Introduction

The ethylene–vinyl acetate copolymer (EVA) is one of the most useful polymers, due to its wide range of applications in, for example, coating of photovoltaic cells, or obtaining tyres and cable, as a consequence of the different varieties of the material, depending on the vinyl acetate percentage [1].

The way the EVA copolymer is industrially obtained is related to the vinyl acetate content. When the vinyl acetate content is lower than 40 %, this material is manufactured by means of high pressure processes, and its main applications are as modifiers [2]; when the vinyl acetate content is between 30 and 40 %, a dissolution process at moderate temperatures and pressures is carried out, and the main applications of the obtained material are as an elastomer (obtaining synthetic elastomers, tyres, …) [2]; finally, when the vinyl acetate content is higher than 70 %, the manufacture is carried out employing an emulsion process, and the main uses of the final material are as adhesives.

Due to the most important applications of EVA-type materials as elastomers, we are focusing in studying dissolution processes [3–5]. In these processes, the main drawback is the separation step, with the aim of getting a pure EVA material; for this reason, it is advisable to accurately study the interactions between the polymer and the solvent.

A polymer–solvent mixture differs from a conventional solvent–solvent mixture, as a consequence of the great difference between the relative sizes of the molecules of both components. Two of the most important parameters of these special systems are the solubility parameter (δ) [6] and the polymer–solvent Flory–Huggins parameter (χ) [7].

Initially, the solubility parameter of a compound i was defined by Hildebrand [6] as the square root of its cohesive energy according to Eq. (1), where Δvap H m,i is the enthalpy of vaporization of the compound i, R is the universal gases constant, T is the absolute temperature and V m,i is the molar volume of the i compound. Cohesive energy can be derived from the enthalpy of vaporization because the intermolecular attractive forces which have to be overcome to vaporize a liquid are the same ones that have to be overcome to dissolve it.

$$\delta_{i} = \sqrt {c_{i} } = \left[ {\frac{{\Updelta_{vap} H_{{{\text{m}},i}} RT}}{{V_{{{\text{m}},i}} }}} \right]^{0.5}$$
(1)

The more important drawback of the traditional defined Hildebrand solubility parameter definition is that it only takes into account dispersive interactions, but no dipole–dipole interactions or hydrogen bonding interactions. This approximation is reliable with non-polar solvents, like cyclohexane, but it does not seem to be accurate with polar solvents, like tetrahydrofuran.

So, with the aim of overcoming this difficulty, Hansen proposed in 1969 [8] to divide the solubility parameter into three different contributions: one due to non-polar or dispersion forces, another due to polar forces, and a last one which takes into account hydrogen bonding effects. Thus, the overall solubility parameter of a compound is now determined with Eq. (2), where δ i indicates the solubility parameter of a solvent i, δ d,i is the apolar contribution, δ p,i is the polar contribution, and δ h,i is the hydrogen bonding contribution.

$$\delta_{i}^{2} = \delta_{d,i}^{2} + \delta_{p,i}^{2} + \delta_{h,i}^{2}$$
(2)

From the solubility parameter of both polymer and solvent, the Flory–Huggins interaction parameter (FH parameter) can be determined, according to Blanks and Prausnitz [9], by applying Eq. (3).

$$\chi = \chi_{\text{S}} + \chi_{\text{H}} = \chi_{\text{S}} + \frac{{V_{{{\text{m}},i}} }}{RT}\left( {\delta_{i} - \delta_{2} } \right)^{2}$$
(3)

In this last equation, χ is the Flory–Huggins parameter, χ S is the entropic contribution of this parameter (usually 0.34, according Blanks and Prausnitz [9]), and χ H is the enthalpic contribution, obtained from the molar volume of a solvent i (V m,i ), the absolute temperature (T), and the Hildebrand solubility parameters of the polymer (δ 2) and the solvent i (δ i ).

However, in order to take into account not only dispersion forces but also polar and hydrogen bonding forces in the calculation of the FH parameter, it is necessary to substitute in Eq. (3) the Hildebrand solubility parameters by a new term which includes the different terms of Hansen solubility parameter (HSP), Eqs. (4) and (5).

$$A_{12} = \left[ {\left( {\delta_{d,i}^{{}} - \, \delta_{d,2}^{{}} } \right)^{2} + 0.25\left( {\delta_{p,i}^{{}} - \, \delta_{p,2}^{{}} } \right)^{2} + 0.25\left( {\delta_{h,i}^{{}} - \, \delta_{h,2}^{{}} } \right)^{2} } \right]$$
(4)
$$\chi = \chi_{\text{S}} + \chi_{\text{H}} = \chi_{\text{S}} + \frac{{V_{{{\text{m}},i}} }}{RT}A_{12}$$
(5)

These parameters (HSP and FH) have been demonstrated to be extremely useful to properly model the separation steps in a polymer obtaining process [10].

