1500
Ind. Eng. Chem. Res. 1991,30, 1500-1506
MATERIALS AND INTERFACES Chemical-Physical Model for the Interfacial Thermodynamics of Liquid Metal Solutions Wayne J. Howellt and Charles A. Eckert* Department of Chemical Engineering, University of Illinois, Urbana, Illinois
61801
The surface thermodynamics using chemical-physical theory (STCPT) model is developed to represent the surface tensions of compound-forming liquid metal alloys. T h e S T C P T model uses an approach to modeling the interfacial thermodynamics wherein the surface is treated not as a separate phase, but rather the system is viewed as consisting of a bulk phase that is acted upon by a surface force. In this treatment, this surface force is analogous to other forces that may act upon the system, e.g., electrical and magnetic forces. The S T C P T model provides an excellent model for the surface tension isotherms for a wide range of binary systems that exhibit various types of behavior. Compared with previous models, the STCPT model provides more accurate representations of the surface tension isotherms for compound-forming systems with fewer model parameters. Possible extensions are also discussed.
Introduction The understanding and exploitation of the interfacial thermodynamic properties of metal alloys is critical to a number of technologies. Important applications include packaging of microelectronic devices (Leonida, 1981; Jackson et al., 1986), production of composite materials (Kingery,1953; Humenik and Kingery, 1954; Humenik and Parikh, 1956; Thomas et al., 1963; Kelly, 1964; Kelly and Davies, 1965a,b; Manning and Stoops, 1968; Humenik, 1973; Vinson and Chou, 1975; Mohn, 1987),fabrication of equipment requiring ceramic-to-metal bonding (Williams and Nielsen, 1959; Morgan et al., 1982; MacKenzie, 1984), and welding, soldering, and brazing (Lancaster, 1980). The efficient development of these and other technologies requires an ability to model the interfacial thermodynamics of liquid metal solutions that are present in the processing steps of the above technologies. In addition, a liquid-phase processing route provides a easy opportunity to make additions of other components to tailor the properties and morphology of the final product. Accurate models provide useful insight and direction into the selection of additives. There is extensive evidence of intermetallic compound formation in liquid metal solution (Wilson, 1965; Steeb and Entress, 1966; Steeb and Hezel, 1966; Bhatia and Thornton, 1970; Bhatia and Hargrove, 1974; Bhatia et al., 1974; Shimoji, 1978; Jordan, 1979; Predel, 1979; Chieux and Ruppersberg, 1980; Stoicos, 1980; Sommer, 1982b). The formation of intermetallic compounds in the liquid phase often times has pronounced effects on the properties of the alloys [e.g.,activity coefficient, enthalpy, and structure factor (Hultgren et al., 1973; Bhatia and Hargrove, 1974; Bhatia et al., 1974; Bhatia and Ratti, 1977)]. Evidence supporting the formation of intermetallic compounds in *Author to whom correspondence should be addread. Preaent address: School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332. Present address: International Business Machines Corporation, Packaging Laboratory, Hopewell Junction, NY 12533.
liquid metal solutions includes the work of Wilson (1965), Steeb and Entress (1966), Steeb and Hezel(1966), Bhatia and Thornton (1970),Bhatia and Hargrove (1974),Bhatia et al. (1974),Shimoji (1978),Jordan (1979), Predel (1979), Chieux and Ruppersberg (19801, Stoicos (19801, and $ommer (1982b). The surface properties are also affected by the formation of intermetallic compounds (Howell and Eckert, 1987). In order to model quantitatively the surface thermodynamics of these systems, it is important to account explicitly for the formation of intermetallic compounds in the liquid state. Chemical theory affords a route for doing this. Chemical theory has been used successfully by a number of investigators to model a wide range of thermodynamic properties of liquid metal solutions that exhibit evidence of compound formation (Hildebrand and Eastman, 1915; Jordan, 1970, 1976, 1979; Bhatia and Hargrove, 1974; Bhatia et al., 1974; Bhatia and Ratti, 1977; Predel and Oehme, 1976; Eckert et al., 1982; Sommer, 1982a,b, 1983; Alger and Eckert, l983,1986a-; Howell and Eckert, 1987, 1988; Howell et al., 1988). In this paper we propose the surface thermodynamics using chemical-physical theory (STCPT) model to represent the surface tension isotherms for compound-forming liquid alloy systems. The STCPT model uses an approach to modeling the interfacial thermodynamics that is different from most previous investigations, in that the surface is not treated as a separate phase, but rather as a force acting on and affecting the bulk phase. The model resulting from this new approach is generally applicable and quite versatile.
