- Email: [email protected]
Dominance of density variations in determining the molecular weight dependence of surface tensions of polymer melts
Advances in Colloid and Interface Science 94 Ž2001. 33᎐38
Dominance of density variations in determining the molecular weight dependence of surface tensions of polymer melts Sanat K. Kumar U , Ronald L. Jones Department of Materials Science and Engineering, Pennsyl¨ ania State Uni¨ ersity, 316 Steidle Building, Uni¨ ersity Park, PA 16802, USA
Abstract We utilize exact thermodynamic relationships to separate the role of density variations and chain end segregation to surfaces in determining the molecular weight dependence of the surface tensions of polymer melts. By utilizing standard mean-field treatments we find that density effects dominate, except in special cases, where the chain ends are strongly attracted or repelled from the surfaces. These results stress that PVT measurements are generally sufficient to properly capture the molecular weight dependence of ␥ for normally encountered polymer melts. 䊚 2001 Elsevier Science B.V. All rights reserved. Keywords: Density; Surface tensions; Polymer melts
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
U
Corresponding author. E-mail address: [email protected] ŽS.K. Kumar.. 0001-8686r01r$ - see front matter 䊚 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 1 - 8 6 8 6 Ž 0 1 . 0 0 0 5 3 - 7
34
S.K. Kumar, R.L. Jones r Ad¨ ances in Colloid and Interface Science 94 (2001) 33᎐38
4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1. Introduction The surface tension of a polymer melt is a primary factor in determining interfacial properties, including adhesion to surfaces, capillarity and mixture compatability w1x. For example, it has been speculated that the lower molecular weight polymers from a melt, which have lower surface tensions, will segregate to a surface w2x. This will cause a deterioration of interfacial mechanical properties due to a reduction in chain entanglement. Similarly, it has been argued that the entropically driven segregation of chain ends to a surface, which has been attributed as a driving force for the segregation of short chains, can also lead to a reduction in the glass transition temperature at the surface w3x. Surface tensions of polymer melts are well known to vary with their molecular weight, M. Experimentally, this question has been investigated in detail. In general, the molecular weight dependence is fit to the form w4᎐7x, ␥ A Myx
Ž1.
where x changes from a value of 2r3 for low molecular weight polymers to 1 in the asymptotic high molecular weight limit w8x. While the experimental results on the variation of ␥ with M are relatively well established, the theoretical interpretation of these results have been the topic of considerable discussion in the current literature. Three separate schools of thought have emerged and these are listed here. Ža. Poser and Sanchez w9x, following on early work of Cahn and Hilliard w10x, suggested that the primary contribution arises from the variation of melt density with molecular weight This line of thought was also elaborated by Dee and Sauer w4,5x who were able to fit experimental data on ␥ by utilizing equations of state Žfit to independent PVT data on these systems.. Žb. DeGennes w11x and others w12,13x have suggested that chain ends will segregate to the surface due purely to entropic factors, even in the absence of energetic factors. This arises since an end, which is connected to one monomer, can be placed more easily at a surface than a middle monomer, where two bonds have to be accomodated under these constraints. Žc. Recently, Aubouy et al. w14x presented an argument based on the loop size distribution profile for an incompressible polymer system near a hard, impenetrable boundary. We have recently questioned the results presented in w14x based on the following two points. First, existing theories accounting for the role of melt density w4,5x and chain end segregation w11᎐13x on surface tension, predict that ⭸␥r⭸Ž1rM .xT s constant as M ª ⬁. In contrast, Aubouy et al. w14x predicts, in the same limit, that ⭸␥r⭸Ž1rM .xT ª ⬁. Experimental results suggest that the derivative discussed above is finite, and constant, as M ª ⬁. Secondly, the formalism of Aubouy features a maximum in ␥ vs. 1rM,
S.K. Kumar, R.L. Jones r Ad¨ ances in Colloid and Interface Science 94 (2001) 33᎐38
35
suggesting that a particular combination of two disparate molecular weights would provide no surface segregation. Since, experimental data apparently supports a monotonic molecular weight dependence for ␥, we shall not consider the role of loop size distributions further. Note that this issue is not completely understood at this time and is a focus of ongoing research. In this paper we shall examine the relative importance of the two other factors, namely the role of density variation and chain end segregation on the molecular weight dependence of the surface tensions of polymer melts. We specifically consider the case where chain ends and the middle monomers are identical. We construct a thermodynamic formalism which allows us to separate the roles played by these two different contributions to the M dependence of ␥. We utilize the Sanchez-Lacombe formalism w15x with the Poser-Sanchez method to evaluate the role of density variations and use the Scheutjens-Fleer approach w16x in the incompressible limit, as implemented by Theodorou w12x, to evaluate the role of chain ends and the modification of chain conformations near the surface. Numerical estimates derived from these formalisms show that the density contributions dominate the M dependence of ␥, in the case where chain ends are not strongly preferred to or repelled from the surfaces. This result is in good agreement with other analytical treatments w4,5,17x.
