ABSTRACT. and since the phase transition of asymmetric one-component fluid or mixture

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1 ABSTRACT Title of Doument: MESOSCOPIC THERMODYNAMICS IN SMOOTH AND CURVED INTERFACES IN ASYMMETRIC FLUIDS Heather Jane St. Pierre, Dotor of Philosophy, 9 Direted By: Professor Mikhail Anisimov Institute for Physial Siene and Tehnology and Department of Chemial and Biomoleular Engineering Phase separation has a signifiant impat on many hemial engineering appliations, and sine the phase transition of asymmetri one-omponent fluid or mixture interfaes an be gradual, or smooth, further analysis is warranted for fluid separation on the mesosale. Complete saling is applied to aount for flutuations in the ritial region and to model the interfaial profiles of one-omponent and dilute binary mixtures, as well as alulate the urvature orretion to the surfae of tension (or Tolman length). Well-established symmetri profiles were seleted for the order parameter and thermal saling densities for use in omplete saling to model these asymmetri fluids. Real fluid asymmetry was applied to these profiles though the saling oeffiients of one-omponent fluids. Saling oeffiients for mixtures were introdued through the experimental ritial parameters of the speifi mixture;

2 harateristis aounted for inluded onsideration of the differene in molar volume between solvent and solute, hanges in ritial temperature and pressure with onentration as well as the Krihevskii parameter. Flory theory was used to approximate the saling oeffiients in dilute polymer solutions to determine the Tolman length. These results indiated that Tolman s length diverges near the ritial point of separation and with an inreasing degree of polymerization. As an infinite degree of polymerization is approahed, the Tolman length beomes half of the width of the interfae. Sine fluid behavior near the ritial point of separation is universal, many of the theoretial expressions shown in this work an be applied to other asymmetri systems. The results of this analysis showed that a large differene in moleular volume led to a higher degree of fluid asymmetry in polymer solutions. The onentration of added solute, speifially in dilute n-heptane-ethane solutions resulted in inreased fluid asymmetry in density profiles, as well as a shift toward the density of the added solute. When onsidering dilute mixtures of aqueous n-hexane and n-heptane-ethane solutions, a slight inrease in onentration of solute, oupled with an inreasing temperature distane to the ritial point of separation, yielded the greatest inrease in fluid asymmetry in onentration profiles.

3 MESOSCOPIC THERMODYNAMICS IN SMOOTH AND CURVED INTERFACES IN ASYMMETRIC FLUIDS By Heather Jane St. Pierre Dissertation submitted to the Faulty of the Graduate Shool of the University of Maryland, College Park, in partial fulfillment of the requirements for the degree of Dotor of Philosophy 9 Advisory Committee: Professor Mikhail A. Anisimov, Dept. of Chemial and Biomoleular Engineering, Chair Professor Robert M. Briber, Dept. of Materials Siene and Engineering Professor Mohammed Al-Sheikhly, Dept. of Materials Siene and Engineering Professor Raymond A. Adomaitis, Dept. of Chemial and Biomoleular Engineering Professor Sheryl H. Ehrman, Dept. of Chemial and Biomoleular Engineering

4 Copyright by Heather Jane St. Pierre 9

5 Aknowledgements I would like to take the opportunity to thank the members of our researh group, as all have provided invaluable thoughts and input. In partiular, I would like to thank Chris Bertrand, as well as Daphne Fuentevilla and Deepa Subramanian for their inredible support throughout this proess. I also wish to thank the faulty and staff of the Chemial and Biomoleular Engineering Department for their assistane throughout this proess. Speifially, I would like to thank my advisor, Professor Mikhail Anisimov for his guidane, patiene and for keeping me motivated throughout this researh, as well as Distinguished University Researh Professor Jan Sengers for his guidane and review during ompletion of this work. Without them, this researh would not have been possible. Lastly, I would like to thank my husband, Eri, for his unending love and support throughout this researh. ii

