water interfaces

Journal of Colloid and Interface Science 258 (2003) 97–101 www.elsevier.com/locate/jcis

Estimation of polymer/water interfacial tensions: hydrophobic homopolymer/water interfaces Yan Dong and Donald C. Sundberg ∗ Polymer Research Group, Chemical Engineering Department, University of New Hampshire, Durham, NH 03824, USA Received 19 July 2001; accepted 4 October 2002

Abstract In this study, quantitative models for monomer/water and the corresponding polymer/water interfaces were developed. The monomer/water interface was modeled within the framework proposed by Fu and Li (Chem. Eng. Sci. 44 (1989) 1519) for organic liquid/water interfaces. We took a similar approach with that which others used to simulate micelle formation of surfactants and block polymers in developing the model for the polymer/water interface. In this model, the Gibbs free energy of interfacial formation was decomposed into two components: transfer free energy of a polymer segment to the interface and mixing free energy within the interface. Interfacial tensions were then estimated using concentration gradient theory. The ratio of the number of total nearest neighbors of a site in a lattice within the interface to that located in each of the adjacent planes was found to be the important correlating factor. It was shown that this ratio for monomer/water interfaces and that for polymer/water interfaces are linearly related. The polymer in the interface is likely to be in the form of short polymer segments (2–4 monomer units) rather than dangling chain ends. The estimation of polymer/water interfacial tension from the models established in this work is in reasonable agreement with experimental observation.  2003 Elsevier Science (USA). All rights reserved. Keywords: Polymer; Monomer; Water; Interfacial; Tensions; Lattice; Prediction; Free energy

1. Introduction Determination of polymer/water interfacial tension is of importance for numerous fields [1,2]. In particular, the information is essential in predicting and ultimately controlling morphology of composite latex particles [3]. Generally speaking, the interfacial tension (γ ) of an immiscible binary system, for a planar interface, can be expressed as [4,5] γ = (Gmix − Gh )/a,

(1)

where Gmix and Gh are, respectively, the Gibbs free energy of the inhomogeneous system with interface and that of the system without the interface (but with the same composition as the inhomogeneous system) and a is the interfacial area. To estimate interfacial tension as defined in Eq. (1), more specific models have to be developed. In estimating liquid/liquid interfacial tension, Shain and Prausnitz [6] simply identified the interfacial zone as a homogeneous layer whose * Corresponding author.

E-mail address: [email protected] (D.C. Sundberg).

composition is intermediate between those of the equilibrium phases. Following their approach, Fu et al. [7] successfully correlated interfacial tensions of aqueous binary systems to mutual solubility. Another approach assumes that composition in the interface changes continuously between the high- and low-composition phases. Cahn–Hilliard gradient theory is of this type [8], which takes account of the concentration–gradient contribution to free energy, in addition to the bulk concentration itself. The theory was applied to organic/water interfaces [9] and more recent research showed that the theory is able to predict organic/water interfacial tension with the maximum deviation less than 1 mN/m over the range ∼2–50 mN/m [1]. Compared to the low molecular weight organic/water interfaces, those for polymer/water are much less understood [2,9]. To our knowledge, there has been little effort to predict polymer/water interfacial tension. However, there exists a great deal of polymer/water interfaces in certain biological, environmental, and chemical engineering systems. It would be significant in understanding and designing the systems if we are able to estimate these interfacial tensions.

0021-9797/03/$ – see front matter  2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(02)00060-7

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Y. Dong, D.C. Sundberg / Journal of Colloid and Interface Science 258 (2003) 97–101

The objective of this work was to estimate interfacial tension between polymer and water from that between the corresponding monomer and water. For simplicity, we considered only insoluble, amorphous homopolymers (see Table 2). Segments for such a polymer in the interface may be considered as a homolog of the corresponding monomer repeat units. In addition, monomer/water interfacial tension is much easier to measure experimentally. We adopted a model proposed by Pliskin and Treybal [9] and Fu and Li [1] for monomer/water interfaces and developed a model for polymer/water interfaces. Our approach to the latter is similar to that used in predicting micelle formation of surfactants and block copolymers [10].