Property Definition Most previous investigators (Taylor, 1956; Hoar and Melford, 1957; Kaufman, 1967; Laty et al., 1976; Bhatia and March, 1978; Okajima and Sakao, 1982; Angal and Roy, 1982; Howell and Eckert, 1987) have based their models of the surface thermodynamics of liquid alloys on the Gibbs or Guggenheim approaches (Defay and Prigogine, 1951; Guggenheim, 1952). These two approaches are
0888-5885/91/2630-1500$02.50/00 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No.7, 1991 1501 similar in that they view the system as consisting of a bulk phase in equilibrium with a surface phase. (The surface phase in the Gibbs model is infinitesimally thin, while in the Guggenheim model the surface phase has some small but finite thickness. The Guggenheim model reduces to the Gibbs model in the limit.) The approach taken in this paper is different in that the system is viewed as consisting only of a bulk phase which is acted upon by a surface force. A separate surface phase is not a part of this model and is, in fact, unnecessary. The STCPT model development begins with a general expression for the change in the Gibbs energy for an open system dG=-SdT+ VdP+&dni+dw (1) The dw term, which can be represented by the general form, u du, represents the work of forces, in addition of PV work, acting on the system. These additional forces include electrical work (dw = -E dQ, where E is the potential difference and Q is the coulombs of electricity), magnetic work (dw = H dMV, where H is the magnetic field, M the magnetization, and V the volume) and, in our case, surface work for which dw is represented by u dA , where u is the interfacial energy, or surface tension, and A is the interfacial area. In this way, many of the concepts usually attributed to a separate surface phase can be captured as bulk-phase properties. All the usual partial derivatives of this equation can be taken, for example (aG/anP,,,,, = -s (2)
(aG/aP)T,,i,, =
v
(3)
and for the case where a surface force is acting on the system caG/aA)Tp,,,i = (4) Continuing with the case in which a surface force is acting on the system, one can develop the expressions for the surface properties analogous to those for systems without surface forces. The interfacial energy, or surface tension, of formation for a compound comprised of atomic elements A and B having the stoichiometric coefficients ai and bi, respectively, is defined by the expression of = ui - aidA - biaB (5) where and UB are the pure component surface tensions for species A and B and ui is the hypothetical surface tension of compound i, hypothetical in the sense that for liquid metal systems the intermetallic compounds are in dynamic equilibrium with the atomic elements that comprise the compounds and, therefore, cannot be isolated so that their surface tensions can be measured experimentally. Eckert and Prausnitz (1964) defined the excess surface tension for a binary system of A and B as u
- XAUA - XBUB
(6) where XA and XB are the atom fractions of components A and B and u is the mixture surface tension at the composition xA. Lastly, the derivative of the surface tension with respect to the number of moles of component i, which we call the partial surface tension, is ai = (du/anihpJj (7) This expression is similar to the definition of a partial molar property for extensive thermodynamic properties. Chemical Theory Chemical theory postulates the existence of intermetallic compounds in dynamic equilibrium with the atomic ele-
ments (or monomers) that comprise the compounds in a reaction of the form aiA biB = Aa,Bbi (8)
+
for which an equilibrium constant for compound i, Ki, can be defined. The solution (called the true solution) is then treated as a multicomponent mixture of the monomers and compounds. The activities of the monomers are related to the experimentally observed activities through an expression derived by Prigogine and Defay (1954) xiyi = ziai (9) where xi is the atom fraction of monomer i in the experimentally observed (or apparent) solution and y i is the experimentally observed activity coefficient. In the true solution zi is the true mole fraction and ai is the true activity coefficient of species i. Because the activity is the product of a composition and an activity coefficient, the equilibrium constant for compound i, Ki, can be expressed as the product of a true composition equilibrium constant, Kzi,and a true activity coefficient equilibrium constant, Kui.
