Ind. Eng. Chem. Res. 2008, 47, 8555–8560
8555
Mathematical Model For Predicting Gel Point in the Process of Manufacturing Alkyd Resins M. A. B. Prashantha, J. K. Premachandra, and A. D. U. S. Amarasinghe* Department of Chemical and Process Engineering, UniVersity of Moratuwa, Sri Lanka
A mathematical model for the prediction of gel formation in the synthesis of alkyd resins is developed. The probability of developing linear polymer molecules is related to the average functionality, Fav. A statistical factor φ is introduced to represent the probability of having free OH groups in the backbone of a linear alkyd molecule. Factor φ increases from 0 to a maximum of φmax when the acid value decreases from AVinitial to AVmax during the polymerization. Branch generation in alkyd molecules becomes significant only after the acid value is reached, AVmax, and it continues until a gel is formed. At φmax, the average functionality (Fav(max)) and the acid value (AVmax) are calculated in terms of the number of moles of OH groups, number of moles of COOH groups, and total number of molecules per unit mass at the beginning of polymerization. The acid value at the gel point, AVgel, is then calculated using the values of Fav(max) and AVmax. The predicted acid values using the new model are in good agreement with the actual data reported for polyesterification in the synthesis of alkyd resins. 1. Introduction
P)
Control of polymerization is essential in making alkyd resins, as gelation may occur before the required extent of polymerization is achieved. Gelation is a common problem in alkyd resin manufacture, and it forms unwanted polymeric materials which cannot be recycled. This results in wastes of raw materials, processing time, and energy. Cleaning vessels also becomes a tedious job as special chemical treatments are required. Therefore, the ability to predict the gel point of a particular polymerization system is of great importance to both research workers and alkyd resin manufacturers. Gelation involves aggregation of clusters in reacting polymers, hydrogels, and colloids.1,2 It is a phase transition process occurring in connecting aggregates of monomers in branched or unbranched chains (each monomer has only one link to each of its nearest neighbors, except at branch points or chain ends) or in networks (monomers are interconnected by several links). Numerous mathematical relations have been derived in modeling both reversible and irreversible gelation.3-10 Gelation in the synthesis of alkyd resins mainly depends on the average functionality of reactants. W. H. Carother developed a mathematical equation11 relating the extent of polymerization (P), initial average functionality (F), and number-average degree of polymerization (DP). An extension to the Carother equation was later proposed by assuming that when a gel product was formed the term DP reached infinity, as no further molecules were available for the reaction to proceed. This argument led to a simple mathematical relation given by eq 1 for predicting the extent of polymerization up to the gel point. P ) 2/F (1) Equation 1 predicts accurate results only if F is close enough to 2, and for higher values of F, the model overestimates the gel point. Flory suggested that Mw (weight-average molar mass) was greater than Mn (number-average molar mass) for polydisperse (Mw * Mn) polymerization systems and therefore Mw achieved infinity before Mn when it formed a gel.12 Flory’s branching treatment led to further modification of the Carother model as * To whom correspondence should be addressed. Tel.: +94 11 2640337. Fax: + 94 11 2650622. E-mail:
[email protected].