2. Theoretical background In taking Pliskin and Treybal’s approach [9], we assume the water content in the monomer/water interface (xW ) increases gradually from the monomer-rich ( phase) to the water-rich ( phase) phases. The interface may consist of several molecular-dimensional, homogeneous planes parallel to the interface, within which the water content varies from one plane to the next (Fig. 1). Assuming that both the bulk phases and the interfacial region between them are described by a lattice model, the interfacial tension can be expressed as [1] 


xw 1/2 mix )1/2 dxW , γ = (2K /V ) (G

(2) 2.1. Monomer/water interfaces

 xw

 mix = G mix (xW ) − G mix (xW G )    xW − xW   mix (xW mix (xW − G )−G )   , xW − xW

(3)

mix is free energy of mixing for a homogeneous where G plane and varies with its composition, K ≈ mV h2 (δ1 − δ2 ) and m = (1/2)(Z − Z  )/Z, h = (V /N)1/3 , N is Avogadro’s number, h is the thickness of the molecular-dimensional planes, V is the average molar volume of water and monomer, and δ1 and δ2 are the solubility parameters

Fig. 1. Water content profile in the interface.

 and x  are, for monomer and water, respectively. xW W respectively, the water solubility (mole fraction) in the monomer- and water-rich phases, Z is the average number of water–monomer molecular contacts, i.e., the number of nearest neighbors to a site in a lattice model. Similarly, Z  is the contact number within the same plane of the site, parallel to the interface (Fig. 1); m is called the “fraction of contact number” in each of the adjacent planes, which is defined as (Z − Z  )/2Z. For a monomer, we designate m as mM . In an octahedral lattice, for example, Z = 6, Z  = 4, and mM = 1/6. The higher value of m indicates a stronger interaction between the adjacent molecular planes shown in Fig. 1. As shown later, mM is an important parameter in determining monomer/water interfacial tensions and varies significantly with different monomers. It is straightforward to extend Eqs. (2) and (3) to amorphous polymer/water interfaces. Considering that a polymer chain is a series of connected monomer repeat units (segments), we applied a lattice model to the polymer, water, and interfacial phases so that the formalism of Eqs. (2) and (3) remains unchanged. However, the parameters in the equations related to a monomer have to change to those for a polymer segment (e.g., replacing mM with mP , the fraction of contact number between polymer segments and water). It is the goal of this work to estimate the terms mix and K in or related to Eqs. (2) and (3), for such as G monomer/water and polymer/water interfaces, and then to calculate the interfacial tensions.

mix defined For immiscible monomer/water systems, G by Eq. (3) can be further written as [1]   fW xW fM xM  Gmix = RT xW ln   + xM ln   , (4) fW xW fM xM  and x  are, respectively, the mole fractions of where xW M water and monomer in the monomer-rich phase, xM and xW are the corresponding mole fraction of monomer and water in the interface, T and R are absolute temperature and the gas constant, and f is the activity coefficient of a component designated by a subscript in monomer-rich or interface phases. The activity coefficients were all taken as unity as a first approximation since the mutual solubilities of monomer and water considered in this work are low. To estimate the parameter K in Eq. (2), the fraction of contact number for monomer (mM ) has to be evaluated. Unfortunately, there seems to be no generally reliable method available. Fu and Li [1] proposed that mM can be correlated to an area parameter for an organic solute molecule, which is proportional to the surface area of the molecule and was first defined by Abrams and Prausnitz [11]. In Fu and Li’s paper [1], however, no correlation coefficient between the two was given, and the data appeared widely scattered around the fitting line. As an alternative, we found that mM was well re ). From a lated to the solubility of monomer in water (xM

Y. Dong, D.C. Sundberg / Journal of Colloid and Interface Science 258 (2003) 97–101

liquid octahedral lattice (Z = 6) model [12], we can have 1   RT ln xM (5) ≈ a ln xM , 6ω where ω is the energy gained on mixing for the creation of a monomer/water contact and a can be a constant for monomer/water systems without significant changes in specific interactions such as hydrogen bonding [13].  may be related to m Equation (5) suggests that xM M for these systems. This is consistent with the linear relationship shown in Fig. 2 for the monomers we considered, the basis for which is discussed later in this paper.

mM ∝

2.2. Polymer/water interfaces Following the treatment used in modeling the micellization of surfactants and block copolymers [10,14–16], we decompose the Gibbs free energy (Gf ) of interface formation between hydrophobic polymer and water into two components: transfer free energy (Gtr ) of polymer segments from the polymer bulk phase (amorphous) into the water and mixing free energy (Gm ) of the segments with water in the interface: Gf = Gtr + Gm .

(6)

The Gtr can be written as [10] Gtr = NW µ0W + NS µ0S ,

(7)

99

where NW and NS are the number of water molecules and number of polymer segments within the interface, respectively, µ0W is the free energy change of the system when a water molecule is added to pure water, and µ0S reflects the free energy change of the system when a single polymer segment is placed at a given position in a polymer/water interface. Note that µ0W and µ0S are commonly referred to as the standard-state chemical potential of water and a polymer segment, respectively. The free energy of mixing is written as Gmix = kT NW ln(xW ) = kT NW ln(1 − xS ),

