errors ϕ. It was shown as a general property of copolymerization kinetics that the absolute error in monomer composition ∆F was proportional to the absolute error in volume addition rate, ε• and * the monomer concentration in the feed solution ci . Depending on the kinetic parameters the analytic solution predicted the occurrence of extreme values of the copolymer compositional error during the reaction, but without specified values of ki,0, γi,0 and f1,0 neither the type of solution curve nor the extend of the extreme values can be obtained. However, with specific values of kinetic parameters describing a copolymerization system at hand, the discussion is obviously bound to this specific comonomer pair and looses its generality. If one has to scarify the general validity of the discussion one should at least not be restricted to small errors. The latter restriction cannot be annulled with analytical techniques, but is easily possible with numerical integration procedures. Of course, the numerical treatment of Eq. 169) requires specific numerical input data on the reaction kinetic parameters. Consequently the numerical solution is valid only for the single comonomer pair under discussion,- but as found above, the analytical solution is not superior with this respect. In this section three examples of copolymerization systems will be discussed that were numerically treated. Equation system 169) was integrated by a Fourth-Order Runge-Kutta algori- thm19 with self-adapting step-width, as implemented in the program COPOSIM1. COPOSIM1 uses the approach outlined in Chapter 3.2 / p. 41 and applies empiric expressions (e.g. Eq. 100 - 102) to calculate the individual rate constants ki for each monomer at any monomer mixture composition f1. The integration yields tables of monomer and initiator molar numbers, molar fraction of monomer 1 as well as the molar fraction of monomer incorporated in the copolymer versus the reaction time. From these data the relative and absolute errors in monomer mixture- and copolymer composition can be obtained. Three monomer pairs have been selected, namely:

(1) Poly(dimethylsiloxane) monomethacrylate / maleic anhydride (=PDMS1k-MA / MSA, r1 = 4.5, r2 = 0). (2) Maleic anhydride / fluorooctyl methacrylate (= MSA / F8H2MA, r1 = 0, r2 = 4.3).

(3) Styrene / methyl acrylate (Sty / MA, r1 = 0.76, r2 = 0.18).

72 The first two pairs are similar in copolymerization parameters, but largely differ in their respective dependence of the rate constants on the monomer composition: While the apparent rate constant kapp of PDMS1k-MA / MSA exhibits a maximum around fPDMS-MA = 0.5 (cf. Figure 24) in mixtures of MSA and F8H2MA kapp continuously decreases with increasing MSA content (cf. Figure 30) The styrene / MA system exhibits an azeotropic copolymerization and on exceeding a styrene content of 10 - 15 mol% the apparent rate constant is only slightly dependent on the monomer mixture composition (cf. Figure 32).

3.3.7.3.1 - PDMS1k-MA / MSA

Figure 15 depicts some primary results of the numerical integration of Eq. 167) for the 1 : 1 mixture of the comonomer pair poly(dimethylsiloxane) monomethacrylate / maleic anhydride (= PDMS1k-MA / MSA). For sake of comparability the time scale is given in units of the addition factor α (remember that α = 1 is equivalent to the reaction time required to convert the initial number of monomers at constant initial rate of polymerization). With perfect addition (Figure 15a, red line) no change in the monomers molar numbers occurs and consequently the respective molar fractions remain unaltered (Figure 10b). The green curves in Figure 10a show the effect of 50% over-dosage (ϕ = • *• ε /(V )0 = 0.5) in monomer feed: The PDMS1k -MA molar number rapidly increases from n1,0 = 0.5 mol to 0.85 mol within two units of alpha and subsequently relaxes slowly to the end a) b) 0.9 0.9

0.8 0.8 0.7 0.7

0.6 PDMS-MA 0 , F /n i 0.5 0.6 n PDMS-MA 0.4 f 0.5

0.3 0.4 0.2 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 alpha alpha Figure 15: Plots of the molar numbers of the monomers PDMS1k-MA and MSA (a) and the monomer molar fraction in the reaction mixture and the copolymer (b) against the addition factor α with different relative monomer feed volume errors ϕ (f0,PDMS-MA = 0.5, n0 = 1.0 mol, nI = 0.04 mol, T = 60 °C).

(a) ϕ = 0.5:  = nPDMS-MA,  = nMSA, ϕ = 0.0:  = nPDMS-MA = nMSA, ϕ = -

0.5:  = nPDMS-MA,  = nMSA, (b) ϕ = 0.5:  = fPDMS-MA,  = FPDMS-MA, ϕ =

0.0:  = fPDMS-MA,  = FPDMS-MA, ϕ = -0.5:  = fPDMS-MA,  = FPDMS-MA.

73 value of 0.75 mol. The maleic anhydride reacts more slowly and steadily increases its concentration to the final value of 0.75 mol without showing a maximum curve. Hence, the final molar composition of the monomer mixture becomes identical to the initial value f0,PDMS-MA, although the content of monomers in the reaction system has considerably increased (n0 = 1 mol, n∞ = 1.5 mol). Under-dosage for -50% has a similar effect, but the deviations from the targeted molar number become negative. Note that the "over-shotage" in the amount of PDMS1k-MA is less pronounced (Figure 10a, blue lines). When the molar fraction fPDMS- MA is calculated from the molar numbers nPDMS-MA and nMSA, one finds that under-dosage has a more severe effect on the deviations in fPSMS-MA than the over-dosage: Due to 50% over (max) dosage a maximum deviation of f PDMS -f0,PDMS = +0.07 develops, but with under-dosage that maximum deviation becomes -0.14 (cf. Figure 15b). The deviations in monomer composition become transferred to the copolymer composition, here are the maximum deviations +0.03 (ϕ = 0.5) and -0.06 (ϕ = 0.5), respectively.

0.06 0.05 0.04 0.03 0.02 0.01 0.00

0,PDMS-MA -0.01 /F -0.02 -0.03

PDMS-MA -0.04 F

∆ -0.05 -0.06 -0.07 -0.08 0 2 4 6 8 101214161820 addition factor α

Figure 16: Plots of the relative copolymer compositional error ∆F1/ F1,0 versus the addition factor α for different relative volume addition rate errors ϕ with the comonomer pair poly(dimethylsiloxane) monomethacrylate / maleic anhydride (=PDMS1k-MA / MA) (f0,PDMS-MA = 0.5, ϕ = -0.5: , ϕ = -0.25: ,ϕ = -0.1: ,ϕ = 0: ,ϕ = 0.1: ,ϕ = 0.25: ,ϕ = 0.5: , ϕ = 1.0: ).

It is remarkable that due to the slow reaction kinetic of the PDMS1k-MSA /MSA copolymerisation the molar numbers and molar fractions do not become constant before α = 10. Since in laboratory scale preparations α values of 10 - 15 are sparsely 74 exceeded, this result means in praxis that the monomer molar fractions cannot be considered as constant during the whole addition period. Figure 16 depicts the time development of the relative error in copolymer composition ∆FPDMS-MA / F0,PDMS-MA of a PDMS1k-MA / MSA = 1 : 1 mixture for eight different values of ϕ. Indepen- dently of the addition error the resulting curves are of similar shape: A maximum deviation is reached between α = 1 - 2 and subsequently the copolymer compositional deviation conver- ges against zero between α = 8 - 10. In Figure 17 the maximum relative error in copolymer composition ∆FPDMS-MA / F0,PDMS-MA is plotted against the relative volume addition rate error ϕ for five different initial monomer compositions. In this copolymer system the copolymer composition becomes more affected in the presence large MSA fractions. This is in accordance to the discussion on the copolymer compositional sensitivity (cf. Figure 12d, black line: r1 = 5, r2 = 0).In any case the effect of an under-dosage (ϕ < 0) is larger that the respective over-dosage (ϕ > 0). However, the total relative error in copolymer composition is moderate with values below |∆FPDMS-MA / F0,PDMS-MA| < 0.06. Figure 18 depicts the copolymerization diagram of PDMS1k- MA / MSA. The different copolymer compositions that can be expected from the maximum deviations are plotted in addition to the Lewis-Mayo copolymerization line. The average copolymer composition arising from an actual feed experiment with an

0.07 0.06 0.05 0.04 0.03 0.02

0.01 (max) ) 0.00 MSA

F -0.01 ∆ ( -0.02 -0.03 -0.04 -0.05 -0.06 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 φ Monomer (max) Figure 17: Plots of the maximum copolymer compositional error ∆F1 versus the relative volume addition rate error ϕ with the comonomer pair poly(dimethylsiloxane) monomethacrylate / maleic anhydride (=

PDMS1k-MA / MA) (f0,MSA = 0.1: , f0,MSA = 0.2: , f0,MSA = 0.5: , f0,MSA

= 0.65: , f0,MSA = 0.8: ). 75 error prone addition rate will be found between the ideal copolymerization composition (violet line in Figure 18) and the line related to the experimental relative error in monomer volume rate, ϕ. Hence, by means of Figure 18 it is possible to estimate whether or not the existing experimental accuracy is sufficient to produce a copolymer of the desired compositional uniformity.

0.5 F 0.5 0,PDMS-MA

F(max) 0.4 φ = 0.1 0.4 φ = 0.5 φ = 1.0 φ = -0.5 0.3 0.3 MSA

F 0.2 0.2

0.1 0.1

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 f 0,MSA (max) Figure 18: Plots of the copolymer compositions F1(∆F1 ) versus the initial monomer composition f0,PDMS-MA for different relative volume addition rate errors ϕ with the comonomer pair poly(dimethylsiloxane) monomethacrylate / maleic anhydride (= PDMS1k-MA / MSA) (ϕ= -0.5: , ϕ = 0: , ϕ = 0.1: , ϕ= 0.5: , ϕ= 1.0: ).