In the literature, several data showing the solubility parameter of a wide range of solvents have been previously published [11, 12]; however, the solubility parameter of an EVA copolymer is not a standard value because could depend on the vinyl acetate content and on the polymer molecular weight distribution. For this reason, it is important to determine this last value experimentally. Concerning the FH parameter, there is no reference reporting this parameter for any kind of EVA. Both parameters are quite important to choose, for example, a suitable solvent for a specific polymer, so that the further purification can be efficiently carried out.

As previously mentioned, a polymer–solvent mixture is quite different from a conventional mixture because there is a great difference between both molecular sizes. For this reason, in order to determine thermodynamic parameters of these mixtures, it is necessary to use non-conventional techniques, because typical ebullometric measurements cannot be directly performed. The most well-known techniques are swelling [13], inverse gas chromatography [14, 15], intrinsic viscosity [16] and turbidimetry [17].

Among all these techniques, intrinsic viscosity (IV) is one of the most widely used because of its reliability and easy starting up. We selected intrinsic viscosity method because, for our purpose, it is very fast and simple to get accurate values indirectly from viscosity measurements of polymer dilute solutions. Since IV is related to an infinite diluted solution, a really small amount of polymer added to pure solvent will change so much its viscosity. In the literature [18, 19], this technique has been widely employed with good results.

This paper reports the results of the HSP and the FH polymer–solvent interaction parameters of two EVA copolymers, with different vinyl acetate contents. The purpose of this study is to assess the dependence of these two parameters on the vinyl acetate content, so that the interactions between each polymer and the studied solvents are clearly defined. This is fundamental to accurately design the polymer–solvent purification steps, which are extremely important in a polymer production process. Besides, the HSP is frequently used as criteria to select the most suitable solvent for a polymer.

Experimental section

Materials

Both EVA copolymers were supplied by REPSOL-YPF Company. While EVA 1 (EVA460) has a vinyl acetate content of 33 % (w/w), EVA 2 (EVA410) has a vinyl acetate content of 18 % (w/w). The weight-average molecular weight and the number-average molecular weight are 61,041 and 18,582 for EVA 1, and 42,460 and 14,008 for EVA 2 determined by gel exclusion chromatography. Finally, the densities of the two polymeric materials are 956 and 937 kg/m3, respectively, as reported by the supplier.

On the other hand, all the employed solvents were analytical grade and were purchased from Aldrich. They were used directly, without any purification step.

Intrinsic viscosity determination

All most concentrated polymer solutions were prepared by adding 200–300 mg of polymer over approx. 60 g of pure solvent, and then shaking until the elastomer became dissolved. The rest of solutions were prepared adding pure solvent.

Viscosity measurements were carried out in a JP–Selecta Ubbelohde 0b type of capillary viscosimeter. Once prepared, each solution was transferred into the viscosimeter, which was immersed in a water bath, thermostated at T ± 0.01 °C. The solutions were allowed to equilibrate at the adequate temperature before starting the measurement. The accuracy of the measurements was 10−2 s. Each flow time was measured 5 times and the average value was taken; from the flow times, relative and specific viscosities were determined.

The IV of a polymer–solvent mixture, [η], is defined as the viscosity of an infinitely diluted polymer solution, and it can be calculated, from the previously described flow time measurements, with Huggins (Eq. 6) and Kraemer (Eq. 7) expressions [20, 21].

$$\frac{{\eta_{\text{sp}} }}{c} = \left[ \eta \right] + K_{\text{H}} \left[ \eta \right]^{2} c$$
(6)
$$\frac{{\ln (\eta_{\text{r}} )}}{c} = \left[ \eta \right] + K_{\text{K}} \left[ \eta \right]^{2} c$$
(7)

In these equations, c is the concentration of polymer solution, K H is the Huggins coefficient, and K K is the Kramer coefficient; relative viscosity (η r) is the relation between the flow time of the polymer solution through the viscosimeter and the flow time of the pure solvent through the apparatus; finally, specific viscosity (η sp) is defined as relative viscosity minus one, and represents the viscosity increasing due to the polymer. So, the intrinsic viscosity ([η]) can be determined as the common intercept of the Kraemer and Huggins relationships, using η r and η sp experimentally determined.