( 1 -2 F ) + ( 2r1 ) + ( FF )
(2)
The term r is the initial molar ratio of total number of COOH groups (including from both dibasic acids and monobasic acids) to the total number of OH groups, and F is the initial molar ratio of COOH groups from carboxylic acids other than dibasic acids to the total COOH groups in the reaction mixture. In deriving this model, the extent of reaction was considered up to the median of the molecular weight distribution curve; hence eq 2 predicts an upper limit to the gel point. Another mathematical relation based on the extent of reaction at infinite molecular weight has been derived (eq 3) for predicting a lower limit to the gel point, using further statistical analysis.11 P ) [r + rF(F - 2)]-1/2 (3) 13 Flory defined a new term, the branching coefficient (R), to describe the incipient of the formation of an infinite network Table 1. Actual and Predicted Gel Point Acid Values gel point acid values calculated by models (mg/g)
alkyd AV final recipes (mg/g) Carother Wiedehorn Flory Stockmayer proposed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
11.57 9.7 5.88 5.27 4.67 7.21 2.41 1.11 0.74 10.1 9 10.2 15 15 18 18 18 29 19 20
60.34 45.76 38.65 31.67 18.54 5.84 22.04 15.84 10.18 15.6 16.4 10.2 29.75 15.39 35.12 29.1 31.17 35.02 30.79 35.02
27.74 19.27 12.43 6.6 -6.15 -16.56 9.28 0.89 -5.83 15.67 9.27 10.87 25.02 9.52 21.1 12.78 15.73 20.97 15.2 20.97
10.1021/ie8005534 CCC: $40.75 2008 American Chemical Society Published on Web 10/16/2008
-2.92 13.47 14.86 18.45 22.22 29.54 43.77 39.81 38.55 68.49 54.59 46.6 62.35 57.87 45.43 47.54 46.82 45.46 46.95 45.46
-21.59 -12.55 -14.9 -14.52 -16.49 -12.37 10.57 3.58 0.11 37.09 21.08 31.56 34.12 23.91 15.94 10.38 12.33 15.85 11.98 15.85
18.94 16.95 13.01 10.13 3.43 -1.37 13.89 8.35 4.08 16.23 13.22 11.84 24.88 13.15 22.25 17.37 19.08 22.17 18.77 22.17
8556 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
Figure 1. Part of an alkyd molecule with a free OH group in the backbone.
COOH groups at the gel point. The outcome of his model consisted of three equations, as given by eqs 7, 8, and 9. AVgel )
18700(p + f) 1.5a - 3.5 + Fn W p+f
(
(
W ) aMa + pMp + fMf - 18
)(
)
(p + f)(Fn - 2) 1.5 -1 Fn - 2 1.5
(7)
) (8)
Fn ) Figure 2. Variation of φ with acid value in the polyesterification stage of monoglyceride mixture.
and developed a relation between the branching coefficient and the extent of polymerization. He assumed that the term R(fb 1) is equal to unity when reaction medium becomes a gel. At that condition R is called the critical branching coefficient (Rc).Therefore Rc ) 1/(fb - 1)
(4)
where fb is the functionality of branching units (molecular species with functionality greater than 2, such as glycerol and pentaerythritol). Flory then related R to a probability parameter PA (probability that an OH group had reacted) and PB (the probability that a COOH group had reacted) as R)
PAPBR 1 - PAPB(1 - R)
(5)
where R is the ratio of the number of OH groups (both reacted and unreacted) in branching units to the total number of OH groups in the reaction mixture. Stockmayer included the effect of ring formation in his analysis and developed another statistical relation to predict the extent of polymerization at the gel point.14-16 Pgel2 )
⁄
(∑ g G ) [∑ (f 2
j
j
2
i
- fi)Hi
∑ (g
2
j
]
- gj)Hj
(6)
where fi and Hi represent the COOH group functionality and the number of moles of i type of carboxylic acid molecules, respectively. gj and Gj represent hydroxyl functionality and the number of moles of j type of polyol molecules, respectively. Wiedehorn followed a method similar to the method of Flory but modified the critical value for branching coefficient by considering an average of 1.5 cross-links per polymer molecule in the reaction mixture when forming the single giant molecule.17 In his model Wiedehorn used the acid value at the gel point (AVgel) as a measure of the number of moles of unreacted
eie - q e
(9)
where a is the number of moles of phthalic anhydride, p is the number of moles of polyols, f is the number of moles of oil, Fn is the number-average functionality of mixed polyols prior to esterification, and Ma, Mp, and Mf are molar masses of dibasic acid, polyol, and oil, respectively. W is the weight of the reaction mixture at the gel point after losing water, e is the number of moles of polyols, ie is the average hydroxyl functionality of polyol, and q is the number of moles of monobasic acid. These models are useful not only in controlling a polymerization process without a gel formation but also in selecting a correct formulation. The objectives of the present study were to develop a new mathematical model for predicting the gel point by considering the probability of forming a highly branched molecule from linear polymer molecules during the polymerization process and to compare its accuracy with the existing models. 2. Basis for Mathematical Model 2.1. Basic Assumptions. The new model is based on the assumption that incipient branching occurs at the unreacted OH groups in branching units (monomers having more than two functional groups) which are in the polymer backbone. Further, ring formation reactions are assumed to be negligible during the polyesterification. The following steps were assumed in the development of the alkyd molecule from a linear structure to a highly branched three-dimensional structure and finally to a single giant three-dimensional molecule at the gel point. 2.1.1. Growth of Polymer Molecule. At the beginning of polyesterification, dibasic acid or its anhydride reacts with various types of monoglycerides, dipentaerythritide, and branching units, and a linear alkyd molecule is formed. As the polymer molecule grows, the probability of the availability of free “OH groups” in the polymer backbone increases and the acid value of the polymerization mixture decreases. Figure 1 shows a part of an alkyd molecule with a free OH group in the polymer backbone. The steric hindrances may prevent these free OH groups in the polymer backbone from taking part in branching.