2. Theory To separate the role of the polymer density and other factors Že.g. end group segregation and loop size distributions. on ␥ we write the exact thermodynamic relationship: ⭸␥ ⭸ Ž 1rM .
s T ,satd
⭸␥ ⭸ Ž 1rM .
q T ,
⭸␥ ⭸
= T ,M
⭸ ⭸ Ž 1rM .
Ž2. T ,satd
where the subscript ‘satd’ on the left side denotes that the derivative is evaluated at gas᎐liquid co-existence. For typical polymers in the M ª ⬁ limit, this corresponds to setting P s 0 in an equation of state, where P is the reduced pressure. It is important to stress that this equation represents the novel contribution of this paper, since it separates the contribution of the M dependence of , the density of the polymer liquid, on ␥ from all other factors. As we shall show below, each of these contributions can then be separately evaluated using relatively standard tools available in the literature. This separation of variables provides an opportunity to provide decisive evidence for the dominant factor in the molecular weight dependence of polymer melt surface tension. We focus on the second term on the right hand side, which derives directly from the M dependence of . We evaluate the first derivative in this second term by using the Cahn-Hilliard square gradient theory w10x. We begin with the expression
S.K. Kumar, R.L. Jones r Ad¨ ances in Colloid and Interface Science 94 (2001) 33᎐38
36
for the surface tension in the square gradient approximation: ␥ s 2
d Ž z .
q⬁
Hy⬁
dz
2
dz
Ž3.
where Ž z . is the local polymer density, is the coefficient of the square gradient term in the Taylor series expansion of the free energy functional and z is the direction normal to the air-polymer interface. We now postulate that the density profile is of the form: Ž z . s L f Ž zr .
Ž4.
where L is the saturated liquid density and is the correlation length for density fluctuations, which we assume to be effectively independent of molecular weight. This yields the simple relationship: ␥ s ␥⬁
L
2
Ž5.
⬁
where the subscript ⬁ denotes the infinite molecular weight limit. This directly yields: 1 ⭸␥ ␥ ⭸
s T ,M
2
Ž6.
L
To evaluate the M dependence of we employ an analytical equation of state, such as the lattice fluid model w15x. This model predicts, in agreement with experiment, that L s ⬁ q
C
Ž7.
M
as M ª ⬁. The quantities ⬁ and C can be derived directly from the SanchezLacombe equation of state. As a consequence, the product which represents the second term in Eq. Ž2. is molecular weight independent in the M ª ⬁ limit. In fact for polystyrene this equation predicts at T s 120⬚C that, a
⭸␥
k B T ⭸
= T ,M
⭸ ⭸ Ž 1rM .
s y2.35
Ž8.
T,sadt
where a is the surface area of a styrene monomer, and k B T is the thermodynamic temperature. We now proceed to evaluate the role of chain distortion and the segregation of chain ends to the surface on ␥. The enumeration of these effects for an incompressible polymer melt near a hard surface has been performed previously by Theodorou w12x using the Scheutjens-Fleer lattice self consistent field theory. In
S.K. Kumar, R.L. Jones r Ad¨ ances in Colloid and Interface Science 94 (2001) 33᎐38
37
this case, the role of density is irrelevant since a completely filled lattice is considered. Theodorou finds w12x, a
⭸␥
k B T ⭸ Ž 1rM .
s y0.19
Ž9.
T ,
where a fully flexible chain has been considered in the lattice analysis. Furthermore, it is assumed that this system is athermal, i.e. chain ends are not energetically preferred to or repelled from the surface. These results immediately illustrate that the variation of ␥ with molecular weight is dominated by equation of state effects. Effects such as the modification of chain conformations by the presence of surfaces, specifically the segregation of chain ends to the surface, contribute less than 10% of the density effect. These results have been illustrated in this case for one polymer, polystyrene, but we believe that these results are generally valid for all flexible polymers. Furthermore, these results are in qualitative agreement with calculations of Helfand et al. w17x where similar conclusions were drawn by perturbatively solving the diffusion equation, which defines polymer chain conformations near a surface.