6 Table of Contents Aknowledgements...ii Table of Contents...iii List of Tables...v List of Figures...vi Chapter 1: Introdution The Smooth Interfae Surfae Tension and Curved Smooth Interfaes Asymmetry and Its Effet on Interfaial Profiles Methodology and Overview...5 Chapter : Mean-Field Approximation of the Smooth Interfae Determining the Loal Helmholtz-Energy Density Landau Expansion and the Landau-Ginsburg Funtional...8. Approximating the Density Profile near the Interfae Determining the Width of the Interfae Position and Density Dependene of the Loal Helmholtz-Energy Density Mean-Field Estimation of Interfaial Tension... Chapter 3: Aounting for Flutuations near the Critial Point Effets of Critial Flutuations and Fluid Asymmetry Saling Theory General Saling Equations Complete Saling for a One-Component Fluid Complete Saling for a Dilute Binary Mixture The Smooth Interfae Estimations of Surfae Tension Renormalization Group Theory Results Dimensional Saling Analysis Comparison of Methodologies...48 Chapter 4: Curved Interfaes and Surfae Tension Tolman s Length: Curvature Corretion to the Surfae Tension Young-Laplae Approximation of Surfae Tension Expression for Tolman s Length Preditions of the Behavior of Tolman s Length Tolman s Length Near the Critial Point General Thermodynami Expression for Tolman s Length...6 Chapter 5: Tolman s Length for Polymer Solutions and Other Dilute Binary Mixtures Polymer Solutions: Theory and Bakground Phase Coexistene Flutuations Tolman s Length in Near-Critial Polymer Solutions Approximation of Saling Coeffiients Approximation of Critial Amplitudes...74 iii

7 5..3 Tolman s Length near the Critial Point Comparison of Terms Tolman s Length at High Degrees of Polymerization Crossover Expression for Tolman s Length in Polymer Solutions Crossover Correlation Length Crossover Tolman-Correlation Length Ratio Crossover Tolman s Length Tolman s Length in Other Polymer Appliations Near-Critial Tolman s Length in Dilute Binary Mixtures...9 Chapter 6: Conentration and Density Profiles for Asymmetri Fluids and Binary Fluid Mixtures Bakground and Introdution Determination of Saling Densities for Use in Complete Saling Symmetri Order Parameter Thermal Saling Density Modeling Density Profiles for One-Component Fluids Appliation to Water Appliation to Ethane Appliation to n-heptane Fisher Renormalization: Temperature Corretion for Conentration Modeling Conentration Profiles for Dilute Binary Solutions Appliation to Dilute Aqueous n-hexane Solutions Appliation to Dilute n-heptane-ethane Solutions Effet of Temperature on Profile Asymmetry Effet of Conentration on Profile Asymmetry Modeling Asymmetri Density Profiles for Dilute Binary Solutions Appliation to Dilute Polystyrene-Cylohexane Solutions Appliation to Dilute Ethane-n-Heptane Solutions Effet of Temperature on Asymmetry in Density Profiles Effet of Conentration on Asymmetry in Density Profiles Conlusions Chapter 7: Summary and Outlook Appendix A: Comparison of Mean-Field Surfae Tension Calulations A.1 Comparison with Landau and Lifshitz A. Comparison with Rowlinson and Widom...14 Referenes iv

8 List of Tables Table 3.1 Table 6.1 Comparison of mean-field and saling methods for the estimation of surfae tension oeffiients.. 48 Comparison of mean-field and saling methods for the estimation of surfae tension oeffiients.. 11 Table 6. Experimental data for ethane 11 Table 6.3 Table 6.4 Experimental data for n-heptane 13 Seleted experimental data for a dilute aqueous n-hexane solution.. 16 Table 6.5 Experimental data for dilute aqueous n-hexane solutions 111 Table 6.6 Table 6.7 Critial temperatures of dilute aqueous n-hexane solutions at various onentrations Critial temperatures of dilute n-hepane-ethane solutions at various onentrations 114 Table 6.8 Experimental data for dilute n-heptane-ethane solutions Table 6.9 Comparison of bulk phase onentration data from dilute aqueous n-hexane solutions at various temperatures... 1 Table 6.1 Comparison of bulk phase onentration data from dilute n- heptane-ethane solutions at various temperatures. 11 v