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8557
K ) initial number of moles of OH groups in branching units which contribute to placing them in segmental positions of polymer backbones expressed as a fraction of the initial number of moles of OH groups in total L ) initial number of moles of OH groups which can generate branches as a fraction of the initial number of moles of OH groups in total R ) probability of the availability of linear polymer molecules in the reaction mixture. 3.1.1. Probability Factor J. The probability factor J is expressed as Figure 3. Comparison of the Wiedehorn model and the proposed model with actual data.
As a result, the average functionality of polymer molecules exceeds the value 2 with further continuation of the polymerization reaction. 2.1.2. Branching. With the increase in the extent of polymerization, the concentration of free OH groups in monomers and oligomers continuously decreases while the concentration of free OH groups in the polymer backbone increases. Consequently, after a certain extent of reaction (or below a certain acid value), the probability of the presence of free OH groups becomes greater in polymer backbones than in monomers and oligomers, and the branching becomes significant. 2.1.3. Gel Formation. When the branching becomes significant, the probability of the presence of free OH groups in the polymer backbone starts to decrease while the acid value further decreases due to the overall reduction in free OH groups. At this stage of polymerization, the probability of the existence of linear polymer molecules significantly decreases and finally a highly branched single giant molecule is formed. At this moment, the reaction medium becomes a solid lump and the corresponding acid value is called the gel point acid value. 2.2. Probability Factor φ. A probability factor φ is introduced to represent the probability of having free OH groups in the backbone of a linear alkyd molecule, which increases from 0 to a maximum (φmax) while the acid value decreases from AVini to AVmax during the early stage of polyesterification (Figure 2). AVmax corresponds to the acid value at which the branching becomes significant, and when the acid value further decreases with the increase in the extent of polymerization, φ starts to decrease. As a result, for a given φ there are two corresponding acid values, as shown in Figure 2. At the higher acid value the reaction mixture has a lower amount of branched polymer molecules, and at the lower acid value the reaction mixture has a higher amount of branched polymer molecules. As the reaction further continues, the acid value approaches AVgel, at which a three-dimensional highly branched gel product is formed. 3. Model Development 3.1. Mathematical Expression for φ. The probability factor φ is expressed as a product of four other probability factors: J, K, L, and R. The factors K and L are constants which depend on the characteristics of the alkyd recipe, while J and R are dependent on the extent of polymerization. φ)J·K·L·R (10) where J ) number of moles of reacted OH groups at a given acid value as a fraction of the initial number of moles of OH groups in total
J)
NOH - nOH NOH
(11)
where NOH is the initial number of moles of OH groups per gram and nOH is the number of moles of OH groups per gram at any moment of polyesterification. Since the stoichiometry between COOH and OH groups is 1:1, the number of moles of reacted OH groups at a given acid value is equal to the number of moles of reacted COOH groups at the same acid value, and is expressed as NOH - nOH ) NCOOH - nCOOH
(12)
and eq 11 can be modified as J)
NCOOH - nCOOH NOH
(13)
where NCOOH is the initial number of moles of COOH groups per gram and nCOOH is the number of moles of COOH groups per gram at any moment of polyesterification. 3.1.2. Probability Factor K. Two of the OH groups in a branching unit are always used in forming a linear molecule. Therefore K can be expressed as K)
2(Ngly + Npen) NOH
(14)