The copolymer pair PDMS1k-MA / MSA is relatively tolerant against monomer addition errors in particular if more monomer is fed than required. If one has determined the initial rate of polymerization at a desired monomer composition with an accuracy of 10 - 20%, the experimental addition error should not exceed ϕ = 0.12 - 0.22: Under these circumstances the copolymers composition should not vary for more than 5 - 10 % in PDMS1k-MA. If one compares this result to a pure batch copolymerization without monomer addition, where a copolymer compositional variation from the starting composition F0,PDMS-MA down to FPDMS-MA = 0.5 is to be expected, the superiority of continuous addition copolymerization procedures becomes visible. 76 3.3.7.3.2 - MSA / F8H2MA

Figure 19 depicts the time dependence of the relative copolymer compositional errors ∆FMSA / F0,MSA with an 1 : 1 mixture of maleic anhydride and perfluorooctyl methacrylate obtained with different errors in monomer feed ϕ. When compared to the analogues plot of MSA/PDMS1k-MA (Figure 16) obviously the curves are of the same type. The errors increase to a maximum value and subsequently fall to zero at large addition times. However, with the MSA/F8H2MA mixture all errors are larger and the deviations need longer addition times to converge against zero: While an addition factor of α = 10 is sufficient to reach the state of negligible deviation with MSA/PDMS1k-MA, (Figure 16) one needs to exceed α = 20 in the present system.

0.25 0.20 0.15 0.10 0.05 0,MSA

/F 0.00 MSA

F -0.05 ∆ -0.10 -0.15 -0.20 0 2 4 6 8 10 12 14 16 18 20 alpha

Figure 19: Plots of the relative copolymer compositional error ∆F1/

F1,0 versus the addition factor α for different relative volume addition rate errors ϕ with the comonomer pair perfluorooctyl methacrylate / maleic anhydride (=F8H2MA / MA) (f0,MSA = 0.5, ϕ = - 0.5: , ϕ = -0.25: ,ϕ = -0.1: ,ϕ = 0: ,ϕ = 0.1: ,ϕ = 0.25: ,ϕ = 0.5: , ϕ = 1.0: ).

Note that the maximum absolute errors in copolymer composition ∆FMSA (Figure 20) arising from ill-fed MSA/F8H2MA mixtures are similar to that of the PDMS1k-MA/MSA system when compared over the whole range of monomer composition and relative feed errors. But one has to bear in mind that because of the slower error-relaxation in the MSA/F8H2MA system the deviations will have a more severe effect on the average copolymer composition. Although the copolymerization parameters of MSA /PDMS1k-MA and MSA/F8H2MA are similar, the differences in reaction kinetics cause the differences in the sensitivity of the resulting copolymer composition on the error in monomer feed. 77 0.05 0.04 0.03 0.02

0.01

(max) 0.00 )

MSA -0.01 F ∆ ( -0.02 -0.03 -0.04 -0.05 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 φ Monomer (max) Figure 20: Plots of the maximum copolymer compositional error ∆F1 versus the relative volume addition rate error ϕ with the comonomer pair perfluorooctyl methacrylate / maleic anhydride (=F8H2MA / MSA)

(f0,MSA = 0.1: , f0,MSA = 0.5: , f0,MSA = 0.75: , f0,MSA = 0.9: ).

Figure 21 depicts the copolymer compositions to be obtained when applying error-prone monomer feeds to MSA/F8H2MA copolymerization reactions. Like with the MSA/PDMS1k-MA copolymerization system it becomes obvious that over-dosage of monomers is less dangerous to the copolymer composition than under -dosage: Without any monomer feed the reaction would proceed until complete consumption of the reactive monomer (PDMS1k-MA or F8H2MA). The residuum, almost entirely consisting of maleic anhydride will produce oligomers containing ≈ 50 mol% MSA, independently of the initial monomer composition. With an 1 : 1 monomer mixture this means a compositional drift of about 35 mol%. Feeding "only" the half of the required amounts of monomers (ϕ = -0.5) dramatically improves this situation. The maximum compositional deviation remains below 5 - 6 mol% and from Figure 14 one knows that this maximum deviation exists over a very limited period of time. When feeding 50% more monomer per time than consumed by polymerization results in a copolymer that contains 2 - 3 mol% less maleic anhydride than aimed for. Again, this represents the peak in compositional deviation and the long time average will be considerably lower. With a rea- listic feed error of 10 - 20% (ϕ = ±0.2) it is expected that the copolymer composition will not differ from the target value for more than 2.5 - 5 mol%. Since this is in the range of error of the NMR compositional analysis it is allowed to state that moderate feed errors will make not much of a difference to perfect monomer addition.

78 0.5 0.5

0.4 0.4

0.3 0.3 MSA

F 0.2 0.2

0.1 0.1

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 f 0,MSA (max) Figure 21: Plots of the copolymer compositions F1(∆F1 ) versus the initial monomer composition f0,MSA for different relative volume addition rate errors ϕ with the comonomer pair maleic anhydride / perfluorooctyl methacrylate (= MSA / F8H2MA) (ϕ = -0.5: , ϕ = 0: , ϕ = 0.1: , ϕ = 0.5: , ϕ = 1.0: ).

3.3.7.3.3 - Styrene / Methyl acrylate

The comonomer pair styrene / methyl methacrylate exhibits an azeotropic copolymerization, with the azeotropic point at fStyrene = 0.774. In the vicinity of the azeotropic point the copolymers compositional drift will be negligible and copolymerizations can produce products of uniform composition without monomer feed. But since the compositional drift rapidly increases with the distance from the azeotropic point, continuous addition becomes mandatory with fStyrene < 0.6. Because of the asymmetry of the copolymerization diagram the azeotropic point "protects" mixtures that are rich in styrene from strong compositional drifts. The plots of the relative compositional error against the monomer feed time look similar to the curves presented in Fi- gure 16. Up to addition factors of α = 10 the maximum deviations, occurring between α = 1 - 1.5, are crossed and relaxed to almost zero. Note that the polymerization is much faster with Styrene / MA that with the previously mentioned monomer pairs.

79 One unit of α represents 1.6 minutes, while the respective time scale is 162 minutes for PDMS1k-MA/MSA and 477 minutes with MSA/F8H2MA (f1 = 0.5, 1 mol% AIBN at 60°C).

1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 Styrene

F 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 f 0,Styrene (max) Figure 22: Plots of the copolymer compositions F1(∆F1 ) versus the initial monomer composition f0,Styrene for different relative volume addition rate errors ϕ with the comonomer pair styrene / methyl acrylate (= Styrene / MA) (ϕ= -0.5: , ϕ = 0: , ϕ = 0.1: , ϕ= 0.5: , ϕ= 1.0: ).

Figure 22, depicting the expected copolymer compositions with different addition errors in the respective maximum of the error / time curve, shows that large copolymer compositional deviations will arise only below fStyrene < 0.3. The pro- tective impact of the azeotropic point at fStyrene = 0.77 is well illustrated by this picture. The copolymer system styrene / MA reacts less sensitive on monomer feed addition errors then both the MSA containing comonomer pairs, although large compositional drifts will occur when a batch polymerization without monomer addition is performed. In summary it is to conclude that even a "badly" controll- led continuous addition copolymerization procedure yields copolymers with superior compositional uniformity as compared to non-addition batch polymerizations. Since many copolymerization systems are of low sensitivity against feed errors, it is simple to restrict copolymer composition deviations below 3 - 5 mol%.

80 4.Empirical Rate Expressions of Comonomer Pairs

4.1. - The System PDMS1k-MA / MSA

The copolymerization diagram of the monomer pair PDMS1k-MA / MSA strongly deviates from the ideal statistic copolymerization (r1 = r2 = 1), since the MSA cannot homopolymerize. The copolymerization parameters are:

rPDMS-MA = 4.5 rMSA = 0

Figure 23: Copolymerization diagram of PDMS1k-MA / MSA. Blue line: rMSA = 0, rPDMS-MA = 4.5, dotted line: r1 = r2 = 1.

At 60 °C the initial rates of copolymerization Rp,0 was measured for different PDMS-MA / MSA mixtures in the presence of the initiator V-601 (Wako). From the observed rates the respective apparent rate constants k(f1) were calculated:

R p,0 R p,0 = q Pt (f1 ) ⋅ k d c I,0 ⋅ c0 ⇒ q Pt (f1 ) = Eq. 199) c0 ⋅ k d c I,0

(Rp,0 = measured initial rate of copolymerization, f1 = molar fraction of PDMS1k-MA in the monomer mixture, c0 = total monomer concentration = cPDMS-MA + cMSA, CI,0 = initiator concentration, kd = decomposition constant of the initiator).

81 Table 2 summarizes the obtained rate constants:

Table 2: Apparent rate constants of the copolymerization of PDMS1k-MA / MSA

-1 -1 -1 f1 = fPDMS-MA kPDMS-MA/min kMSA/min kapp/min qPt(f1) / (L/mol⋅min)1/2 1 5⋅10-5 0 5⋅10-5 0.01828 0.8 0.005 0.0012 0.00424 1.39044 0.5 0.017 0.003 0.01 2.63093 0.2 0.036 0.0038 0.01024 1.74865 0 0 0 0 0

3.0 q = Rp / {([I] k )1/2 [M]} Pt d

2.5

2.0 Data: Data1_kapp Model: CoPoTest

Chi^2/DoF = 0.00033

/min] R^2 = 0.99994

1/2 1.5 a0 0.01828 ±0 a1 10.21852 ±0.29278 a2 -5.55895 ±1.52744 a3 -13.05046 ±2.46033 a4 8.39089 ±1.23208

[(L/mol) 1.0 Pt q

0.5

0.0 0.0 0.2 0.4 0.6 0.8 1.0 f = f(PDMS -MA) 1 1k

Figure 24: Dependence of the apparent rate constant qpt(f1) on the monomer composition in the system PDMS1k-MA / MSA at 60 °C.