(8)

where xW = NW /(NW + NS ) is the mole fraction of water and xS = NS /(NW + NS ) is the mole fraction of polymer segments in the interface; k is the Boltzmann constant. Note that polymer segments have no contribution to the mixing energy in interfacial formation as shown in Eq. (8). This is because the segments lack translational motions in the interfaces [16]. Combining Eqs. (6)–(8) yields Gf = NW µW + NS µS + kT NW ln(xW )

(9)

and substituting Gf in Eq. (9) for Gmix (since the two terms  ≈ 0 and x  ≈ 1, i.e., no are identical) in Eq. (3), with xW W mixing of polymer and water in either the bulk water or the bulk polymer phases, mix = RT xW ln(xW ). G

(10)

Therefore, the polymer/water interfacial tension from Eq. (2) can be rewritten as
0 γP /W = 2(KRT )

1/2

/V

xW ln(xW ) dxW 1

= 0.964(KRT )1/2 /V .

(11)

Note that the integral in Eq. (11) is a constant equal to ∼0.482 for all the polymers listed in Table 1.

3. Results and discussion Fig. 2. Relationship between the fraction of contact number of monomer  ). (mM ) and monomer solubility in water (xM

The interfacial tension data used in this article were produced in this lab via the “pendant drop” method [17]. In

Table 1 Calculated and measured monomer/water interfacial tensions (γM/W ) Monomers

Mutual solubility [23]

MA MMA BA BMA St

 xM

 xW

0.0113 0.0027 0.00028 0.00008 0.00005

0.1179 0.0702 0.0477 0.0498 0.0040

δW − δM [24] (cal/cm3 )1/2

12.3 12.1 12.4 12.9 12.0

V

A [25]

(cm3 /mol)

(×109 cm2 /mol)

50.2 59.5 77.7 85.0 66.3

7.26 9.91 11.31 13.15 8.27

  xW  xW

mix )1/2 dxW (G

mM

γM/W (mN/m)

0.637 0.970 0.889 0.8781 1.373

MA, methyl acrylate; MMA, methyl methacrylate; BA, n-butyl acrylate; PBMA, n-butyl methacrylate; St, styrene.

0.0048 0.0075 0.0132 0.0166 0.0189

Observed

Calculated

8.1 12.2 17.5 19.2 31.9

7.7 12.7 17.9 20.2 31.4

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Table 2 Calculated and measured polymer/water interfacial tensions (γP /W ) Polymers PMAc PMMA PBA PBMA PS

δW − δP [24]

Va

(cal/cm3 )1/2

(cm3 /mol)

14.8 13.9 14.2 14.5 14.3

41.5 48.8 67.0 73.2 57.4

0



1 (Gmix )

1/2 dx W

0.482 0.482 0.482 0.482 0.482

mP 0.0053 0.0397 0.0945 0.1251 0.1347

γP /W (mN/m) Observedb

Calculated

– 16 22 26 32

7 16 24 28 31

a Calculated with the density of the polymers. b Standard deviation: ∼ ±1 mN/m. c PMA, poly(methyl acrylate); PMMA, poly(methyl methacrylate); PBA, poly(n-butyl acrylate); PBMA, poly(n-butyl methacrylate); PS, polystyrene.

measuring polymer/water interfacial tension, low molecular weight polymers (Mn = 20,000–40,000) were synthesized via solution polymerization. The polymers were dissolved in 1-bromopropane (CH3 CH2 CH2 Br), and then interfacial tensions of a series of solutions with polymer concentrations up to ∼60% against water were measured. The polymer/water interfacial tension was estimated by extrapolating the polymer composition to 100% [18]. 3.1. Monomer/water interfacial tension Monomer/water interfacial tensions (γM/W ) can be calculated from Eqs. (2) and (4) if we can estimate mM since all the rest of the parameters involved in the equations are available in the literature (Table 2). We thus back-calculated mM values from experimental γM/W data (25 ◦ C) with Eqs. (2) and (4), which are also listed in Table 1. The exact same procedure was taken by Fu and Li [1] in developing the correlation discussed previously. As shown in Table 1, mM increases consistently with respective interfacial tension, from methyl acrylate (MA) to styrene (St). This suggests that mM is a dominant variable in Eq. (2) in determining monomer/water interfacial tension. On the other hand, mM did not correlate well to the calculated van der Waals surface areas (A) of the monomers, particularly for St (see columns for A and mM in Table 1). This is inconsistent with the correlation proposed by Fu and Li [1]. The high mM value for styrene may be caused by the preferential orientation of the benzene rings: they are tilted in the plane direction (Fig. 1). To make the γM/W calculation predictive, it is necessary to estimate mM independently. Using Eq. (2), Fu and Li [1] showed that the interfacial tension between organic liquid and water can be predicted very well. In the procedure used for their predictions, a parameter similar to the mM we defined here, was taken as a fitting parameter to relate to the external surface area of an organic molecule. They stated that the prediction of interfacial tension is greatly affected by the choices of that parameter. They further suggested that the impact reflects the effect of molecular orientation on interfaces. As suggested by Eq. (5), however, mM can be related to monomer solubility in water by curve fitting as shown in Fig. 2. According to Amidon et al. [13],  ), where G Ghydr/RT ≈ ln(xM hydr is the Gibbs free