where Ngly is the initial number of moles of free glycerol per gram and Npen is the initial number of moles of free pentaerythritol per gram. 3.1.3. Probability Factor L. Since glycerol contributes one OH group and pentaerythritol contributes two OH groups for branching, the factor L can be expressed as L)
Ngly + 2Npen NOH
(15)
3.1.4. Probability Factor R. The average functionality at any moment is defined as the ratio of the total number of moles of OH and COOH groups to the total number of moles of all the molecules, and can be expressed as Fav )
nOH + nCOOH n
(16)
The average functionality (Fav) of molecules in a mixture of dicarboxylic acid and dihydroxy derivatives like monoglycerides is exactly 2; hence such a mixture forms entirely linear alkyd molecules without forming a gel. On the other hand, polyols having more than two OH groups react with dicarboxylic acid to form linear polymer molecules consisting of unreacted OH groups. In such a system Fav is greater than 2 and branching occurs due to the presence of these free OH groups in polymer backbones. Hence the probability of generating a linear polymer molecule in a polyesterification system is assumed to be 2/Fav. Therefore, the probability factor R is expressed as
8558 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
R)
2 2n ) Fav nOH + nCOOH
(17)
moles of COOH groups per gram of the reaction mixture at a given acid value (AVx) is given by
The total number of molecules in the polyesterification mixture at any moment of polyesterification, n, is given by n ) N - (nest + 1) + 1
(18)
where nest is the number of moles of ester bonds per gram of the reaction mixture at any moment of the polyesterification, and is given by nest ) NCOOH - nCOOH
(19)
Substitution of eqs 12, 18, and 19 in eqs 16 and 17 gives the alternative forms of these equations as (NOH - NCOOH) + 2nCOOH Fav ) N - NCOOH + nCOOH R)
2(N - NCOOH + nCOOH) NOH - NCOOH + 2nCOOH
2(N - NCOOH + nCOOH) NCOOH - nCOOH KL NOH NOH - NCOOH + 2nCOOH
(22)
nCOOH(max) ) Substituting for nCOOH(max) AVmax )
(
AVmax
5.61 × 104 and rearranging eq 27 yields
(29)
)
AVmax )
(
)
5.61 × 104 [(NCOOH - NOH) + 2 √(NCOOH + NOH)[(NCOOH + NOH) - 2N] ] (31)
Further, the average functionality Fav(max) at φ ) φmax can be obtained from eq 20 as Fav(max) )
(24)
Note that φ varies only with Q as N, NOH, and NCOOH are characteristic constants for a given alkyd recipe. At φ ) φmax the branching becomes significant and, therefore, by equating the first derivative of φ to 0, the corresponding Qmax can be found: Qmax ) +(NCOOH + NOH) ( √(NCOOH + NOH)[(NCOOH + NOH) - 2N] (25) 2 Since Q ) NCOOH - nCOOH, Qmax can be expressed as Qmax ) NCOOH - nCOOH(max)
Hence the acid value at φ ) φmax, AVmax, can be calculated from
Since the acid value is always positive, the negative result of eq 30 is ignored and the equation becomes
3.2. Significance of φmax. As explained in section 2.2, the probability factor φ varies with the extent of polymerization, which is a direct function of the acid value (see Figure 2). Note that the term NCOOH - nCOOH in eq 23 is a measure of the number of ester bonds formed at any moment (see eq 19). By assigning Q and C for the terms NCOOH - nCOOH and 2KL/NOH, respectively, eq 23 can be modified as C(-Q2 + NQ) NOH + NCOOH - 2Q
(28)
(21)
(2KL/NOH)[NCOOH - nCOOH][N - (NCOOH - nCOOH)] NOH + NCOOH - 2(NCOOH - nCOOH) (23)
φ)
5.61 × 104
(20)
Rearranging eq 22 gives φ)
AVx
5.61 × 104 [(NCOOH - NOH) ( 2 √(NCOOH + NOH)[(NCOOH + NOH) - 2N] ] (30)
Since K and L are constants for a given recipe, substituting for J using eq 13 and R using eq 21, in eq 10 yields a mathematical expression for φ as φ)
nCOOH )
(26)
where nCOOH(max) is the total number of moles of COOH groups per gram at φ ) φmax. Substituting for Qmax in eq 25 yields nCOOH(max) ) (NCOOH - NOH) ( √(NCOOH + NOH)[(NCOOH + NOH) - 2N] (27) 2 The acid value is the required number of moles of KOH to neutralize the COOH groups per gram; hence the number of
(NOH - NCOOH) + 2nCOOH(max) N - NCOOH + nCOOH(max)
(32)
Substituting for nCOOH(max) from eq 29 yields Fav(max) )
(5.61 × 104)(NOH - NCOOH) + 2(AVmax) (5.61 × 104)(N - NCOOH) + AVmax
(33)