3. Discussion We now consider experimental evidence which supports our assertion that the M dependence of dominates the molecular weight dependence of the surface tension. First, Dee and Sauer w4,5x found quantitative agreement with a large body of experimental data on polymer melt surface tensions using PVT measurements and an appropriate equation of state. The equation of state does not account for factors such as the segregation of chain ends to the surface, suggesting that over a range of molecular weight where data are of sufficient quality, density variations are sufficient to capture the M dependence of ␥. Second, Hariharan et al. w18x have measured the difference in surface energy parameters between isotopic polystyrenes using neutron reflectivity measurements. Applying the incompressible Schmidt-Binder formalism w19x to their data, these workers found that the surface energy parameter derived from the experiments varied as 1rM1 ᎐1rM2 , where Mi is the molecular weight of species i in the binary mixture. This surface energy parameter, which is an intrinsic property of the substrate, should be molecular weight independent. This suggests that the incompressible free energy utilized in the Schmidt-Binder formalism miss an important term. This result persisted even for high molecular weights and was rationalized as arising from differences between the component surface tensions, driven by density variations with molecular weight. Completely ignoring this effect, by considering a mixture of the two polymers in the framework of an incompressible lattice model, does not permit for the proper understanding of the experimentally observed segregation. Rather, this incompressible model predicts that the surface energy difference, which is dominated by the isotopic labeling, is essentially unaffected by
38
S.K. Kumar, R.L. Jones r Ad¨ ances in Colloid and Interface Science 94 (2001) 33᎐38
molecular weights, in qualitative disagreement with experiment. Using a variable density model, qualitative agreement with this ‘isotopic reversal’ effect has been achieved w20x. In summary, the experimental results point to the fact that the molecular weight dependence of ␥ is dominated by density variations with molecular weight. Of course, these results will be modified strongly if one considers situations where chain ends are strongly preferred or repelled from the surface w6x.
4. Conclusions The important conclusion we derive from our work is that the molecular dependence of ␥ is dominated by the variation of the saturated liquid density with M. Chain ends play a minor role in this context, except in special cases where they are strongly attracted or repelled from surfaces. Our results, therefore, stress the importance of equation of state measurements in determining the molecular weight dependence of the ␥ of polymer melts, a result which is in good agreement with past work of Helfand and coworkers and Dee and Sauer.
Acknowledgements Financial support from the National Science Foundation wDMR-9804327x is gratefully acknowledged.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x
S. Wu, Polymer Interfaces and Adhesion, Marcel Dekker, New York, 1982. A. Hariharan, S.K. Kumar, T.P. Russell, Macromolecules 23 Ž1990. 3584. A. Mayes, Macromolecules 27 Ž1994. 3114. G.T. Dee, B.B. Sauer, Adv. Phys. 47 Ž1998. 161. B.B. Sauer, G.T. Dee, Macromolecules 27 Ž1994. 6112. C. Jalbert et al., Macromolecules 30 Ž1997. 4481. Q. Bhatia, D.H. Pan, J.T. Koberstein, Macromolecules 21 Ž1988. 2166. S.H. Anastasiadis, I.C. Gancarz, J.T. Koberstein, Macromolecules 21 Ž1988. 2980. C.I. Poser, I.C. Sanchez, J. Coll. Int. Sci. 69 Ž1979. 539. J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 Ž1958. 258. P.G. deGennes, C. R. Acad. Sci. II ŽParis. 307 Ž1988. 1841. D.N. Theodorou, Macromolecules 21 Ž1988. 1411. S.K. Kumar, M. Vacatello, D.Y. Yoon, J. Chem. Phys. 89 Ž1988. 5206. M. Aubouy, M. Manghi, E. Raphael, Phys. Rev. Lett. 84 Ž2000. 4858. I.C. Sanchez, R.H. Lacombe, J. Phys. Chem. 80 Ž1976. 2352. J.M.H.M. Scheutjens, G.J. Fleer, J. Phys. Chem. 83 Ž1979. 1619. E. Helfand, S.M. Bhattacharjee, G.H. Fredrickson, J. Chem. Phys. 91 Ž1989. 7200. A. Hariharan, S.K. Kumar, T.P. Russell, J. Chem. Phys. 98 Ž1993. 4163. I. Schmidt, K. Binder, J. Phys. ŽParts. 46 Ž1985. 1631. A. Hariharan, S.K. Kumar, T.P. Russell, J. Chem. Phys. 99 Ž1993. 4041.