Solubility parameter calculation

The three contributions of the HSP of the two EVA copolymers were calculated following Segarceanu and Leca [22] procedure. In this method, the intrinsic viscosities values are normalized by the intrinsic viscosity of that solvent giving the highest value, according to Eqs. (8), (9) and (10). Finally, the overall solubility parameter can be calculated using Eq. (2).

$$\delta_{d,2}^{{}} = \frac{{\sum\nolimits_{i}^{n} {\delta_{d,i}^{{}} \left[ \eta \right]_{i} } }}{{\sum\nolimits_{i}^{n} {\left[ \eta \right]_{i} } }}$$
(8)
$$\delta_{p,2} = \frac{{\sum\nolimits_{i}^{n} {\delta_{p,i} \left[ \eta \right]_{i} } }}{{\sum\nolimits_{i}^{n} {\left[ \eta \right]_{i} } }}$$
(9)
$$\delta_{h,2} = \frac{{\sum\nolimits_{i}^{n} {\delta_{h,i} \left[ \eta \right]_{i} } }}{{\sum\nolimits_{i}^{n} {\left[ \eta \right]_{i} } }}$$
(10)

In these equations, the subscript 2 refers to polymer, and the subscript i refers to each solvent from 1 to n (n = 5 in this case). The subscripts d, p and h refers to dispersion, polar and hydrogen bonding contributions, respectively.

FH parameter calculation

The FH parameter values were directly determined from IV values, by means of Stockmayer and Fixman [23] (Eq. 11) and Berry [24] (Eq. 12) relationships. These equations relate the intrinsic viscosity and the Flory–Huggins parameter in theta conditions, with the intrinsic viscosity and the Flory–Huggins parameter in non-theta conditions. It allows calculating the value of the Flory–Huggins parameter in any condition, taking into account that the value of this parameter at theta conditions is 0.5. These expressions have been already used in the literature [25] to obtain polymer solvent interaction parameters, with good results.

$${\raise0.7ex\hbox{${\left[ \eta \right]}$} \!\mathord{\left/ {\vphantom {{\left[ \eta \right]} {M_{2}^{1/2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${M_{2}^{1/2} }$}} = K_{0} + 0.51B\phi_{0} M_{2}^{1/2}$$
(11)
$${\raise0.7ex\hbox{${\left[ \eta \right]^{1/2} }$} \!\mathord{\left/ {\vphantom {{\left[ \eta \right]^{1/2} } {M_{2}^{1/4} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${M_{2}^{1/4} }$}} = K_{0}^{1/2} + 0.42K_{0}^{1/2} B\phi_{0} M_{2} \left[ \eta \right]^{ - 1} \,$$
(12)

In these last expressions, [η] is the IV value of the polymer–solvent couple, K 0 is the unperturbed dimension parameter, which is related to the intrinsic viscosity under theta conditions ([η] θ ) by means of Eq. (13), Φ 0 is the universal viscosity constant, being equal to 2.5 × 10−21 mol−1 if the unit of [η] is dl g−1, M 2 is number-average polymer molecular weight, and B is the parameter for the polymer–solvent interactions, which is related to the Flory–Huggins parameter (χ) by means of Eq. (14).

$$K_{0} = {\raise0.7ex\hbox{${\left[ \eta \right]_{\theta } }$} \!\mathord{\left/ {\vphantom {{\left[ \eta \right]_{\theta } } {M_{2}^{1/2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${M_{2}^{1/2} }$}} \,$$
(13)
$$B = \frac{{v_{{^{2} }}^{2} \left( {1 - 2\chi } \right)}}{{V_{1} N_{\text{A}} }}$$
(14)

In these last equations, v 2 is the partial specific volume of the polymer, V 1 is molar volume of the solvent, and N A is Avogadro’s number.

Intrinsic viscosity under theta conditions determination

To determine the intrinsic viscosity under theta conditions, several techniques have been proposed; however, the most reliable ones are the turbidimetric-based or cloud-point techniques: cloud-point titration and cloud temperature titration [12]. The first one consists on titrating dilute polymer solutions of different compositions, with a non-solvent at constant temperature. The second one implies cooling or heating a polymer solution until the appearance of cloudiness. In the literature, these techniques have been accurately employed with a wide range of polymeric materials [26].

In this work, we have applied the cloud-point titration according to the “Elias method” [27]. Following this method, five diluted solutions (D1–D5) of both EVA copolymers (component 2) in cyclohexane (component 1) were prepared, with compositions ranging between 0.0005 and 0.003 v/v. The solutions were initially prepared in weight basis and later on, the volumetric fractions were calculated from the mass fractions, taking into account the density of both components.