This allows one to separate the excess Gibbs free energy into two contributions: chemical and physical. The chemical interactions are contained in Kziand the physical interactions are in Kui.
Model Development Howell et al. (1988) developed the linear chemicalphysical theory (LCPT) model to represent the Gibbs energy and enthalpy of a wide range of systems with a minimum of model parameters. In the LCPT model the physical interactions are a function of the apparent atom fractions and are modeled using a single-parameter Scatchard-type expression (Scatchard, 1931) gsys= A@A@B(XAUA
+ WJB)
(11) where A is the physical-interaction model parameter, ai is the volume fraction of component i based on the apparent solution composition, and ui is the molar volume of component i. Expressions for a1and a2resulting from (11)are R T In al = Au1@22 (12)
R T In a2 = Au2aI2 (13) The chemical interactions are modeled using a group contribution approach in which the Gibbs energies of formation for the postulated compounds are linearly related to the number of atoms comprising the compounds gf,chem = -RT In Ki = (ni - 1)G,
(14) where Gp is the single chemical-interaction model parameter, ni is the number of atoms comprising compound i , and g&hem is the Gibbs energy of formation for compound i. Only a single chemical-interactionparmeter is required, regardless of the number of compounds postulated to exist in the true solution. Thus, there are only two model parameters for each binary, G, and A, and, as with other chemical theory models, these two parameters are determined by fitting experimental activity coefficient data. The enthalpy can also be represented using the LCPT model by taking the temperature derivatives of (11)and (14) hSp = B~A@B(xAuA+ X& (15)
1502 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991
= (ni - 1)Hg
(16) where B and H,are the physical and chemical parameters used to represent the enthalpy. These two parameters are determined by fitting either experimental partial molar enthalpy data or integral molar enthalpy of mixing data. Using the definitions of interfacial energy properties given above, we can develop expressions similar to (11)and (14) for the area derivative of the Gibbs energy, or the interfacial energy. The area derivative of the apparent solution activity coefficient is given by the expression (17) (a In Y ~ / ~ A ) ~=?ai/RT , hfdem
Introducing chemical theory, the partial surface tension can be divided into two contributions: a chemical contribution and physical contribution. ai/RT = (a In + (8 In cui/aA),,, (18)
ai = al; + ap
(19) where the superscripts in (19) denote the chemical and physical contributions, respectively, to the partial surface tension. In this model we chose to represent these contributions using expressions analogous to those used to represent the Gibbs energy and enthalpy in the LCPT model usys= C@A@B(XAUA
+ xBuB)
(20)
where C is the single physical-interaction parameter and S, is the group chemical-interaction parameter. These two parameters are determined from a fit of the excess surface tension isotherm using the methodology developed by Alger and Eckert (1986) for the determination of the enthalpy and heat capacity with the SCPT model. Analogous expressions can be developed for the surface tension. The chemical contribution to the partial surface tension is given by
I
[ ii ] [
&ti
Dz.