Dominance of density variations in determining the molecular weight dependence of surface tensions of polymer melts Sanat K. Kumar U , Ronald L. Jones Department of Materials Science and Engineering, Pennsyl¨ ania State Uni¨ ersity, 316 Steidle Building, Uni¨ ersity Park, PA 16802, USA
Abstract We utilize exact thermodynamic relationships to separate the role of density variations and chain end segregation to surfaces in determining the molecular weight dependence of the surface tensions of polymer melts. By utilizing standard mean-field treatments we find that density effects dominate, except in special cases, where the chain ends are strongly attracted or repelled from the surfaces. These results stress that PVT measurements are generally sufficient to properly capture the molecular weight dependence of ␥ for normally encountered polymer melts. 䊚 2001 Elsevier Science B.V. All rights reserved. Keywords: Density; Surface tensions; Polymer melts
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
U
Corresponding author. E-mail address: [email protected] ŽS.K. Kumar.. 0001-8686r01r$ - see front matter 䊚 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 1 - 8 6 8 6 Ž 0 1 . 0 0 0 5 3 - 7
34
S.K. Kumar, R.L. Jones r Ad¨ ances in Colloid and Interface Science 94 (2001) 33᎐38
4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1. Introduction The surface tension of a polymer melt is a primary factor in determining interfacial properties, including adhesion to surfaces, capillarity and mixture compatability w1x. For example, it has been speculated that the lower molecular weight polymers from a melt, which have lower surface tensions, will segregate to a surface w2x. This will cause a deterioration of interfacial mechanical properties due to a reduction in chain entanglement. Similarly, it has been argued that the entropically driven segregation of chain ends to a surface, which has been attributed as a driving force for the segregation of short chains, can also lead to a reduction in the glass transition temperature at the surface w3x. Surface tensions of polymer melts are well known to vary with their molecular weight, M. Experimentally, this question has been investigated in detail. In general, the molecular weight dependence is fit to the form w4᎐7x, ␥ A Myx
Ž1.
where x changes from a value of 2r3 for low molecular weight polymers to 1 in the asymptotic high molecular weight limit w8x. While the experimental results on the variation of ␥ with M are relatively well established, the theoretical interpretation of these results have been the topic of considerable discussion in the current literature. Three separate schools of thought have emerged and these are listed here. Ža. Poser and Sanchez w9x, following on early work of Cahn and Hilliard w10x, suggested that the primary contribution arises from the variation of melt density with molecular weight This line of thought was also elaborated by Dee and Sauer w4,5x who were able to fit experimental data on ␥ by utilizing equations of state Žfit to independent PVT data on these systems.. Žb. DeGennes w11x and others w12,13x have suggested that chain ends will segregate to the surface due purely to entropic factors, even in the absence of energetic factors. This arises since an end, which is connected to one monomer, can be placed more easily at a surface than a middle monomer, where two bonds have to be accomodated under these constraints. Žc. Recently, Aubouy et al. w14x presented an argument based on the loop size distribution profile for an incompressible polymer system near a hard, impenetrable boundary. We have recently questioned the results presented in w14x based on the following two points. First, existing theories accounting for the role of melt density w4,5x and chain end segregation w11᎐13x on surface tension, predict that ⭸␥r⭸Ž1rM .xT s constant as M ª ⬁. In contrast, Aubouy et al. w14x predicts, in the same limit, that ⭸␥r⭸Ž1rM .xT ª ⬁. Experimental results suggest that the derivative discussed above is finite, and constant, as M ª ⬁. Secondly, the formalism of Aubouy features a maximum in ␥ vs. 1rM,
S.K. Kumar, R.L. Jones r Ad¨ ances in Colloid and Interface Science 94 (2001) 33᎐38
35
suggesting that a particular combination of two disparate molecular weights would provide no surface segregation. Since, experimental data apparently supports a monotonic molecular weight dependence for ␥, we shall not consider the role of loop size distributions further. Note that this issue is not completely understood at this time and is a focus of ongoing research. In this paper we shall examine the relative importance of the two other factors, namely the role of density variation and chain end segregation on the molecular weight dependence of the surface tensions of polymer melts. We specifically consider the case where chain ends and the middle monomers are identical. We construct a thermodynamic formalism which allows us to separate the roles played by these two different contributions to the M dependence of ␥. We utilize the Sanchez-Lacombe formalism w15x with the Poser-Sanchez method to evaluate the role of density variations and use the Scheutjens-Fleer approach w16x in the incompressible limit, as implemented by Theodorou w12x, to evaluate the role of chain ends and the modification of chain conformations near the surface. Numerical estimates derived from these formalisms show that the density contributions dominate the M dependence of ␥, in the case where chain ends are not strongly preferred to or repelled from the surfaces. This result is in good agreement with other analytical treatments w4,5,17x.