In the next step, from each solution, a 5 mL aliquot was taken and put in a thermostated bath until reaching constant temperature (measured with an Hg thermometer with an accuracy of ±0.1 °C). Finally, while magnetically stirring, the aliquot was titrated with acetone (component 3) until cloud point was achieved, employing an analogic burette whose accuracy is 0.05 mL; the cloud point was visually detected.

This experiment was repeated five times for each solution so the final acetone volume was considered to be the arithmetic mean of the five titrations. The experiments were carried out at temperature values of 30, 40 and 50 °C.

From the acetone consumption, the average volume fraction of acetone at cloud point (Φ 3,cp) of each solution was calculated. According to the “Elias method”, this parameter is plotted, at each temperature, against the logarithm of the polymer volume fraction at cloud point (Φ 2,cp), and the intercept of the plot is the volume fraction of the titrating agent (Φ 3,θ ) in a solvent-titrating agent mixture (cyclohexane-acetone in this case) which behaves as “theta solvent” at the specified temperature (Eq. 15).

$$\phi_{{3,{\text{cp}}}} = \phi_{3,\theta } - B_{cp} \cdot \,\ln \phi_{{2,{\text{cp}}}}$$
(15)

Once the “theta solvent” was known at each temperature for both EVA materials, their intrinsic viscosity was measured at 30, 40 and 50 °C following the procedure already described in "Intrinsic viscosity determination" and "Solubility parameter calculation".

Results and discussion

Intrinsic viscosity under theta conditions

Tables 1 and 2 show the acetone consumption (expressed as Φ 3,cp, volumetric fraction of acetone at cloud point), as a function of the volumetric fraction of EVA (Φ 3) in the initial solution.

Table 1 Turbidimetric results for EVA 1 (460)
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Table 2 Turbidimetric results for EVA 2 (410)
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As it can be observed, the acetone consumption is clearly higher for EVA460 polymer than for EVA410 polymer. This can be related to the vinyl acetate content: the more vinyl acetate content, the more polar character of the copolymer and the more affinity towards acetone, which implies that a higher amount of acetone is needed to reach the immiscibility.

It can also be noticed that the acetone consumption increases with the temperature. Taking into account that cyclohexane and acetone are completely miscible in the whole temperature range and that both EVA copolymers are completely miscible with cyclohexane, this indicates that it is the solubility of the two polymeric materials in acetone what really increases with the temperature.

Figures 1 and 2 show the Elias plots for the two polymeric materials at the three different temperatures.

Fig. 1
figure 1

Elias plot for EVA 1—cyclohexane/acetone system

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Fig. 2
figure 2

Elias plot for EVA 2—cyclohexane/acetone system

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From the intercepts of Figs. 1 and 2, the acetone/cyclohexane volumetric proportion which behaves as theta solvent for each copolymer and at each temperature can be determined; the values are shown in Table 3. Finally, the IV values under theta conditions are shown in Table 4.

Table 3 Volumetric composition of theta solvent for both EVA’s copolymers
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Table 4 Intrinsic viscosity under theta conditions
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Flory–Huggins and solubility parameters from intrinsic viscosity measurements

In this work, the intrinsic viscosity of five different solvents (cyclopentane, cyclohexane, toluene, p-xylene and tetrahydrofuran) in two commercial EVA copolymers supplied by REPSOL-YPF Company and at three different temperatures (30, 40, 50 °C) has been measured; all of them are potential solvents to carry out the synthesis of an EVA material following a dissolution process. The intrinsic viscosity results along with the difference between Huggins and Kraemer constants are summarized in Table 5. Moreover, as an example, the Huggins and Kraemer plots for both toluene–EVA systems at 30 °C are shown in Fig. 3.

Table 5 Intrinsic viscosity results
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Fig. 3
figure 3

Huggins and Kraemer plots for Toluene—EVA mixtures at T = 30 °C

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To analyze the dependence of intrinsic viscosity upon temperature and vinyl acetate content, it is important to point out that the behavior of a polymer in solution is related mainly to its nature, but also on the kind of solvent; these two parameters will influence the dimensions of the polymer, which are related to the intrinsic viscosity.