The important apparent rate constant qPt(f1) is well approximated by an fourth order polynomial in f1:

2 3 4 k( f1 ) ≈ 0.0183 +10.218⋅ f1 − 5.559 ⋅ f1 −13.050 ⋅ f1 + 8.391⋅ f1 Eq. 200)

This equation allows to calculate the rate of copolymerisation of PDMS1k-MA / MSA at 60 °C for any monomer- and initiator concentration. Together with the Lewis-Mayo equation that con- nects the expected composition of the copolymer to the composition of the monomer mixture, this equation allows to predict the conditions to prepare P[PDMS1k-MAx-co-MSAy] copolymers of arbitrary compositions (as long as y < 0.5).

82 4.2. - The System MSA / EA

The copolymerization of maleic anhydride (MSA) and ethyl acrylate (EA) was investigated by Rätzsch.20 The data from ref [20] were used to calculate the copolymerization diagram and the values of Table 3. rMSA = 0 rEA = 3.69

0.5 0.5

0.4 0.4

min] 0.3 0.3 1/2

0.2 0.2 [ (L/mol) Pt q 0.1 0.1

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 f MSA Figure 25: Copolymerization diagram of MSA / EA. Blue line: rMSA = 0, rEA = 3.69, dotted line: r1 = r2 = 1.

The polymerizations were carried out at 50 °C in 1,2- dichloroethane with a total initial monomer concentration of c0 = 3 mol/L, initiated with AIBN. Table 3 summarizes the relevant data to calculate the apparent rate constants qPt(f1).

Table 3: Apparent rate constants of the copolymerization of MSA / Ethyl Acrylate (T = 50 °C in 1,2-dichloro ethane, AIBN, 0.001 mol/L)

f1 = fEA RP0 qPt(f1) / [mol/L min] (L/mol⋅min)1/2 0.0000 0.00996 8.402 0.0483 0.00866 7.313 0.0992 0.00716 6.039 0.2020 0.00534 4.510 0.3012 0.00346 2.921 0.4093 0.00208 1.753 0.5015 0.00124 1.044 0.6007 4.97792E-4 0.420 0.6982 1.63338E-4 0.138 1.0000 0 0.000 83

q = R / {c (k c )1/2} 10 Pt P0 0 d I0 [01.06.2004 14:23 "/qPt" (2453157)] 9 Polynomielle Regression für Auswertung_qPt: Y = A + B1*X + B2*X^2 + B3*X^3 + B4*X^4 8 Parameter Wert Fehler ------] 7 A 8.380 ± 0.1 1/2 B1 -24.093 ± 1.8 B2 22.325 ± 8.5 6 B3 -7.427 ± 14.1 B4 0.817 ± 7.3 5 ------

R-Square(COD) SD N P 4 ------0.99938 0.10407 10 <0.0001 [ (L/molmin) [ 3 ------Pt q 2 1 0 0.0 0.2 0.4 0.6 0.8 1.0 f MSA Figure 26: Dependence of the apparent rate constant qpt(f1) on the monomer composition in the system MSA / EA at 50 °C.

The apparent rate constant qPt(f1) is well approximated by an fourth order polynomial in f1 between 0 < f1 < 0.8 (see Fi- gure 26). For f1 > 0.8, qPt is virtually zero, but the fit function predicts a qPt ≈ 0.06 at fMSA = 1. Hence, in this region care must be taken to check whether this accuracy is sufficient.

2 3 4 k( f1 ) ≈ 8.38 − 24.093⋅ f1 + 22.325⋅ f1 − 7.427 ⋅ f1 − 0.817 ⋅ f1 Eq. 201)

84 4.3. - The System MSA / AN

The copolymerization of MSA and acrylo nitril (AN) was investigated by Rätzsch.20 The data from ref [20] were used to calculate the copolymerization diagram and the values of Table 4. rMSA = 0 rAN = 7.52

0.5 0.5

0.4 0.4

min] 0.3 0.3 1/2

0.2 0.2 [ (L/mol) Pt q 0.1 0.1

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 f MSA Figure 27: Copolymerization diagram of MSA / AN. Blue line: rMSA = 0, rAN = 7.52, dotted line: r1 = r2 = 1.

The polymerizations were carried out at 50 °C in 1,2- dichloroethane with a total initial monomer concentration of c0 = 3 mol/L, initiated with AIBN. Table 4 summarizes the relevant data to calculate the apparent rate constants qPt(f1).

Table 4: Apparent rate constants of the copolymerization of MSA / Acrylo nitrile (T = 50 °C in 1,2-dichloro ethane, AIBN, 0.001 mol/L)

f1 = fAN RP0 qPt(f1) / [mol/L min] (L/mol⋅min)1/2 0.0000 0.00506 4.882 0.0992 0.00337 3.247 0.2046 0.00236 2.272 0.2986 0.00181 1.747 0.4040 9.72250 E-40.937 0.4989 9.02248 E-40.870 0.6007 6.37796 E-40.615 0.6982 1.63338 E-40.157 0.0000 0.00000 0.000

q = R / {c (k c )1/2} 5 Pt P0 0 d I0

Data: Data1_qPt [01.06.2004 14:11 "/Graph3" (2453157)] Model: CoPoTest Polynomielle Regression für Data7_qPt: Y = A + B1*X + B2*X^2 + B3*X^3 + B4*X^4 4 Chi^2/DoF = 0.01448 Parameter Wert Fehler ] R^2 = 0.99506 ------1/2 r1 0 A 4.851 ± 0.16 B1 -17.915 ± 3.7 r2 7.52 3 B2 28.724 ± 24.2 d1 4.849 ± 0.33315 B3 -17.326 ± 53.9 d2 1.553 ± 0.03646 B4 -1.119 ± 38.3 ------2 R-Square(COD) SD N P ------

[ (L/mol min) 0.99562 0.15982 8 7.21562E-4

Pt ------q 1

0 0.0 0.2 0.4 0.6 0.8 1.0 f MSA

Figure 28: Dependence of the apparent rate constant qpt(f1) on the monomer composition in the system MSA / AN at 50 °C.

The apparent rate constant qPt(f1) is well approximated by an fourth order polynomial in f1 between 0 < f1 < 0.8 (see Fi- gure 28). For f1 > 0.8, qPt is virtually zero, but the fit function predicts a qPt ≈ 0.016 at fMSA = 1. Hence, in this region care must be taken. One has to check whether this accuracy is sufficient.

2 3 4 k( f1 ) ≈ 4.851−17.915⋅ f1 + 28.724 ⋅ f1 −17.326 ⋅ f1 −1.119 ⋅ f1 Eq. 202)

A better fit was achieved by means of Eq. 203):

2 ⋅ f ⋅ (1− f ) + 7.52 ⋅ (1− f ) 2 q (f ) = 1 1 1 Pt 1 4.849 ⋅ f +1.553⋅ (1− f ) 1 1 Eq. 203)

86 4.4. - The System MSA / F8H2-MA

The copolymerization of MSA and perfluorooctyl methacrylate (F8H2-MA) was investigated by Kraus.21 The data from ref [21] were used to calculate the copolymerization diagram and the values of Table 5. rMSA = 0 rAN = 4.3 1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 MSA F 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f MSA Figure 29: Copolymerization diagram of MSA / F8H2-MA. Blue line: rMSA = 0, rAN = 4.3, dotted line: r1 = r2 = 0, --- : r1 = r2 = 1.

The polymerizations were carried out at 60 °C in 2-butano- ne / hexafluoroxylene (1 : 1), initiated with AIBN. Table 5 summarizes the relevant data to calculate the apparent rate constants qPt(f1).

Table 5: Apparent rate constants of the copolymerization of MSA / F8H2-MA (T = 50 °C in 1,2-dichloro ethane, AIBN, 0.001 mol/L)

c0 c0I f1 = fMSA RP0 qPt(f1) / [mol/L] [mol/L] [mol/L min] (L/mol⋅min)1/2 0.0000 6.5(a) 0.23460 0.00468 0.2495 7.5108E-4 2.076 0.34882 0.00696 0.5000 4.0335E-4 0.615 0.67847 0.01353 0.5000 5.8723E-4 0.895 0.87868 0.01753 0.5000 17.9E-4 1.933 0.34882 0.00696 0.5000 4.0335E-4 0.154 0.34882 0.01392 0.7500 5.6944E-4 0.320 1.0000 0.000 (a) extrapolated from experiments (2) - (7)

q = R / {c (k c )1/2} 7 Pt P0 0 d I0

[28.05.2004 15:06 "/Rp0vonf1" (2453153)] 6 Polynomielle Regression für Data3_qPt: Y = A + B1*X + B2*X^2 + B3*X^3 + B4*X^4

Parameter Wert Fehler 5 ------A 6.5 ± 10.8 B1 -30.110 ± 14.9

min] 4 B2 64.455 ± 68

1/2 B3 -65.703 ± 106 B4 24.859 ± 53 3 ------R-Square(COD) SD N P

[(L/mol) ------Pt 0.94685 0.75402 8 0.02966 q 2 ------1

0 0.00.10.20.30.40.50.60.70.80.91.0 f MSA Figure 30: Dependence of the apparent rate constant qpt(f1) on the monomer composition in the system MSA / F8H2-MA at 60 °C.