&ipiZi
Zbiii~i]
-C
Z.ZiAOi*
= L ~ ~ z ~ A o :(22) ]
where the summations are from i = 1 to N and Bi = (ai + bi)rl- airesults from a mass balance on component 1. The Aui* term results from the area derivative of the true composition equilibrium constant Aui* = RT(d In K J C ~ A ) , ~ , (23) Since the true composition equilibrium constant is the ratio of the equilibrium constant, Ki, and the true activity coefficient equilibrium constant, Kai, then Aui* can be divided into chemical and physical contributions AuF = RT(d In K i / d A ) , p , = uf,,he,,, = (ni- 1)S,
Au? = RT(d In Kai/i3A)Tp,
(24) (25)
where the single chemical-interaction parameter is introduced in (24) and A< can be expressed as a function of the physical-interaction parameter, C, as follows. The normalization condition for the true activity coefficients for the compounds is that a t the stoichiometric composition of the compound the true activity coefficient is equal to unity (akBbi = 1,at x,*i = ai/(ai + b J ) . Substituting this into the expression for the area derivative of the true activity coefficient equilibrium constant yields where the expression is evaluated at the stoichiometric
Table I. STCPT Model Parameters
S,
c2,
mJ/(mz m3) -182.3
system compd T, K 1373 AI-Cu AlCu
mJ/m2 129
AICU~ A12C~3 Mg-Pb MgPb 943
-422
37.44
1073
53
-21.07
Eremenko et al. (1977)
data source Eremenkoet al. (1961) Korol’kov and Igumnova (1961)
Mg-Sn
$kb
Mg-Zn
MgzSn MgZn
973
-1599
114.3
Pelzel and Sauerwald (1941)
Au-Sn
MgZnz AuSn
773
-1547
188.3
Mg-Al
MgzAl3
Zn-Sb In-Sn
973 1073 Zn,Sb3 908 573 InSn
-562 -844 -231 -151
-44.13 -29.29 -47.85 -2.80
Kaufman and Whalen (1965) Pelzel (1949)
In-Sb
In,Sb
Pb-Bi
PbBi
623 823 973 673
-156 -47 -29 316
5.70 -6.90 -1.44 -18.26
Pb-Sb
PbSb
908
2861
-33.61
Matuyama (1927) Howell et al. (1991) Lazarev (1964) Kazakova et al. (1984) Matuyama (1927)
composition of the compound. The number and stoichiometry of the postulated compounds are determined by using the LCPT model of the Gibbs energy (Eckert et al., 1982; Howell et al., 1988; Howell and Eckert, 1988). No new or additional compounds are introduced in the surface tension or enthalpy representations. In addition, the only parameters varied in representing the surface tension isotherm are those associated with the STCPT model: C and S . The Gibbs energy and enthalpy parameters from the kCPT model ( A and G,, B and H,,respectively) are not adjusted in the interfacial energy representation. Once the two model parameters are determined, the surface tension or excess surface tension isotherm can be constructed by reversing the procedure presented in this section.
Application of STCPT Model The STCPT model was used to determine the surface tension isotherms for 11liquid alloy systems. The behavior of the systems ranges from those in which compound formation is the dominant effect to systems in which both chemical and physical interactions are important. Table I lists the STCPT model parameters for the systems investigated. Figures 1-7 are plots of the surface tension isotherms and excess surface tension isotherms for six representative systems. The excess surface tension is helpful in that scatter in the experimental data is amplified and, more importantly, the effect of compound formation on the interfacial properties is shown more clearly. The error bar on the excess surface tension plots represents an estimate of the experimental error for surface tension measurements for liquid metal systems (&2%). The STCPT model representation for the Pb-Bi system is shown in Figure 1. This system forms only a weak compound with little effect on the isotherm; therefore it is nearly linear. Since the excess surface tensions for weak-compound-forming systems are usually small, the error bar appears quite large relative to the excess surface tension.