3.3. Prediction of Gel Point. The polymerization reaction forms two types of molecules: linear and branched. The formation of linear molecules predominates at the beginning of the reaction until the acid value decreases to AVmax, and the branching becomes significant once AVmax is reached. If the total number of molecules per gram at AVmax is N′max and if all the molecules are assumed to be taking part in the reaction at the moment of gelation, then the number of ester bonds formed during the period from AVmax to AVgel is N′max - 1 and the number of reacted functional groups is 2(N′max - 1). The number of functional groups reacting during the period from AVmax to AVgel as a fraction of the total number of functional groups available at φ ) φmax is used to denote the extent of reaction, X, during the period from AVmax to AVgel, and is given by X)
2(N′max - 1) n′OH(max) + n′COOH(max)
(34)
Since N′max is very much greater than 1, eq 34 further simplifies to X)
2N′max n′OH(max) + n′COOH(max)
(35)
Using eq 16, the average functionality at φ ) φmax can be expressed as Fav(max) )
n′OH(max) + n′COOH(max) N′max
(36)
where n′OH(max) and n′COOH(max) are the number of OH groups and number of COOH groups respectively at φ ) φmax.
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8559 Table 2. Average of Absolute Differences of Model Data from Actual Data model average of absolute variation
Carother Wiedehorn Flory Stockmayer proposed 13.6
5.5
25.1
11.3
4.1
Comparing eqs 35 and 36, a simple relation between the extent of reaction during the period from AVmax to AVgel and the average functionality at φ ) φmax can be obtained as X)
2 Fav(max)
(37)
The extent of a polyesterification reaction is generally expressed in terms of acid value; hence the extent of reaction during the period from AVmax to AVgel is expressed as X)
AVmax - AVgel AVmax
(38)
A final equation relating the acid value at the gel point to AVmax and Fav( max) can be obtained using eqs 37 and 38 and is given by
(
AVgel ) AVmax 1 -
2 Fav(max)
)
functional groups such as the OH groups in glycerol have unequal reactivity due to the effects of steric hindrance. Stockmayer included the ring formation effects in deriving his model. The over- and underestimations of the ring formation effects might have caused the observed positive and negative results of the predicted acid values. Table 2 shows that the Wiedehorn model is in good agreement with experimental final acid values. This suggests that his assumption for the number of cross-links per polymer molecule, 1.5, was in close agreement with the actual average of the number of cross-links per molecule for synthesizing alkyd resins. Table 2 indicates that the proposed model gave lower deviations from actual acid values observed at the gel point for polyesterification reactions in the synthesis of alkyd resins. A comparison of the proposed model with the Wiedehorn model which gave the lowest deviations from actual values is shown in Figure 3. These results suggest that the new model is useful in predicting the acid values at the gel point for the polyesterification reactions in synthesizing alkyd resins. However, further study is required to examine the validity of the new model for other polymerization reactions. 5. Conclusion
(39)
The values of AVmax and Fav(max) can be calculated from eqs 31 and 33, and therefore eq 39 predicts the acid value at the gel point of a polyesterification reaction in the synthesis of alkyd resins. 4. Comparison of the New Model with Existing Models The experimental data published in the literature 18-21 were used to evaluate the mathematical models available for predicting the onset of gel point in the synthesis of alkyd resins, including the proposed model. The mathematical models used in the analysis were the Carother model, the equation for the lower limit gel point, the equation for the upper limit gel point, the Flory equation, the Stockmayer equation, and the Wiedehorn equation. The extent of polymerization of the experimental data used in this analysis was found to be above 94%. Therefore the actual acid values at the gel point corresponding to each of these recipes were assumed to be just below the