The apparent rate constant qPt(f1) was tried to approximate by an fourth order polynomial in f1 between 0 < f1 < 1 (see Fi- gure 30). The quality of the data is quite low, in particular for f1 < 0.2 and f1 > 0.8 no experimental data exist. The value qPt(f1=0) was estimated only from a semi-logarithmic plot of qPt versus f1 and extrapolation of a linear fit to f1 = 0. Hence, the fit can only expect to be of qualitative character.

2 3 4 k( f1 ) ≈ 6.5 − 30.11⋅ f1 + 64.455⋅ f1 − 65.703⋅ f1 + 24.859 ⋅ f1 Eq. 204)

88 4.5. - The System Styrene / MA

The copolymerization of styrene and methyl acrylate (MA) was investigated by Walling.12 The data from ref. [12] were used to calculate the copolymerization diagram and the values of Table 6. The copolymerization is represented by an "azeotropic" diagram. The azeotropic point is found at fStyrene = 0.766.

rStyrene = 0.75 rMA = 0.18 1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 Styrene

F 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f Styrene

Figure 31: Copolymerization diagram of Styrene / MA. Blue line: rStyrene = 0.75, rMA = 0.18, diagonal line: r1 = r2 = 1.

The polymerizations were carried out at 60 °C in ethyl acetate : monomer = 1 : 1 (vol : vol) mixtures, initiated with AIBN. Table 6 summarizes the relevant data to calculate the apparent rate constants qPt(f1).

Table 6: Apparent rate constants of the copolymerization of Styrene / MA (ethyl acetate, AIBN, cI0 = 0.0003045 mol/L, T = 60 °C)

f1 = fStyrene RP0 fMA⋅MMA / c0 qPt(f1) / [mol/L min] [mol / L] 1/2 fSty⋅MSty (L/mol⋅min) 1.00000 1.28571 E-5 0.00000 4.34950 0.00224 0.77070 1.28571 E-5 0.24593 4.57594 0.00213 0.54459 2.00000 E-5 0.69125 4.82358 0.00315 0.34713 2.42857 E-5 1.55461 5.06284 0.00364 0.16561 2.92857 E-5 4.16477 5.30475 0.00419 0.08917 3.78571 E-5 8.44309 5.41366 0.00531 0.00000 4.12143 E-4 --- 5.54652 0.05640

89 Styrene / Methyl acrylate

Data: Auswertung_qPt 5 Model: CoPoTest

Chi^2/DoF = 0.00427

] R^2 = 0.99911 [01.06.2004 14:54 "/Graph1" (2453157)]

1/2 Polynomielle Regression für Auswertung_qPt: 4 r1 0.75 Y = A + B1*X + B2*X^2 r2 0.18 d1 5.6110 ± 0.4 Parameter Wert Fehler d2 0.03192 ± 0.00037 ------3 A 0.56428 0.03901 B1 -0.70359 0.18463 B2 0.35826 0.16805

)[ (L/mol min) ------2 R-Square(COD) SD N P Styrene

(f ------0.95319 0.03379 6 0.01013 app k 1 ------ACHTUNG: Fit nur für f >= 0.1 Styrol

0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f Styrene Figure 32: Dependence of the apparent rate constant qpt(f1) on the monomer composition in the system Styrene / MA at 60 °C.

Methyl acrylate polymerizes much faster than styrene, but fairly small amounts of styrene reduce the rate of copolymerization considerably (cf. Figure 32). The calculated apparent rate constants qPt(f1) were not well represented by polynomials, especially not for 0 < fStyrene < 0.1. Hence, the rate law Eq. 100) was used as fit-function to yield Eq. 205):

2 2 0.75⋅ f1 + 2 ⋅ f1 ⋅ (1− f1 ) + 0.18⋅ (1− f1 ) q Pt (f1 ) = Eq. 205) 5.61⋅ f1 + 0.0319 ⋅ (1− f1 )

90 4.6. - The System Styrene / MMA

The copolymerization of styrene and methyl methacrylate (MMA) was investigated by Walling.12 The data from ref. [12] were used to calculate the copolymerization diagram and the values of Table 7. The copolymerization is represented by an "azeotropic" diagram. The azeotropic point is found at fStyrene = 0.490. rStyrene = 0.52 rMA = 0.46 1.0 1.0 0.9 0.9

] 0.8 0.8 1/2 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 ) [ (L/mol min) ) [ (L/mol 0.3 0.3

Styrene 0.2 0.2 (f Pt

q 0.1 0.1 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f Styrene Figure 33: Copolymerization diagram of Styrene / MMA. Blue line: rStyrene = 0.52, rMA = 0.46, diagonal line: r1 = r2 = 1.

The polymerizations were carried out at 60 °C in bulk, initiated with AIBN. Table 6 summarizes the relevant data to calculate the apparent rate constants qPt(f1).

Table 7: Apparent rate constants of the copolymerization of Styrene / MMA (in bulk, AIBN, cI0 = 0.00609 mol/L, T = 60 °C)

f1 = fStyrene RP0 fMA⋅MMA / c0 qPt(f1) / [mol/L min] [mol / L] 1/2 fSty⋅MSty (L/mol⋅min) 1.0000 0.00355 0.00000 8.69899 0.21916 1.0000 0.00278 0.00000 8.69899 0.17177 1.0000 0.00327 0.00000 8.69899 0.20176 0.7580 0.00277 0.30691 8.86288 0.16751 0.7580 0.00266 0.30691 8.86288 0.16097 0.5790 0.00289 0.69898 8.98814 0.17270 0.5790 0.00285 0.69898 8.98814 0.17019 0.3780 0.00340 1.58183 9.13307 0.19993 0.3780 0.00323 1.58183 9.13307 0.18970 0.1906 0.00467 4.08227 9.27248 0.27020 0.1906 0.00477 4.08227 9.27248 0.27611

91 0.0913 0.00696 9.56778 9.34809 0.39961 0.0913 0.00612 9.56778 9.34809 0.35138 0.0000 0.01188 --- 9.41870 0.67699 0.0000 0.01164 --- 9.41870 0.66331

0.7

0.6 Data: Auswertung_kapp Model: Diffusion ]

1/2 0.5 Chi^2 = 0.00027 R^2 = 0.99267

0.4 r1 3.132 ± 0.7 r2 1.212 ± 0.4 d1 16.235 ± 3.1 0.3 d2 1.811 ± 0.6 ) [ (L/mol min) 0.2 Styrene (f

Pt 0.1 q 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 f Styrene Figure 34: Dependence of the apparent rate constant qpt(f1) on the monomer composition in the system Styrene / MMA at 60 °C.

Methyl methacrylate polymerizes faster than styrene and similar to styrene / MA small amounts of styrene reduce the rate of copolymerization considerably (cf. Figure 19). Like with the copolymer system styrene / MA the calculated apparent rate constants qPt(f1) were not well represented by polynomials, and the rate law Eq. 100) was used as fit-function. Note that this time the fitting procedure did not reproduce the copolymerization parameter in the denominator of the equation.

2 2 3.13⋅ f1 + 2 ⋅ f1 ⋅ (1− f1 ) +1.21⋅ (1− f1 ) q Pt (f1 ) = Eq. 206) 16.2 ⋅ f1 +1.8⋅ (1− f1 )

92 5. The Program COPODOS

5.1. - System description COPODOS is a program system for calculation of the required experimental conditions for continuous addition copolymerization preparations. At an arbitrary monomer composition f1 the program calculates the instantaneous copolymer composition by means of the Lewis-Mayo equation (Eq. 90) and the relevant rates of polymerization from empiric expressions (either Eq. 96) or Eq. 100)). Until now the copolymerization of the following monomer pairs can be calculated:

Table 8: Monomer pairs that can be used for COPODOS calculations.

Monomer 1 Monomer 2 Temperature / °C PDMS1k-MA Maleic anhydride 60 Maleic anhydride Ethyl acrylate 50 Maleic anhydride Acrylo nitril 50 Maleic anhydride Perfluorooctylmethacrylate 60 Styrene Methyl acrylate 60 Styrene Methyl methacrylate 60

The user is guided though the calculation by a series of ques- tions and all calculated information's are immediately presented at the screen. The final results are saved in an ASCII file and can be read and printed with any text system.

The program system consists of the program COPODOS and a series of data files: The program COPODOS.EXE is located in the COPODOS folder, together with the data files Initiatr.DAT, Monomere.DAT and REPORT.DAT (cf. Figure 35).

Figure 35: Organization of the program system COPODOS (COPODOS.EXE = program, Initiatr.dat = list of initiators, Monomere.dat = list of monomer pairs, DATA = sub folder containing monomer pair parameters, Report.dat = ASCII file containing the results of the calculation).