energy of monomer hydration, a measure of interaction between monomer and water molecules. In concept, mM is proportional to the number of contacts between adjacent planes in the interface (Fig. 1), which includes the contacts between water–water, monomer–monomer, and monomer– water. The good relationship (R 2 = 0.99) between mM and  ) in Fig. 2) suggests that Ghydr/RT (shown as ln(xM the interaction between monomer and water may be the predominant factor in determining mM , rather than the other two like molecule interactions. This may be because of the well-mixed nature of interfaces. Finally, we estimated γM/W from Eqs. (2) and (4) in combination with the regression equation for mM shown within the plot in Fig. 2. As shown in Table 1, the maximum deviation from the experimental values is 1 mN/m (over the range 6.8–32), suggesting the model provides a reasonable basis for estimating monomer/water interfacial tensions. 3.2. Polymer/water interfacial tension Similar to the case with monomer/water interfacial tension discussed above, the fraction of contact number of polymer segments (mP ) has to be estimated before we can calculate γP /W using Eq. (11). mP values were first backcalculated (Table 2) in the same manner as in the mM calculation, using γP /W data produced in this lab. V in Eq. (11) is the average molar volume of water and a polymer segment, estimated here by the average molar volume of water and monomer units [1]. Interestingly, the values of mP vary linearly (R 2 = 0.97) with mM (Fig. 3), suggesting that mP may be estimated directly from mM via the correlation. A linear relationship has been found, for a homologous series of hydrocarbon/water solutions (
C 8 ), between the contact number and the length of hydrocarbon molecules [19]. It is also known that the contact number is proportional to the excluded volume or surface area of a polymer segment in water [20], if the chains are short so that conformation effects are negligible. Unfortunately, there are little data available regarding the length of polymer segments protruding from the polymer surface into the interface. However, the following information may still be helpful. On the sodium dodecyl sulfate micelle/water interface, there is only one CH2 group adjacent to the fully hydrated sulfate

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(mM ) or a polymer segment (mP ) is not well documented in the literature. However, we found that mM for monomers considered in this work can be estimated from the water solubility of the monomer for the systems under consideration and that mP for a polymer segment can be simply related to mM for the corresponding monomer. The estimation of both monomer/water and polymer/water interfacial tensions from the models established in this work are in good agreement with experimental observation. This will allow us to obtain reasonable estimates for previously unmeasured polymer/water interfacial tensions by utilizing easily measured monomer/water interfacial tensions. Fig. 3. Relationship between the fraction of contact number of polymer segment (mP ) and that of monomer (mM ).

Acknowledgment group [19]. Moreover, in emulsion polymerization, polymer segments with sulfated end groups, which may be often introduced from water-soluble initiators, are first formed in the water phase. The maximum length of the oligmers dissolved in water has been estimated by Maxwell et al. [21]: ∼2–4 monomer units for most of the monomers considered in this work. It should be noticed that we considered only hydrophobic polymers produced with benzoyl peroxide initiator. Therefore, the average length of the tethered segments in water must be shorter than ∼2–4 monomer units. In brief, the short length of polymer segments in polymer/water interfaces may be one of the causes for the linear relationship between mM and mP (Fig. 3). Thus, we calculated γP /W using Eq. (11) and the correlation in Fig. 3. The results are listed in Table 2 and generally are in reasonable agreement with experimental data measured in this laboratory by the pendant drop method [18]. The maximum deviation was ∼2 mN/m. Both calculated and experimental values correlated closely with mP , suggesting that mP is the predominant factor in determining hydrophobic polymer/water interfacial tension. It should be pointed out that there is much greater experimental error in γP /W than γM/W . Reported data for γP /W are sparse and vary significantly from different sources in the literature [22]. This may be because of the variety in experimental procedures, the presence of the residue end groups, impurities and high viscosity in the polymer samples, and the dynamic nature of polymer segment motions. Nevertheless, it is important to have developed a method to estimate γP /W , instead of only relying on difficult experimental measurements.

4. Conclusion We demonstrated that both polymer/water and monomer/ water interfacial tensions can be estimated from concentration gradient theory. Unlike other parameters in Eq. (2), determination of the fraction of contact number for a monomer

We are grateful for the financial support from the UNH Latex Morphology Industrial Consortium (Atofina, Mitubishi Chemicals, NeoResins, UCB Chemicals).

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