experimentally observed final acid values. Table 1 shows the actual data of both single oil systems and oil blends available in the literature with calculated acid values at the gel point using different types of models. Table 2 shows the average values of the absolute differences between the calculated and actual data for each model. The majority of the predicted values calculated using the models of the lower limit gel point and upper limit gel point (eqs 2 and 3) were negative. Hence the predicted data from these two models were not included in Table 1. The models of Carother, Flory, and Stockmayer are widely used in predicting the gel points of polymerization reactions due to their simplicity and applicability to a wide variety of polymers. However, the predicted values using these models deviated considerably from the actual values for the polyesterification reactions in the synthesis of alkyd resins (see Table 2). The possible reasons for these deviations are discussed below. The Carother model was based on the assumption that Mn reached infinity at the gel point. However, in reality, Mw reaches infinity before Mn. Further, Carother did not consider effects other than initial functionality, such as structural changes during the polymerization and side reactions. Flory assumed equal reactivity of functional groups in deriving his model. However,
The new model gave better predictions of acid values at the gel point than the existing models for the polyesterification reactions in the synthesis of alkyd resins. Since the new model was developed on the basis of the availability of OH groups in the polymer backbone during the polyesterification stage of the manufacturing of alkyd resins, further study is required to examine its applicability for other polymerization reactions. Nomenclature AVini ) initial acid value of the polyesterification of alkyd resin AVx ) acid value at any moment AVmax ) acid value at φ ) φmax AVgel ) acid value at gel point Fav ) average functionality at any moment given by ratio of total moles of OH and COOH to total moles of molecules Fav(max) ) average functionality at φ ) φmax J ) fraction of reacted moles of OH groups at a given acid value to initial moles of OH K ) OH groups in branching units which contribute to place them in segmental positions of polymer backbones expressed as a ratio to initial moles of total OH groups L ) ratio of branch generating OH groups to total moles of OH groups N ) total number of moles of molecules per gram of reaction medium before starting polyesterification NCOOH ) initial moles of COOH groups per gram NOH ) initial moles of OH groups per gram Ngly ) moles of free glycerol per gram of reaction medium before starting polyesterification Npen ) moles of free pentaerythritol per gram of reaction medium before starting polyesterification Noil ) initial moles of oil per gram N′max ) total number of molecules per gram at φmax n ) total number of moles of molecules per gram of reaction medium at any moment after starting polyesterification nCOOH ) total moles of COOH groups per gram at any acid value after starting polyesterification nCOOH(max) ) total number of moles of COOH groups per gram at φmax n′COOH (max) ) total number of COOH groups per gram at φmax
8560 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 nest ) number of moles of ester bonds per gram at a given acid value after starting polyesterification nOH ) total moles of OH groups per gram at any acid value after starting polyesterification n′OH (max) ) total number of OH groups per gram at φmax R ) probability of availability of linear polymer molecules in reaction mixture X ) functional groups reacted during polyesterification reaction from AVmax to AVgel as a fraction of total number of functional groups at φmax Greek Symbols φ ) probability of having free OH groups in backbone of linear alkyd molecule φmax ) maximum value of φ for a given recipe of making alkyds
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ReceiVed for reView April 7, 2008 ReVised manuscript receiVed August 11, 2008 Accepted September 3, 2008 IE8005534