93 The file Initiatr.DAT contains a list of molecular weights and decomposition constants of free radical initiators together with the respective decomposition temperature. Presently CO PODOS "knows" the azo initiators Wako V-601 (60 °C), AIBN (50 °C, 60 °C) and the peroxide initiator BPO (80 °C):

The file "INITIATR.DAT": "Name, Molgew, Tdec, kdec/min-1 V-601, 230.24, 60.00, 5.660D-04 AIBN, 164.21, 60.00, 5.080D-04 AIBN, 164.21, 50.00, 1.195D-04 BPO, 242.23, 80.00, 1.160D-03

Users are invited to add further initiator lines to INITIA- TR.DAT, but keep uniform units throughout: molecular weight in g/mol, decomposition temperatures in degree Celsius and decomposition rate constants in minutes-1. Note that the COPODOS allows the combination of any monomer pair with any initiator, but it is highly recommended to use the initiator that is mentioned in the experimental conditions line with each monomer pair. Care must be taken to choose the initiator decomposition temperature identical to the polymerization temperature ! Monomere.DAT contains the list of all monomer pairs that can be handled by COPODOS (cf. Table 8). Within the folder CO- PODOS a second folder is located, named DATA. In DATA each monomer pair is represented by a single ASCII file, containing the comonomers molecular weight, densities, copolymerization parameter, kinetic parameter to describe the qPt(f1) curve and two text lines describing the experimental conditions of the rate- and copolymer composition measurements as well as the quality of the fit functions. The user is free to add new parameter files of other comonomer pairs or initiators. The struc- ture of the monomer pair parameter file can be seen from the following example file "STYMMA.dat": Only the first 12 lines are of importance to COPODOS, the rest has explanatory character. The first two numbers are the copolymerization parameters r1 and r2, the second pair of numbers gives the monomers densities ρ1 and ρ2, while the last number pair represents the monomers molecular weight M1 and M2. Note that the sequence of numbers always corresponds to "Monomer 1 / Monomer 2". The next line contains a single integer that allows to select the empiric formula to be used for calculation of qPt(f1). Presently only the values "1" - Eq. 106) and "2" - Eq. 104) are allowed. The next four lines each contain a single floating point number, representing the parameters of the empiric qPt(f1) relation. The next line is a pure text line, containing the experimental parameters that were used to measure the copolymer parameters and the rate constants that were used to obtain the qPt(f1) curve. The next line is intended to hold "remarks" on the quality and applicability limits of the fits. The information bearing part of the file is finished with "ENDE", all subsequent lines are ignored by COPODOS and can be used for all kind of notes. 94

The file "StyMMA.DAT":

0.5200, 0.4600 0.9060, 0.9430 104.15, 100.12 2 +0.6602 -3.6793 10.3469 -12.3312 +5.1779 "T = 60 deg C, AIBN" "ACHTUNG: fuer f(Styrol) > 0.8 ungenau" ENDE

Styrene / MMA r1, r2 Dichte1, Dichte2 Molmasse1, Molmasse2 Modell a b c d e "Reaktionsbedingungen" "Bemerkungen"

fuer (1) kapp(f1) = a + b*f1 + c*f1^2+.... (2) kapp(f1) = (r1*f1^2+2*f1*f2+r2*f2^2)/(d1*f1+d2*f2)

Whenever a new file with a monomer pair was added, the name of the monomers and the path where to find the parameter file must be inserted in the file "MONOMERE.DAT". The organization of the MONOMERE.DAT file is depicted below:

The file "MONOMERE.DAT": "PDMS-MA", "MSA ", C:\Copodos\Data\PDMSA.DAT "MSA ", "EA ", C:\Copodos\Data\MSAEA.DAT "MSA ", "AN ", C:\Copodos\Data\MSAAN.DAT "MSA ", "F8H2-MA", C:\Copodos\Data\MSAF8MA.DAT "Styrol ", "MMA ", C:\Copodos\Data\StyMMA.DAT "Styrol ", "MA ", C:\Copodos\Data\StyMA.DAT "XXX ", "YYY ", C:\Copodos\Data\XXXYYY.DAT

In each line the file contains the name of monomer 1, the name of monomer 2 and the path where to find the monomer pair parameter file. Note that the length of the monomer abbreviations - seven letters - in MONOMERE.DAT must not be altered, because the text format of the COPODOS output relies on this constant length of the monomer name string variables. The length of the path is not restricted,- the new monomer file can be placed everywhere. But for sake of convenience it is recommended to place the new monomer pair parameter file in the COPODOS/DATA folder.

Click here to load down COPODOS source code and files

95 5.2. - Using COPODOS

The list below is a guide through the COPODOS calculations.

Nr. User Action Program Reaction 1) Start the program COPODOS.EXE A text menu appears, showing the list of known monomer pairs. COPODOS asks for your selection of a monomer pair. 2) Key in the number referring COPODOS shows the parameters to the desired pair and press and experimental conditions the enter-key. characterizing the monomer pair.

A second text menu appears, showing a list of possible initiators. COPODOS asks for your selection of an initiator. 3) Key in the number referring to the desired initiator and press the enter-key. COPODOS asks for the required content of monomer 1 in the copolymer 4) Key in the value of F1 and COPODOS shows the required press the enter-key. molar composition of the stock solution.

COPODOS asks for the total monomer concentration c0 in the stock solution. 5) Key in the value of c0 and press the enter-key.

COPODOS asks for the total mass of monomer m0 to place in the stock solution. 6) Key in the value of m0 and press the enter-key.

COPODOS asks for the initiator concentration cI0 to place in the stock solution. 7) Key in the value of cI0 (in COPODOS prints the recipe of mol% relative to the monomer) the stock solution, the and press the enter-key. specific and total rates of polymerization, the rate if initiation, the theoretical degree of polymerization and the time required to convert

96 the monomer stock.

COPODOS asks for the addition factor α. 8) Key in the value of α and press the enter-key. COPODOS asks for the ratio of monomer concentration between monomer-feed and * stock- and solution c /c0. * 9) Key in the value of c /c0 and COPODOS prints the recipe of press the enter-key. the monomer feed solution and the recipe for the initiator feed solution inclu- ding the volume addition rates as well as the total addition time.

COPODOS asks if a Table is required. 10) Key in "j" ------> COPODOS plots a table of reaction time, reaction mixture volume and specific rate of polymerization. Key in "n" ------> ---

(followed by the enter-key) A text menu appears. It is possible to do a calculation with the same monomer pair, to select an new calculation or to end the program.

COPODOS writes the results of the calculation to the file "REPORT.DAT". The results from the last calculations are deleted. 11) Key in "w", "n" or "q" "w" ----> COPODOS starts with step 3) "n" ----> COPODOS starts with step 2) "q" ----> COPODOS stops.

The whole calculated results are written to the file RE- PORT.DAT, hence this file contains the recipes for the stock solution, the initiator- and mixed monomer feed solution as well as the volume addition rate and the addition time τ. Ex- amples of COPODOS output files can be found in section 6. Note that the REPORT.DAT file is overwritten at the end of each new calculation. To store the data either rename the file or copy the output file to a different location prior to run the next recipe calculation. 97 6. COPODOS Example Recipes

6.1. - COPODOS Recipes for P[PDMS1k-MA-co-MSA]

6.1.1. P[PDMS1k-MA0.95-co-MSA0.05] COPODOS Ansatzberechnung fuer Copolymerisationen mit Monomer- und Initiator - Nachdosierung

Zielpolymer: F(PDMS-MA) = 0.9500 Zielpolymer: F(MSA ) = 0.0500

06-17-2004 12:29:23 ------REZEPT FUER STOCK LOESUNG ------f1 = 0.8889 f2 = 0.1111 [M1] = 0.4444 mol/L [M2] = 0.0556 mol/L [I0] = 0.0200 mol/L m1 = 1.9758 g PDMS-MA m2 = 0.0242 g MSA mI = 0.0205 g V-601 VLm = 2.3991 mL Loesungsmittel V0 = 4.4455 mL Gesamtvolumen ------Polymerisationsgeschwindigkeiten - total: PDMS-MA - total = 5.554D-06 mol/ min = Rtp1 MSA - total = 2.923D-07 mol/ min = Rtp2 SUMME - total = 5.846D-06 mol/ min = Rtp Polymerisationsgeschwindigkeiten - spezifisch: PDMS-MA - spezifisch = 1.249D-03 mol/ L min = Rp1 MSA - spezifisch = 6.575D-05 mol/ L min = Rp2 SUMME - spezifisch = 1.315D-03 mol/ L min = Rp Initiierungsgeschwindigkeit: - total = 5.032D-05 mol/ min = Ri - spezifisch = 1.132D-05 mol/ L min = Rit

Polymerisationsgrad Xn = 5.1642D-01 Molekulargewicht Mn = 4.9313D+02 g/mol

Zeit fuer einfachen Stock-Umsatz: tau = 380.23 min

Stock-Umsatzfaktor alpha = 10 Nachdosierungs - Konzentrationsfaktor beta = 1

------REZEPT FUER MONOMER NACHDOSIERUNGSLOESUNG ------[M1] = 0.4444 mol/L [M2] = 0.0556 mol/L m1 = 19.7580 g m2 = 0.2420 g MSA VLm = 23.9908 mL Loesungsmittel V0 = 44.4554 mL Gesamtvolumen Volumendosierungsrate: dV/dt = 0.01169 mL/min-1 ------REZEPT FUER INITIATOR NACHDOSIERUNGSLOESUNG ------[MI] = 0.3999 mol/L [I]/[I]0 = 19.9968 mI = 0.0441 g V-601 VLm = 0.4446 mL Loesungsmittel V0 = 0.4784 mL Gesamtvolumen dV/dt(I) = 0.0001258 mL/min ------Gesamte Nachdosierzeit: td = 3.802D+03 min = 63.372 h

98 6.1.2. P[PDMS1k-MA0.90-co-MSA0.10]

COPODOS Ansatzberechnung fuer Copolymerisationen mit Monomer- und Initiator - Nachdosierung

Monomer 1 = PDMS-MA Monomer 2 = MSA Initiator = V-601 T = 60 deg C, V-601

Zielpolymer: F(PDMS-MA) = 0.9000 Zielpolymer: F(MSA ) = 0.1000

06-17-2004 12:28:22 ------REZEPT FUER STOCK LOESUNG ------f1 = 0.7805 f2 = 0.2195 [M1] = 0.3902 mol/L [M2] = 0.1098 mol/L [I0] = 0.0200 mol/L m1 = 1.9464 g PDMS-MA m2 = 0.0536 g MSA mI = 0.0230 g V-601 VLm = 2.9469 mL Loesungsmittel V0 = 4.9875 mL Gesamtvolumen ------