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1503 T460
I
I
I
I
I
I
r
c
I
-
KAZAKOV, ET AL. (1984)
>
:::1 700
1
0
-50O
-1 6 0.0
0.2
0.4
0.6
0.8
ATOM FRACTION Pb
Figure 1. STCPT model representation for Pb-Bi system at 673 K. Surface tension and excess surface tension isotherms. T570
~
0.0
1.0
I
I
A
h
0.2
0.4 0.6 0.8 ATOM FRACTION AI
1.0
Figure 3. STCPT model representation for Al-Cu system at 1373 K. Surface tension and excess surface tension isotherms. ~
~
~
~
~
HOWELL, ET AL. (1989) v
2
P
3490
-
600
2
500
E
? 470 -
J
$ 400
r
1
lo
0.0
700
E 0
K
- "V
2 W
0
800
r
-250'
0.2
0.4
0.6
0.8 ATOM FRACTION In
1.0
Figure 2. STCPT model representation for In-Sn system at 573 K.
0.0
1
"
0.2
'
I
"
"
0.4 0.6 0.8 ATOM FRACTION Mg
"
1.0
Surface tension and excess surface tension isotherms.
Figure 4. STCPT model representation for Mg-A1 system at two temperatures. Surface tension and excess surface tension isotherms.
A weak-compound-formingsystem that exhibits unusual behavior is the In-Sn system in which a minimum is found in the surface tension isotherm (Figure 2). Since the surface tensions of the two pure components are approximately equal, the compound, InSn, has a more dramatic effect on the isotherm than if the two pure component surface tensions were different, as in the Pb-Bi system. The effect of the somewhat stronger compounds in the Al-Cu system as seen in Figure 3 is interesting in that the excess surface tension is much greater and the compounds produce a perturbation (or change in curvature) near the stoichiometric composition of the dominant compounds AICua and A12Cu3(0.2 < xu < 0.4). Surface tension data are available at two temperatures (973 and 1073 K)and are far enough apart to distinguish the data two isotherms clearly on a plot for the Mg-A1 system (Figure 4). This system has a moderate-strength compound that causes the surface tension isotherm to have a weak minimum at about x = 0.5. This figure also demonstrates clearly the b e n s t of plotting the excess surface tension in that the minimum is found at the stoichiometric composition of the compound (x% = 0.4). Often the surface tension isotherm will exhibit extrema
or perturbations in a composition region shifted slightly from the stoichiometric composition of the compounds, while the extrema or perturbations in the excess surface tension isotherm will be centered at the stoichiometric composition of the compounds. A more dramatic example of surface tension minima is seen in the strong-compound-forming system Mg-Pb (Figure 5 ) . A pronounced minimum exists a t the stoichiometric composition of the compound. The Mg-containing systems are some of the strongest compoundforming systems, and abrupt variations in the thermodynamic properties, including the surface tension, near the stoichiometriccomposition of the compounds are exhibited for many of these systems. The Mg-Zn system (Figure 6) is another example. Au-Sn, though only a moderate-strength compoundforming systems, also exhibits rather unusual behavior (Figure 7). First note the dramatic decrease in the surface tension (approximately 40%) with the addition of Sn (up to about 30%). Couple this with the liquidus temperature decreasing from that of pure Au (1064 O C ) to the eutectic temperature of 278 O C at xAu= 0.71, and it is easy to see why this system makes an excellent choice for soldering
1504 Ind. Eng. Chem. Res., Vol. 30,No. 7, 1991
T 550
2500 v
z 450
P
400
Y
I-
w
350
0
2 300 K 3 VI
250
1
-500' 0.0
"
0.2
"
'
I
'
I
0.4 0.6 0.8 ATOM FRACTION Au
" 1.0
Figure 7. STCPT model representation for Au-Sn system at 773 K. Surface tension and excess surface tension isotherms.
:
(
5700
50
4
PEUEL & SAUERWALO (1941)
r
1
-6""
0.0
0.2
0.4 0.6 0.8 ATOM FRACTION M.3
1.0
Figure 6. STCPT model representation for Mg-Zn system at 973 K. Surface tension and excess surface tension isotherms.
applications. Secondly, a local maximum occurs in the composition range 0.15 C XA" < 0.50.