Although, the unperturbed dimension of a polymer (< r 2 > 0) is supposed to be independent of the temperature (due to its own definition: the dimension of a macromolecule in solution, in absence of long-range interactions), the dimension of a macromolecule, related to the intrinsic viscosity, is clearly influenced by the interactions between the polymer chains and between the polymer and the solvent [28]. These interactions can be short-range ones, between adjacent atoms, and long-range ones, which are attractive or repulsive forces between segments of a polymer chain that are far from each other (although in some cases they can be close, due to the excluded volume effect), and between polymer segments and solvent molecules.

This way, the intrinsic viscosity rises up, whenever the molecules of the polymer are more opened, as a consequence of the polymer–solvent interactions. On the other hand, the intrinsic viscosity decreases, whenever the interactions between the segments of the polymer are stronger than the interactions between the polymer segment and the solvent and, as a result, the polymer molecules are more closed.

As it can be observed in Table 5, there is no clear dependence of intrinsic viscosity with temperature. In the case of EVA 1, there is a tendency of the intrinsic viscosity to decrease with temperature; in the case of EVA 2, the tendency is the opposite. This means that when the vinyl acetate content is high, the interactions between the segments of the polymer and the solvent tend to diminish with temperature, and when the vinyl acetate content is lower, the interactions between the segment of the polymer and the solvent tend to become stronger as temperature increases.

Regarding the overall dependence with vinyl acetate content, it can be observed that the intrinsic viscosity values are lower in the EVA elastomer having less vinyl acetate percentage. This means that the vinyl acetate segment is the main responsible of the interactions between the polymer and the solvents; so, when the amount of vinyl acetate segments decreases, the interactions between polymer and solvent become less important than the interaction between the segments of the polymer. As a consequence the polymer coils itself and the intrinsic viscosity goes down.

Regarding the Flory–Huggins parameter, Table 6 shows their calculated values following the Stockmayer–Fixman relationship [23], and Table 7 shows their calculated values according to Berry relationship [24]. Both Tables also show the values of the enthalpic [calculated with Eqs. 4 and 5] and entropic (calculated as the difference between the overall value and the enthalpic term) contributions to the FH parameter.

Table 6 Flory–Huggins parameter results, following the Stockmayer and Fixman method [23]
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Table 7 Flory–Huggins parameter results, following the Berry method [24]
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As it can be observed, the values of the FH interaction parameters calculated with both methods are very close, being the values lower in the case of the EVA material with more vinyl acetate content (the one with more polar character), although all the solvents are clearly compatible with both elastomers.

It can also be noticed that except for THF, the main contribution to the FH parameter is the entropic ones. This seems to be related to the low polar character of the solvents. In the case of THF, its high polarity makes the enthalpic contribution to be more important.

Concerning the temperature dependence in the literature has been described that the Flory–Huggins parameter can decrease [29] but also increase [19] with temperature. The Flory–Huggins parameter is obtained by adding two components, one entropic and one enthalpic [9]. The first one, mainly due to the free volume of the solvent, is expected to increase with temperature; the free volume of the solvent also increases with temperature, so this compound will be less accessible to polymer lattice. The second one is expected to decrease with temperature, due to the decreasing of intermolecular forces between polymer and solvent. Therefore, the overall dependence of χ with temperature will depend on the prevailing effect. In this case, the temperature dependence is almost negligible.

Finally, the three contributions to the Hansen solubility parameter of both elastomers were calculated according to Eqs. (2), (8), (9) and (10), from the values of the Hansen solubility parameter of the studied solvents (Table 8). The obtained results, shown in Table 9, indicate that the different vinyl acetate content, although having quite influence in the value of the FH parameter, is not a factor that affects the HSP. On the other hand, although the main contribution to the HSP in both cases is the dispersion one, the hydrogen bonding contribution is also relatively important, due to the relatively polar character of the vinyl acetate monomer.

Table 8 Non-polar, polar and hydrogen bonding contributions, and overall solubility parameter of the studied solvents [12]
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Table 9 Non-polar, polar and hydrogen bonding contributions, and overall solubility parameter of the studied copolymers, at T = 30 °C
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Conclusion

The theta solvent composition, the Hansen solubility parameter and the Flory–Huggins parameter have been obtained for two EVA copolymers with different vinyl acetate content, by means of turbidimetric and intrinsic viscosity measurements.

The obtained values of the theta solvent composition indicate that the more vinyl acetate content the polymer has, the more polar the theta solvent is, as a consequence of the increasing polar character. The vinyl acetate content is a variable which has a good influence over the Flory–Huggins parameter (the higher the vinyl acetate content, the lower the Flory–Huggins parameter), although its influence over the Hansen solubility parameter is almost unnoticeable.