Polymerisationsgeschwindigkeiten - total: PDMS-MA - total = 1.145D-05 mol/ min = Rtp1 MSA - total = 1.272D-06 mol/ min = Rtp2 SUMME - total = 1.272D-05 mol/ min = Rtp

Polymerisationsgeschwindigkeiten - spezifisch: PDMS-MA - spezifisch = 2.296D-03 mol/ L min = Rp1 MSA - spezifisch = 2.551D-04 mol/ L min = Rp2 SUMME - spezifisch = 2.551D-03 mol/ L min = Rp

Initiierungsgeschwindigkeit: - total = 5.646D-05 mol/ min = Ri - spezifisch = 1.132D-05 mol/ L min = Rit

Polymerisationsgrad Xn = 1.1238D+00 Molekulargewicht Mn = 1.0224D+03 g/mol

Zeit fuer einfachen Stock-Umsatz: tau = 196.03 min

Stock-Umsatzfaktor alpha = 10 Nachdosierungs - Konzentrationsfaktor beta = 1

------REZEPT FUER MONOMER NACHDOSIERUNGSLOESUNG ------[M1] = 0.3902 mol/L [M2] = 0.1098 mol/L m1 = 19.4635 g m2 = 0.5365 g MSA VLm = 29.4694 mL Loesungsmittel V0 = 49.8753 mL Gesamtvolumen Volumendosierungsrate: dV/dt = 0.02544 mL/min-1 ------REZEPT FUER INITIATOR NACHDOSIERUNGSLOESUNG ------[MI] = 0.2135 mol/L [I]/[I]0 = 10.6760

mI = 0.0255 g V-601 VLm = 0.4988 mL Loesungsmittel V0 = 0.5184 mL Gesamtvolumen dV/dt(I) = 0.0002644 mL/min ------Gesamte Nachdosierzeit: td = 1.960D+03 min = 32.672 h

99 6.1.3. P[PDMS1k-MA0.80-co-MSA0.20]

COPODOS Ansatzberechnung fuer Copolymerisationen mit Monomer- und Initiator - Nachdosierung

Monomer 1 = PDMS-MA Monomer 2 = MSA Initiator = V-601 T = 60 deg C, V-601

Zielpolymer: F(PDMS-MA) = 0.8000 Zielpolymer: F(MSA ) = 0.2000

06-17-2004 12:27:13 ------REZEPT FUER STOCK LOESUNG ------f1 = 0.5714 f2 = 0.4286 [M1] = 0.2857 mol/L [M2] = 0.2143 mol/L [I0] = 0.0200 mol/L m1 = 1.8631 g PDMS-MA m2 = 0.1369 g MSA mI = 0.0300 g V-601 VLm = 4.4967 mL Loesungsmittel V0 = 6.5207 mL Gesamtvolumen ------

Polymerisationsgeschwindigkeiten - total: PDMS-MA - total = 2.195D-05 mol/ min = Rtp1 MSA - total = 5.488D-06 mol/ min = Rtp2 SUMME - total = 2.744D-05 mol/ min = Rtp

Polymerisationsgeschwindigkeiten - spezifisch: PDMS-MA - spezifisch = 3.367D-03 mol/ L min = Rp1 MSA - spezifisch = 8.417D-04 mol/ L min = Rp2 SUMME - spezifisch = 4.208D-03 mol/ L min = Rp

Initiierungsgeschwindigkeit: - total = 7.381D-05 mol/ min = Ri - spezifisch = 1.132D-05 mol/ L min = Rit

Polymerisationsgrad Xn = 2.4242D+00 Molekulargewicht Mn = 1.9869D+03 g/mol

Zeit fuer einfachen Stock-Umsatz: tau = 118.81 min

Stock-Umsatzfaktor alpha = 10 Nachdosierungs - Konzentrationsfaktor beta = 1

------REZEPT FUER MONOMER NACHDOSIERUNGSLOESUNG ------[M1] = 0.2857 mol/L [M2] = 0.2143 mol/L m1 = 18.6306 g m2 = 1.3694 g MSA VLm = 44.9672 mL Loesungsmittel V0 = 65.2073 mL Gesamtvolumen Volumendosierungsrate: dV/dt = 0.05488 mL/min-1 ------REZEPT FUER INITIATOR NACHDOSIERUNGSLOESUNG ------[MI] = 0.1314 mol/L [I]/[I]0 = 6.5682

mI = 0.0202 g V-601 VLm = 0.6521 mL Loesungsmittel V0 = 0.6676 mL Gesamtvolumen dV/dt(I) = 0.0005619 mL/min ------Gesamte Nachdosierzeit: td = 1.188D+03 min = 19.802 h

100 6.1.4. P[PDMS1k-MA0.70-co-MSA0.30]

COPODOS Ansatzberechnung fuer Copolymerisationen mit Monomer- und Initiator - Nachdosierung

Monomer 1 = PDMS-MA Monomer 2 = MSA Initiator = V-601 T = 60 deg C, V-601

Zielpolymer: F(PDMS-MA) = 0.7000 Zielpolymer: F(MSA ) = 0.3000

06-17-2004 12:26:15 ------REZEPT FUER STOCK LOESUNG ------f1 = 0.3721 f2 = 0.6279 [M1] = 0.1860 mol/L [M2] = 0.3140 mol/L [I0] = 0.0200 mol/L m1 = 1.7162 g PDMS-MA m2 = 0.2838 g MSA mI = 0.0425 g V-601 VLm = 7.2298 mL Loesungsmittel V0 = 9.2245 mL Gesamtvolumen ------

Polymerisationsgeschwindigkeiten - total: PDMS-MA - total = 2.758D-05 mol/ min = Rtp1 MSA - total = 1.182D-05 mol/ min = Rtp2 SUMME - total = 3.940D-05 mol/ min = Rtp

Polymerisationsgeschwindigkeiten - spezifisch: PDMS-MA - spezifisch = 2.990D-03 mol/ L min = Rp1 MSA - spezifisch = 1.281D-03 mol/ L min = Rp2 SUMME - spezifisch = 4.272D-03 mol/ L min = Rp

Initiierungsgeschwindigkeit: - total = 1.044D-04 mol/ min = Ri - spezifisch = 1.132D-05 mol/ L min = Rit

Polymerisationsgrad Xn = 3.4809D+00 Molekulargewicht Mn = 2.5390D+03 g/mol

Zeit fuer einfachen Stock-Umsatz: tau = 117.05 min

Stock-Umsatzfaktor alpha = 10 Nachdosierungs - Konzentrationsfaktor beta = 1

------REZEPT FUER MONOMER NACHDOSIERUNGSLOESUNG ------[M1] = 0.1860 mol/L [M2] = 0.3140 mol/L m1 = 17.1619 g m2 = 2.8381 g MSA VLm = 72.2975 mL Loesungsmittel V0 = 92.2450 mL Gesamtvolumen Volumendosierungsrate: dV/dt = 0.07881 mL/min-1 ------REZEPT FUER INITIATOR NACHDOSIERUNGSLOESUNG ------[MI] = 0.1295 mol/L [I]/[I]0 = 6.4732

mI = 0.0281 g V-601 VLm = 0.9224 mL Loesungsmittel V0 = 0.9441 mL Gesamtvolumen dV/dt(I) = 0.0008066 mL/min ------Gesamte Nachdosierzeit: td = 1.171D+03 min = 19.509 h

101 6.1.5. P[PDMS1k-MA0.60-co-MSA0.40]

COPODOS Ansatzberechnung fuer Copolymerisationen mit Monomer- und Initiator - Nachdosierung

Monomer 1 = PDMS-MA Monomer 2 = MSA Initiator = V-601 T = 60 deg C, V-601

Zielpolymer: F(PDMS-MA) = 0.6000 Zielpolymer: F(MSA ) = 0.4000

06-17-2004 12:24:51 ------REZEPT FUER STOCK LOESUNG ------f1 = 0.1818 f2 = 0.8182 [M1] = 0.0909 mol/L [M2] = 0.4091 mol/L [I0] = 0.0200 mol/L m1 = 1.3879 g PDMS-MA m2 = 0.6121 g MSA mI = 0.0703 g V-601 VLm = 13.3378 mL Loesungsmittel V0 = 15.2672 mL Gesamtvolumen ------

Polymerisationsgeschwindigkeiten - total: PDMS-MA - total = 2.501D-05 mol/ min = Rtp1 MSA - total = 1.667D-05 mol/ min = Rtp2 SUMME - total = 4.169D-05 mol/ min = Rtp

Polymerisationsgeschwindigkeiten - spezifisch: PDMS-MA - spezifisch = 1.638D-03 mol/ L min = Rp1 MSA - spezifisch = 1.092D-03 mol/ L min = Rp2 SUMME - spezifisch = 2.730D-03 mol/ L min = Rp

Initiierungsgeschwindigkeit: - total = 1.728D-04 mol/ min = Ri - spezifisch = 1.132D-05 mol/ L min = Rit

Polymerisationsgrad Xn = 3.6825D+00 Molekulargewicht Mn = 2.3538D+03 g/mol

Zeit fuer einfachen Stock-Umsatz: tau = 183.12 min

Stock-Umsatzfaktor alpha = 10 Nachdosierungs - Konzentrationsfaktor beta = 1

------REZEPT FUER MONOMER NACHDOSIERUNGSLOESUNG ------[M1] = 0.0909 mol/L [M2] = 0.4091 mol/L m1 = 13.8793 g m2 = 6.1207 g MSA VLm = 133.3782 mL Loesungsmittel V0 = 152.6718 mL Gesamtvolumen Volumendosierungsrate: dV/dt = 0.08337 mL/min-1 ------REZEPT FUER INITIATOR NACHDOSIERUNGSLOESUNG ------[MI] = 0.2000 mol/L [I]/[I]0 = 9.9977

mI = 0.0729 g V-601 VLm = 1.5267 mL Loesungsmittel V0 = 1.5828 mL Gesamtvolumen dV/dt(I) = 0.0008643 mL/min ------Gesamte Nachdosierzeit: td = 1.831D+03 min = 30.520 h