Discussion and Conclusions The STCPT model achieves excellent quantitative representation of the surface tension isotherms for a wide range of systems with a minimum of model parameters. The approach taken in the development of the model is somewhat different than previous investigators in that the surface is not treated as a separate phase, but rather as a force acting on and affecting the bulk properties of the system. A linear chemical theory approach, successfully applied to the representation of other thermodynamic properties of liquid alloy systems, has been used to reduce the number of chemical parameters to one regardless of the number of postulated compounds. Compared with previous models, most of which are based upon either a Gibbs or a Guggenheim approach, for the surface thermodynamics of liquid alloy systems (Taylor, 1956; Hoar and Melford, 1957; Kaufman, 1967; Laty et al., 1976; Bhatia and March, 1978; Okajima and Sakao, 1982; Angal and Roy, 1982; Howell and Eckert, 1987) the STCPT model provides more accurate repre-
sentations of the surface tension isotherms for a wider range of compound-forming systems with fewer model parameters. The primary reason is that previous models, with the exception of the STICT model developed by Howell and Eckert (1987), did not account explicitly for intermetallic compound formation. When compound formation is included (usually through the use of chemical theory), as in the STCPT model, the abrupt changes in the mixture surface tension near the stoichiometric composition of the intermetallic compounds, exhibited by may systems, can be represented accurately. In addition, treating the system as a bulk phase being acted upon by a surface force allows one to incorporate many of the concepts usually attributed to interfacial phases into the bulk-phase representation. This in turn provides a route by which many of the methods developed for representing the bulk-phase properties of compound-forming liquid metal systems can be readily extended to the representation of interfacial properties. Therefore, the strength of this model rests upon the consideration of chemical theory coupled with the concept of a surface force acting on a bulk system. The reduced number of model parameters required in the STCPT model is a direct result of the exploitation of the relationship between the surface tensions of formation for the compounds and the number of atoms comprising the compound. This group contribution approach, first applied to liquid metal systems in the LCPT model, is the first application to surface thermodynamics known to the authors. This approach allows for the incorporation of physical interactions while maintaining a low number of model parameters, thus increasing the number of systems to which the model is applicable. The STCPT model is extremely versatile, representing quantitatively systems in which chemical interactions dominate and those in which both chemical and physical forces are important. In addition, it represents a wide range of surface tension isotherm behavior, from nearly linear to systems that exhibit extrema. Possible extensions to this model include the representation of multicomponent systems. The equations necessary to model the Gibbs energy and enthalpy using the LCPT model have been developed (Howell and Eckert, 1989), and it would be an easy task to extend this work to the representation of surface tension. This model would be important for the screening of systems for specific ap-
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1505 plications in which there is only a limited experimental database. Another logical extension is to apply the approach outlined here to modeling the thermodynamic properties of a system that was being acted upon by forces other than surface forces (e.g., magnetic and electric). The equations would again be quite easy to develop. In total, we feel that the STCPT is an acceptable approximation, given the precision of the available data, when representing the interfacial energy of liquid metal systems. Acknowledgment We gratefully acknowledge financial support from Eastman Kodak Company and Standard Oil (Indiana). Nomenclature A = LCPT model Gibbs energy physical-interaction parameter A = area A = atomic element designation a = stoichiometric coefficient B = LCPT model enthalpy physical-interaction parameter B = atomic element designation b = stoichiometric coefficient C = STCPT model physical-interaction parameter E = potential difference G = Gibbs energy G, = LCPT model Gibbs energy chemical-interaction parameter H = magnetic field Hg= LCPT model enthalpy chemical-interaction parameter K = equilibrium constant M = magnetization n = number of moles P = pressure Q = coulombs of electricity R = gas constant S = entropy S = STCPT model chemical-interaction parameter = temperature u = generalized displacement V = volume u = molar volume, generalized force x = atom fraction z = true mole fraction Greek Letters a = true solution activity coefficient = constant defined in text y = apparent solution activity coefficient p = chemical potential Q = volume fraction 0 = interfacial energy or surface tension Subscripts chem = chemical contribution i = index for species phys = physical contribution Superscripts C = chemical contribution E = excess quantity f = formation property P = physical contribution * = chemical and physical contributions combined 3 = stoichiometric composition of compound
fl
Literature Cited Alger, M. M.; Eckert, C. A. Thermodynamics of Highly Solvated Liquid Metal Solutions. Ind. Eng. Chem. Fundam. 1989,22,249. Alger, M. M.; Eckert, C. A. Simplified Chemical-PhyaicalTheory for Temperature Dependence of the Liquid Metal Solutions-I.