2. Theory To separate the role of the polymer density and other factors Že.g. end group segregation and loop size distributions. on ␥ we write the exact thermodynamic relationship: ⭸␥ ⭸ Ž 1rM .
s T ,satd
⭸␥ ⭸ Ž 1rM .
q T ,
⭸␥ ⭸
= T ,M
⭸ ⭸ Ž 1rM .
Ž2. T ,satd
where the subscript ‘satd’ on the left side denotes that the derivative is evaluated at gas᎐liquid co-existence. For typical polymers in the M ª ⬁ limit, this corresponds to setting P s 0 in an equation of state, where P is the reduced pressure. It is important to stress that this equation represents the novel contribution of this paper, since it separates the contribution of the M dependence of , the density of the polymer liquid, on ␥ from all other factors. As we shall show below, each of these contributions can then be separately evaluated using relatively standard tools available in the literature. This separation of variables provides an opportunity to provide decisive evidence for the dominant factor in the molecular weight dependence of polymer melt surface tension. We focus on the second term on the right hand side, which derives directly from the M dependence of . We evaluate the first derivative in this second term by using the Cahn-Hilliard square gradient theory w10x. We begin with the expression
S.K. Kumar, R.L. Jones r Ad¨ ances in Colloid and Interface Science 94 (2001) 33᎐38
36
for the surface tension in the square gradient approximation: ␥ s 2
d Ž z .
q⬁
Hy⬁
dz
2
dz
Ž3.
where Ž z . is the local polymer density, is the coefficient of the square gradient term in the Taylor series expansion of the free energy functional and z is the direction normal to the air-polymer interface. We now postulate that the density profile is of the form: Ž z . s L f Ž zr .
Ž4.
where L is the saturated liquid density and is the correlation length for density fluctuations, which we assume to be effectively independent of molecular weight. This yields the simple relationship: ␥ s ␥⬁
L
2
Ž5.
⬁
where the subscript ⬁ denotes the infinite molecular weight limit. This directly yields: 1 ⭸␥ ␥ ⭸
s T ,M
2
Ž6.
L
To evaluate the M dependence of we employ an analytical equation of state, such as the lattice fluid model w15x. This model predicts, in agreement with experiment, that L s ⬁ q
C
Ž7.
M
as M ª ⬁. The quantities ⬁ and C can be derived directly from the SanchezLacombe equation of state. As a consequence, the product which represents the second term in Eq. Ž2. is molecular weight independent in the M ª ⬁ limit. In fact for polystyrene this equation predicts at T s 120⬚C that, a
⭸␥
k B T ⭸
= T ,M
⭸ ⭸ Ž 1rM .
s y2.35
Ž8.
T,sadt
where a is the surface area of a styrene monomer, and k B T is the thermodynamic temperature. We now proceed to evaluate the role of chain distortion and the segregation of chain ends to the surface on ␥. The enumeration of these effects for an incompressible polymer melt near a hard surface has been performed previously by Theodorou w12x using the Scheutjens-Fleer lattice self consistent field theory. In
S.K. Kumar, R.L. Jones r Ad¨ ances in Colloid and Interface Science 94 (2001) 33᎐38
37
this case, the role of density is irrelevant since a completely filled lattice is considered. Theodorou finds w12x, a
⭸␥
k B T ⭸ Ž 1rM .
s y0.19
Ž9.
T ,
where a fully flexible chain has been considered in the lattice analysis. Furthermore, it is assumed that this system is athermal, i.e. chain ends are not energetically preferred to or repelled from the surface. These results immediately illustrate that the variation of ␥ with molecular weight is dominated by equation of state effects. Effects such as the modification of chain conformations by the presence of surfaces, specifically the segregation of chain ends to the surface, contribute less than 10% of the density effect. These results have been illustrated in this case for one polymer, polystyrene, but we believe that these results are generally valid for all flexible polymers. Furthermore, these results are in qualitative agreement with calculations of Helfand et al. w17x where similar conclusions were drawn by perturbatively solving the diffusion equation, which defines polymer chain conformations near a surface.