102 6.1.6. P[PDMS1k-MA0.55-co-MSA0.45]

COPODOS Ansatzberechnung fuer Copolymerisationen mit Monomer- und Initiator - Nachdosierung

Monomer 1 = PDMS-MA Monomer 2 = MSA Initiator = V-601 T = 60 deg C, V-601

Zielpolymer: F(PDMS-MA) = 0.5500 Zielpolymer: F(MSA ) = 0.4500

06-17-2004 12:24:06 ------REZEPT FUER STOCK LOESUNG ------f1 = 0.0899 f2 = 0.9101 [M1] = 0.0449 mol/L [M2] = 0.4551 mol/L [I0] = 0.0200 mol/L m1 = 1.0039 g PDMS-MA m2 = 0.9961 g MSA mI = 0.1029 g V-601 VLm = 20.4837 mL Loesungsmittel V0 = 22.3366 mL Gesamtvolumen ------

Polymerisationsgeschwindigkeiten - total: PDMS-MA - total = 1.825D-05 mol/ min = Rtp1 MSA - total = 1.493D-05 mol/ min = Rtp2 SUMME - total = 3.318D-05 mol/ min = Rtp

Polymerisationsgeschwindigkeiten - spezifisch: PDMS-MA - spezifisch = 8.169D-04 mol/ L min = Rp1 MSA - spezifisch = 6.684D-04 mol/ L min = Rp2 SUMME - spezifisch = 1.485D-03 mol/ L min = Rp

Initiierungsgeschwindigkeit: - total = 2.528D-04 mol/ min = Ri - spezifisch = 1.132D-05 mol/ L min = Rit

Polymerisationsgrad Xn = 2.9307D+00 Molekulargewicht Mn = 1.7412D+03 g/mol

Zeit fuer einfachen Stock-Umsatz: tau = 336.64 min

Stock-Umsatzfaktor alpha = 10 Nachdosierungs - Konzentrationsfaktor beta = 1

------REZEPT FUER MONOMER NACHDOSIERUNGSLOESUNG ------[M1] = 0.0449 mol/L [M2] = 0.4551 mol/L m1 = 10.0389 g m2 = 9.9611 g MSA VLm = 204.8369 mL Loesungsmittel V0 = 223.3655 mL Gesamtvolumen Volumendosierungsrate: dV/dt = 0.06635 mL/min-1 ------REZEPT FUER INITIATOR NACHDOSIERUNGSLOESUNG ------[MI] = 0.3570 mol/L [I]/[I]0 = 17.8490

mI = 0.1960 g V-601 VLm = 2.2337 mL Loesungsmittel V0 = 2.3844 mL Gesamtvolumen dV/dt(I) = 0.0007083 mL/min ------Gesamte Nachdosierzeit: td = 3.366D+03 min = 56.106 h

103 6.2. - COPODOS Recipes for P[MSA-co-EA]

6.2.1. P[MSA0.05-co-EA0.95] COPODOS Ansatzberechnung fuer Copolymerisationen mit Monomer- und Initiator - Nachdosierung

Zielpolymer: F(MSA ) = 0.0500 Zielpolymer: F(EA ) = 0.9500

06-17-2004 12:22:36 ------REZEPT FUER STOCK LOESUNG ------f1 = 0.0930 f2 = 0.9070 [M1] = 0.2789 mol/L [M2] = 2.7211 mol/L [I0] = 0.0300 mol/L m1 = 0.1824 g MSA m2 = 1.8176 g EA mI = 0.0329 g AIBN VLm = 4.5540 mL Loesungsmittel V0 = 6.6718 mL Gesamtvolumen ------

Polymerisationsgeschwindigkeiten - total: MSA - total = 1.199D-05 mol/ min = Rtp1 EA - total = 2.278D-04 mol/ min = Rtp2 SUMME - total = 2.398D-04 mol/ min = Rtp

Polymerisationsgeschwindigkeiten - spezifisch: MSA - spezifisch = 1.797D-03 mol/ L min = Rp1 EA - spezifisch = 3.414D-02 mol/ L min = Rp2 SUMME - spezifisch = 3.594D-02 mol/ L min = Rp

Initiierungsgeschwindigkeit: - total = 2.392D-05 mol/ min = Ri - spezifisch = 3.585D-06 mol/ L min = Rit

Polymerisationsgrad Xn = 6.6885D+01 Molekulargewicht Mn = 6.6894D+03 g/mol

Zeit fuer einfachen Stock-Umsatz: tau = 83.47 min

Stock-Umsatzfaktor alpha = 10 Nachdosierungs - Konzentrationsfaktor beta = 1

------REZEPT FUER MONOMER NACHDOSIERUNGSLOESUNG ------[M1] = 0.2789 mol/L [M2] = 2.7211 mol/L m1 = 1.8236 g m2 = 18.1764 g EA VLm = 45.5396 mL Loesungsmittel V0 = 66.7181 mL Gesamtvolumen Volumendosierungsrate: dV/dt = 0.07993 mL/min-1 ------REZEPT FUER INITIATOR NACHDOSIERUNGSLOESUNG ------[MI] = 0.0298 mol/L [I]/[I]0 = 0.9938

mI = 0.0033 g AIBN VLm = 0.6672 mL Loesungsmittel V0 = 0.6697 mL Gesamtvolumen dV/dt(I) = 0.0008023 mL/min ------Gesamte Nachdosierzeit: td = 8.347D+02 min = 13.912 h 104 6.2.2. P[MSA0.10-co-EA0.90]

COPODOS Ansatzberechnung fuer Copolymerisationen mit Monomer- und Initiator - Nachdosierung

Monomer 1 = MSA Monomer 2 = EA Initiator = AIBN T = 50 deg C, AIBN, 1,2-Dichlorethan

Zielpolymer: F(MSA ) = 0.1000 Zielpolymer: F(EA ) = 0.9000

06-17-2004 12:21:38 ------REZEPT FUER STOCK LOESUNG ------f1 = 0.1874 f2 = 0.8126 [M1] = 0.5622 mol/L [M2] = 2.4378 mol/L [I0] = 0.0300 mol/L m1 = 0.3683 g MSA m2 = 1.6317 g EA mI = 0.0329 g AIBN VLm = 4.6149 mL Loesungsmittel V0 = 6.6852 mL Gesamtvolumen ------

Polymerisationsgeschwindigkeiten - total: MSA - total = 1.747D-05 mol/ min = Rtp1 EA - total = 1.572D-04 mol/ min = Rtp2 SUMME - total = 1.747D-04 mol/ min = Rtp

Polymerisationsgeschwindigkeiten - spezifisch: MSA - spezifisch = 2.614D-03 mol/ L min = Rp1 EA - spezifisch = 2.352D-02 mol/ L min = Rp2 SUMME - spezifisch = 2.614D-02 mol/ L min = Rp

Initiierungsgeschwindigkeit: - total = 2.397D-05 mol/ min = Ri - spezifisch = 3.585D-06 mol/ L min = Rit

Polymerisationsgrad Xn = 4.8736D+01 Molekulargewicht Mn = 4.8691D+03 g/mol

Zeit fuer einfachen Stock-Umsatz: tau = 114.79 min

Stock-Umsatzfaktor alpha = 10 Nachdosierungs - Konzentrationsfaktor beta = 1

------REZEPT FUER MONOMER NACHDOSIERUNGSLOESUNG ------[M1] = 0.5622 mol/L [M2] = 2.4378 mol/L m1 = 3.6833 g m2 = 16.3167 g EA VLm = 46.1492 mL Loesungsmittel V0 = 66.8520 mL Gesamtvolumen Volumendosierungsrate: dV/dt = 0.05824 mL/min-1 ------REZEPT FUER INITIATOR NACHDOSIERUNGSLOESUNG ------[MI] = 0.0409 mol/L [I]/[I]0 = 1.3646

mI = 0.0045 g AIBN VLm = 0.6685 mL Loesungsmittel V0 = 0.6720 mL Gesamtvolumen dV/dt(I) = 0.0005854 mL/min ------Gesamte Nachdosierzeit: td = 1.148D+03 min = 19.131 h

105 6.2.3. P[MSA0.20-co-EA0.80]

COPODOS Ansatzberechnung fuer Copolymerisationen mit Monomer- und Initiator - Nachdosierung

Monomer 1 = MSA Monomer 2 = EA Initiator = AIBN T = 50 deg C, AIBN, 1,2-Dichlorethan

Zielpolymer: F(MSA ) = 0.2000 Zielpolymer: F(EA ) = 0.8000

06-17-2004 12:20:52 ------REZEPT FUER STOCK LOESUNG ------f1 = 0.3808 f2 = 0.6192 [M1] = 1.1424 mol/L [M2] = 1.8576 mol/L [I0] = 0.0300 mol/L m1 = 0.7515 g MSA m2 = 1.2485 g EA mI = 0.0331 g AIBN VLm = 4.7406 mL Loesungsmittel V0 = 6.7128 mL Gesamtvolumen ------

Polymerisationsgeschwindigkeiten - total: MSA - total = 1.563D-05 mol/ min = Rtp1 EA - total = 6.253D-05 mol/ min = Rtp2 SUMME - total = 7.816D-05 mol/ min = Rtp