Gibbs Energy. Chem. Eng. Sci. 1986a, 41,2829. Alger, M. M.; Eckert, C. A. Simplified Chemical-PhysicalTheory for Liquid Metal Solutions-11. Composition Dependence of the Gibbs Energy. Chem. Eng. Sci. 1986b, 41,2839. Alger, M. M.; Eckert, C. A. Thermodynamics of Highly Solvated Ternary Liquid Metal Solutions. Znd. Eng. Chem. Fundam. 1986c, 25, 416. Angal, R. D.; Roy, D. L. The Surface Tension of Binary Liquid Alloys. 2.Metallkde. 1982, 73, 428. Bhatia, A. B.; Thornton, D. E. Structural Aspects of the Electrical Resistivity of Binary Alloys. Phys. Reo. B. 1970, B4, 3004. Bhatia, A. B.; Hargrove, W. H. Concentration Fluctuations and Thermodynamic Properties of Some Compound-Forming Binary Molten Systems. Phys. Reu. 1974,lO (8),3186. Bhatia, A. B.; Rhatti, V. K. Number-Concentration Structure Factors and Their Long Wavelength Limit in Multicomponent Fluid Mixtures. Phys. Chem. Liq. 1977, 6, 201. Bhatia, A. B.; March, N. H. Surface Tension, Compressibility, and Surface Segregation. J. Chem. Phys. 1978, 68, 4651. Bhatia, A. B.; Hargrove, W. H.; Thornton, D. E. Concentration Fluctuations and Partial Structure Factors of Compound-Forming Binary Molten Alloys. Phys. Reo. 1974, 9 (2), 435. Chieux, P.; Ruppersberg, H. The Observation of Chemical ShortRange Order in Liquid and Amorphous Metallic Systems by Diffraction Methods. J. Phys. Colloq., C8, 1980, 41, 145. Defay, R.; Prigogine, I. Surface Tension and Adsorption; translated by D. H. Everett; Wiley: New York, 1951. Eckert, C. A.; Prausnitz, J. M. Statistical Surface Thermodynamics for Simple Liquid Mixtures. AZChE J. 1964, 10, 667. Eckert, C. A.; Smith, J. S.; Irwin, R. B.; Cox, K. R. A Chemical Theory for the Thermodynamics of Highly-Solvated Metal Mixtures. AZChE J. 1982,28, 325. Eremenko, V. N.; Nishenko, V. I.; Naiditsch, Yu. V. The Surface Tension of Melts of Some Intermetallic Compounds. Zzu. Akad. Nauk SSSR, Met. Topl. 1961, No. 3, 150. Eremenko, V. N.; Ivashchenko, Yu. N.; Khilya, G. P. Density and Free Surface Energy of Magnesium-Tin Melts. Izu. Akad. Nauk SSSR, Met. 1977, No. 3, 38. Guggenheim, E. A. Mixtures; Oxford: London, 1952; Chapter 9. Hildebrand, J. H.; Eastman, E. D. The Vapor Pressure of Thallium Amalgams. J. Am. Chem. SOC.1915,37, 2452. Hoar, T. P.; Melford, D. A. The Surface Tension of Binary Liquid Mixtures: Lead + Tin and Lead + Indium Alloys. Trans. Faraday SOC.1957,53, 315. Howell, W. J.; Eckert, C. A. STICT Model for Surface Thermodynamics of Liquid Metal Solutions. AZChE J. 1987,33, 1612. Howell, W. J.; Eckert, C. A. Solution Thermodynamics for Liquid Metals. ACC.Chem. Res. 1988,21, 168. Howell, W. J.; Eckert, C. A. A Linear Chemical-Physical Theory Model for Ternary Liquid Metal Solutions. 2.Metallkde. 