3. Discussion We now consider experimental evidence which supports our assertion that the M dependence of dominates the molecular weight dependence of the surface tension. First, Dee and Sauer w4,5x found quantitative agreement with a large body of experimental data on polymer melt surface tensions using PVT measurements and an appropriate equation of state. The equation of state does not account for factors such as the segregation of chain ends to the surface, suggesting that over a range of molecular weight where data are of sufficient quality, density variations are sufficient to capture the M dependence of ␥. Second, Hariharan et al. w18x have measured the difference in surface energy parameters between isotopic polystyrenes using neutron reflectivity measurements. Applying the incompressible Schmidt-Binder formalism w19x to their data, these workers found that the surface energy parameter derived from the experiments varied as 1rM1 ᎐1rM2 , where Mi is the molecular weight of species i in the binary mixture. This surface energy parameter, which is an intrinsic property of the substrate, should be molecular weight independent. This suggests that the incompressible free energy utilized in the Schmidt-Binder formalism miss an important term. This result persisted even for high molecular weights and was rationalized as arising from differences between the component surface tensions, driven by density variations with molecular weight. Completely ignoring this effect, by considering a mixture of the two polymers in the framework of an incompressible lattice model, does not permit for the proper understanding of the experimentally observed segregation. Rather, this incompressible model predicts that the surface energy difference, which is dominated by the isotopic labeling, is essentially unaffected by
38
S.K. Kumar, R.L. Jones r Ad¨ ances in Colloid and Interface Science 94 (2001) 33᎐38
molecular weights, in qualitative disagreement with experiment. Using a variable density model, qualitative agreement with this ‘isotopic reversal’ effect has been achieved w20x. In summary, the experimental results point to the fact that the molecular weight dependence of ␥ is dominated by density variations with molecular weight. Of course, these results will be modified strongly if one considers situations where chain ends are strongly preferred or repelled from the surface w6x.
4. Conclusions The important conclusion we derive from our work is that the molecular dependence of ␥ is dominated by the variation of the saturated liquid density with M. Chain ends play a minor role in this context, except in special cases where they are strongly attracted or repelled from surfaces. Our results, therefore, stress the importance of equation of state measurements in determining the molecular weight dependence of the ␥ of polymer melts, a result which is in good agreement with past work of Helfand and coworkers and Dee and Sauer.
Acknowledgements Financial support from the National Science Foundation wDMR-9804327x is gratefully acknowledged.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x
S. Wu, Polymer Interfaces and Adhesion, Marcel Dekker, New York, 1982. A. Hariharan, S.K. Kumar, T.P. Russell, Macromolecules 23 Ž1990. 3584. A. Mayes, Macromolecules 27 Ž1994. 3114. G.T. Dee, B.B. Sauer, Adv. Phys. 47 Ž1998. 161. B.B. Sauer, G.T. Dee, Macromolecules 27 Ž1994. 6112. C. Jalbert et al., Macromolecules 30 Ž1997. 4481. Q. Bhatia, D.H. Pan, J.T. Koberstein, Macromolecules 21 Ž1988. 2166. S.H. Anastasiadis, I.C. Gancarz, J.T. Koberstein, Macromolecules 21 Ž1988. 2980. C.I. Poser, I.C. Sanchez, J. Coll. Int. Sci. 69 Ž1979. 539. J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28 Ž1958. 258. P.G. deGennes, C. R. Acad. Sci. II ŽParis. 307 Ž1988. 1841. D.N. Theodorou, Macromolecules 21 Ž1988. 1411. S.K. Kumar, M. Vacatello, D.Y. Yoon, J. Chem. Phys. 89 Ž1988. 5206. M. Aubouy, M. Manghi, E. Raphael, Phys. Rev. Lett. 84 Ž2000. 4858. I.C. Sanchez, R.H. Lacombe, J. Phys. Chem. 80 Ž1976. 2352. J.M.H.M. Scheutjens, G.J. Fleer, J. Phys. Chem. 83 Ž1979. 1619. E. Helfand, S.M. Bhattacharjee, G.H. Fredrickson, J. Chem. Phys. 91 Ž1989. 7200. A. Hariharan, S.K. Kumar, T.P. Russell, J. Chem. Phys. 98 Ž1993. 4163. I. Schmidt, K. Binder, J. Phys. ŽParts. 46 Ž1985. 1631. A. Hariharan, S.K. Kumar, T.P. Russell, J. Chem. Phys. 99 Ž1993. 4041.