Polymerisationsgeschwindigkeiten - spezifisch: MSA - spezifisch = 2.329D-03 mol/ L min = Rp1 EA - spezifisch = 9.314D-03 mol/ L min = Rp2 SUMME - spezifisch = 1.164D-02 mol/ L min = Rp

Initiierungsgeschwindigkeit: - total = 2.407D-05 mol/ min = Ri - spezifisch = 3.585D-06 mol/ L min = Rit

Polymerisationsgrad Xn = 2.1801D+01 Molekulargewicht Mn = 2.1735D+03 g/mol

Zeit fuer einfachen Stock-Umsatz: tau = 257.67 min

Stock-Umsatzfaktor alpha = 10 Nachdosierungs - Konzentrationsfaktor beta = 1

------REZEPT FUER MONOMER NACHDOSIERUNGSLOESUNG ------[M1] = 1.1424 mol/L [M2] = 1.8576 mol/L m1 = 7.5154 g m2 = 12.4846 g EA VLm = 47.4055 mL Loesungsmittel V0 = 67.1280 mL Gesamtvolumen Volumendosierungsrate: dV/dt = 0.02605 mL/min-1 ------REZEPT FUER INITIATOR NACHDOSIERUNGSLOESUNG ------[MI] = 0.0913 mol/L [I]/[I]0 = 3.0436

mI = 0.0102 g AIBN VLm = 0.6713 mL Loesungsmittel V0 = 0.6791 mL Gesamtvolumen dV/dt(I) = 0.0002636 mL/min ------Gesamte Nachdosierzeit: td = 2.577D+03 min = 42.945 h

106 6.2.4. P[MSA0.30-co-EA0.70]

COPODOS Ansatzberechnung fuer Copolymerisationen mit Monomer- und Initiator - Nachdosierung

Monomer 1 = MSA Monomer 2 = EA Initiator = AIBN T = 50 deg C, AIBN, 1,2-Dichlorethan

Zielpolymer: F(MSA ) = 0.3000 Zielpolymer: F(EA ) = 0.7000

06-17-2004 12:19:45 ------REZEPT FUER STOCK LOESUNG ------f1 = 0.5805 f2 = 0.4195 [M1] = 1.7415 mol/L [M2] = 1.2585 mol/L [I0] = 0.0300 mol/L m1 = 1.1505 g MSA m2 = 0.8495 g EA mI = 0.0332 g AIBN VLm = 4.8714 mL Loesungsmittel V0 = 6.7415 mL Gesamtvolumen ------

Polymerisationsgeschwindigkeiten - total: MSA - total = 6.400D-06 mol/ min = Rtp1 EA - total = 1.493D-05 mol/ min = Rtp2 SUMME - total = 2.133D-05 mol/ min = Rtp

Polymerisationsgeschwindigkeiten - spezifisch: MSA - spezifisch = 9.493D-04 mol/ L min = Rp1 EA - spezifisch = 2.215D-03 mol/ L min = Rp2 SUMME - spezifisch = 3.164D-03 mol/ L min = Rp

Initiierungsgeschwindigkeit: - total = 2.417D-05 mol/ min = Ri - spezifisch = 3.585D-06 mol/ L min = Rit

Polymerisationsgrad Xn = 5.9503D+00 Molekulargewicht Mn = 5.9196D+02 g/mol

Zeit fuer einfachen Stock-Umsatz: tau = 948.10 min

Stock-Umsatzfaktor alpha = 10 Nachdosierungs - Konzentrationsfaktor beta = 1

------REZEPT FUER MONOMER NACHDOSIERUNGSLOESUNG ------[M1] = 1.7415 mol/L [M2] = 1.2585 mol/L m1 = 11.5054 g m2 = 8.4946 g EA VLm = 48.7135 mL Loesungsmittel V0 = 67.4154 mL Gesamtvolumen Volumendosierungsrate: dV/dt = 0.00711 mL/min-1 ------REZEPT FUER INITIATOR NACHDOSIERUNGSLOESUNG ------[MI] = 0.3259 mol/L [I]/[I]0 = 10.8634

mI = 0.0376 g AIBN VLm = 0.6742 mL Loesungsmittel V0 = 0.7031 mL Gesamtvolumen dV/dt(I) = 0.0000742 mL/min ------Gesamte Nachdosierzeit: td = 9.481D+03 min = 158.016 h

107 6.2.5. P[MSA0.35-co-EA0.65]

COPODOS Ansatzberechnung fuer Copolymerisationen mit Monomer- und Initiator - Nachdosierung

Monomer 1 = MSA Monomer 2 = EA Initiator = AIBN T = 50 deg C, AIBN, 1,2-Dichlorethan

Zielpolymer: F(MSA ) = 0.3500 Zielpolymer: F(EA ) = 0.6500

06-17-2004 12:17:27 ------REZEPT FUER STOCK LOESUNG ------f1 = 0.6828 f2 = 0.3172 [M1] = 2.0484 mol/L [M2] = 0.9516 mol/L [I0] = 0.0300 mol/L m1 = 1.3563 g MSA m2 = 0.6437 g EA mI = 0.0333 g AIBN VLm = 4.9388 mL Loesungsmittel V0 = 6.7564 mL Gesamtvolumen ------

Polymerisationsgeschwindigkeiten - total: MSA - total = 2.027D-06 mol/ min = Rtp1 EA - total = 3.765D-06 mol/ min = Rtp2 SUMME - total = 5.792D-06 mol/ min = Rtp

Polymerisationsgeschwindigkeiten - spezifisch: MSA - spezifisch = 3.000D-04 mol/ L min = Rp1 EA - spezifisch = 5.572D-04 mol/ L min = Rp2 SUMME - spezifisch = 8.573D-04 mol/ L min = Rp

Initiierungsgeschwindigkeit: - total = 2.422D-05 mol/ min = Ri - spezifisch = 3.585D-06 mol/ L min = Rit

Polymerisationsgrad Xn = 1.6156D+00 Molekulargewicht Mn = 1.6056D+02 g/mol

Zeit fuer einfachen Stock-Umsatz: tau = 3499.49 min

Stock-Umsatzfaktor alpha = 10 Nachdosierungs - Konzentrationsfaktor beta = 1

------REZEPT FUER MONOMER NACHDOSIERUNGSLOESUNG ------[M1] = 2.0484 mol/L [M2] = 0.9516 mol/L m1 = 13.5628 g m2 = 6.4372 g EA VLm = 49.3880 mL Loesungsmittel V0 = 67.5636 mL Gesamtvolumen Volumendosierungsrate: dV/dt = 0.00193 mL/min-1 ------REZEPT FUER INITIATOR NACHDOSIERUNGSLOESUNG ------[MI] = 1.0830 mol/L [I]/[I]0 = 36.0984

mI = 0.1392 g AIBN VLm = 0.6756 mL Loesungsmittel V0 = 0.7827 mL Gesamtvolumen dV/dt(I) = 0.0000224 mL/min ------Gesamte Nachdosierzeit: td = 3.499D+04 min = 583.249 h

108 7. References

1 G. Odian, "Principles of Polymerization", Chap. 3, p. 298 - 334, 3rd Ed., John Wiley & Sons, Inc., New York / Chichester / Brisbane / Toronto / Singapore 1991. 2 cf. Ref. 1, Chap. 3-6, pp. 243 - 266. 3 cf. Ref. 1, Chap. 3-3b, pp. 206 - 208. 4 cf. Ref. 1, Chap. 3-5, pp. 241 - 243. 5 cf. Ref. 1, Chap. 3-6a, pp. 243 - 244. 6 B. Volmert, "Grundriss der Makromolekularen Chemie", Bd. 1, S. 172 - 173, E. Volmert Verlag, Karlsruhe 1985. 7 G. Odian, "Principles of Polymerization", Chap. 6.2-e, p. 467, 3rd Ed., John Wiley & Sons, Inc., New York / Chichester / Brisbane / Toronto / Singapore 1991. 8 A. E. Hamielec, J. F. MacGregor, A. Penlidis, Makromol. Chem., Macromol. Symp. 10/11, 521 - 570 (1987). 9 F. R. Mayo, C. Walling, Chem. Rev. 46, 191 - 287 (1950). 10 G. E. Ham, "Theory of Copolymerization" in "Copolymeri zation" (G. E. Ham, ed.), Chap. 1, p. 1 - 65, Interscience Publishers, New York - London - Sydney 1964. 11 G. Odian, "Principles of Polymerization", Chap. 6, p. 452 - 531, 3rd Ed., John Wiley & Sons, Inc., New York / Chichester / Brisbane / Toronto / Singapore 1991. 12 C. Walling, J. Amer. Chem. Soc. 71, 1930-1935 (1949). 13 J. N. Atherton, A. M. North, Trans. Faraday Soc. 58, 2049 - 2057 (1962). 14 A. M. North, Polymer 4, 134 - ??? (1963). 15 E. H. DeButts, J. Amer. Chem. Soc. 72, 411 - 414 (1950). 16 V. E. Meyer, G. G. Lowry, J. Polym. Sci. A3, 2843 - 2851 (1965) 17 B. Vollmert, "Grundriss der Makromolekularen Chemie", Vol. I, p. 133 - 136, E. Vollmert Verlag, Karlsruhe 1985. 18 A. D. Polyanin, V. F. Zaitsev, "Handbuch der linearen Differentialgleichungen", p.6, p.9 (No. 11), Spektrum Akademischer Verlag, Heidelberg, Berlin, Oxford 1996. 19 H. R. Schwarz, "Numerische Mathematik", p. 371 - 387, 2nd ed., B. G. Teubner, Stuttgart 1988. 20 M. Rätzsch, M. Arnold, J. Macromol. Sci. - Chem. A24(5), 507 - 515 (1987). 21 M. Kraus, Dissertation, University Ulm, 2002.

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