1990, 81, 335. Howell, W. J.; Lira, C. T.; Eckert, C. A. A Linear Chemical-Physical Theory Model for Liquid Metal Solution Thermodynamics. AZChE J. 1988,34, 1477. Howell, W. J.; Katsikopoulos, P.; Eckert, C. A. Surface Tensions for Liquid Metal Systems: Bismuth-Tin and Indium-Tin. Submitted for publication in High Temp. Sci. 1991. Hultgren, R.; Desai, P. D.; Hawkins, D. T.; Gleiser, M.; Kelly, K. K. Selected Values of the Thermodynamic Properties of Binary Alloys; American Society of Metals: Metals Park, OH, 1973. Humenik, M.Ceramic-Metal Composites and Their Uses. The Science of Materials Used in Adoanced Technologies; Parker, E. R., Colombo, U., Eds.; Wiley: New York, 1973. Humenik, M.; Kingery, W. D. Metal-Ceramic Interactions 111.: Surface Tension and Wettability of Metal-Ceramic Systems. J. Am. Ceram. SOC.1954,37, 18. Humenik, M.; Parikh, N. M. Cermets. I.: Fundamental Concepts Related to Microstructure and Physical Properties of Cermet Systems. J. Am. Ceram. SOC.1956,39,60. Jackson, K. A.; Pohanka, R. C.; Uhlmann, D. R.; Ulrich, Electronic Packaging Materials Science II; Materials Research Society Symposia Proceedings 72; Elsevier: New York, 1986. Jordan, A. S. A Theory of Regular Associated Solutions Applied to the Liauidus Curves on the Zn-Te and Cd-Te Svstems. Metall. Trans.'1970, 1, 239. Jordan, A. S. Calculation of Phase Equilibria in the Ga-Bi and GaP-Bi Systems Based on the Theory of Regular Associated Solutions. -Metall. Trans. B 1976, 7B,-191. Jordan, A. S. A Review of Semiconductor Phase Diagram Calculations Employing the Regular Associated Solution (RAS)Model.
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Prediction of Solute Partition Coefficients between Polyolefins and Alcohols Using the Regular Solution Theory and Group Contribution Methods Albert L. Baner and Otto 0.Piringer* Fraunhofer Institute for Food Technology and Packaging, Munich, Federal Republic of Germany
The regular solution theory using group contribution solubility parameter estimation methods was applied to the estimation of partition coefficients of solutes between polyolefii polymers and alcohol solvents. Quantitative prediction was improved by using only the Hansen dispersive type solubility parameters and adding an empirical correction term to account for polar type interactions between the solute and the solvent and polymer phases. The method fails to fully account for multiple functional groups and stearic hindrances. The correction term is a function of the solute’s functional groups, the solute molecular weight, and the solvent and polymer phases. The group contribution methods of Hoy and of Van Krevelen and Hoftyzer gave equivalent results.
Introduction The prediction of solute partition coefficients between polymers and liquids is important in a number of applied fields such as protective clothing (Mansdorf et al., 1988), biomedical studies (Dunn et al., 1986), chromatography (Barton, 1983),chemical separations (Lee et al., 1989) and,
of major interest for this study, packaging (Hotchkiss, 1988). Solubility coefficients are important in package design and food shelf-life prediction because they are used in modeling the migration of substances from the packaging into the food and from the food into the package (Vom Bruck et al., 1986; Reid et al., 1980; Chatwin and
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