9 List of Figures Figure 1.1 Figure 1. Figure.1 Figure. Figure.3 Figure.4 Figure.5 Figure 3.1 Simulated urved smooth interfae of vapor-liquid separation in a droplet Shemati of an asymmetri vapor-liquid oexistene urve.. 4 Loal Helmholtz- energy density from Landau expansion (in terms of the order parameter) given in Eq. (.1.14) for u = 9/16 and varying temperature distanes, ˆT Loal Helmholtz- energy density from Landau expansion (in terms of the order parameter) given in Eq. (.1.15) whih inludes the referene value for the loal free energy density for u = 9/16 and varying temperature distanes, ˆT 13 Mean-field dimensionless density profile (a) and gradient (b) as a funtion of height.. 17 Contour plot of the interfae between two fluid phases as predited by mean-field theory. 18 Three-dimensional universal plot of the dimensionless loal 4 Helmholtz-energy density ( f / ϕ ) and its dependene on height and density. Shemati of an asymmetri vapor-liquid oexistene urve omparing mean-field and saling approximations near the ritial point. 5 Figure 3. Shemati of a T ρ diagram indiating the loation of zero ordering field (h 1 = ).. 9 Figure 3.3 Figure 3.4 Figure 3.5 Comparison of mean-field and renormalization group theory density profiles as a funtion of height. 4 Contour plot of the interfae between liquid and vapor phases as predited by renormalization group theory.. 4 Simulation of a smooth interfae for a droplet near the ritial point of fluid-fluid separation using RG theory 43 vi

10 Figure 3.6 Figure 4.1 Representation of the interfae thikness onsidered for the dimensional saling analysis alulation of surfae tension in Eqs. (.4.11) and (.4.13)-(.4.15) 47 Simple shemati illustrating the properties of the variables used in the Young-Laplae relation for surfae tension at the interfae between two phases, α and β. 51 Figure 4. Sketh of the onept of the Tolman length ( δ ) for a droplet of liquid. 5 Figure 4.3 Sketh of the onept of using the oexistene diameter to approximate the ratio of the Tolman length to the interfae thikness in Eq. (4.3.1). 61 Figure 5.1 Phase diagram for a polymer solution ( N = 14 ) near the ritial point.. 67 Figure 5. Figure 5.3 Figure 5.4 Plots of aeff and b against inverse degree of polymerization (1/N) for the range N = 1 to Three-dimensional plot of the dimensionless near-ritial Tolman length for a droplet of polymer in solution (from Eq. (5..18)) 76 Plots of the two terms from the dimensionless Tolman length for a droplet of polymer in solution at three temperatures 77 Figure 5.5 Comparison of N-dependent amplitudes ˆB and Γˆ for a moderate to high degree of polymerization.. 8 Figure 5.6 Figure 5.7 Universal behavior of the dimensionless Tolman length with respet to degree of polymerization and temperature distane to phase separation for a polymer-rih droplet as predited by Eq. (5.4.1) 88 Dimensionless Tolman length for a polymer-rih droplet with N = 1 4 exhibiting rossover between the ritial and polymer regimes from Eq. (5.4.1) Figure 6.1 Shemati of an asymmetri vapor-liquid oexistene urve Figure 6. Normalized symmetri mean field and Renormalization Group theory profiles 96 vii

11 Figure 6.3 Figure 6.4 Figure 6.5 Figure 6.6 Density profile for pure water at a temperature distane of Tˆ = 1, plotted with respet to height (z-oordinate).. 14 Density profile for ethane at a temperature distane of Tˆ = 1, plotted with respet to height (z-oordinate).. 1 Density profile for n-heptane at a temperature distane of Tˆ = 1, plotted with respet to height (z-oordinate)...14 Plots of the differene (a) and normalized differene (b) between experimental ( Tˆ ( x)) and theoretial ( T ˆ( µ )) temperature distanes as predited by Eq. (5.4.) and data in Table 6.4 for a dilute aqueous n-hexane solution where x = Figure 6.7 Critial lous for aqueous n-hexane solutions.. 11 Figure 6.8 Critial lous for a mixture of ethane and n-heptane 113 Figure 6.9 Figure 6.1 Figure 6.11 Figure 6.1 Figure 6.13 Conentration profiles for n-hexane in water at the vapor-liquid interfae, at a ritial and average onentration of x = x =.15, and at various temperatures, plotted with respet to height (z-oordinate). 116 Conentration profiles for n-hepane in ethane at the vapor-liquid interfae, at a ritial and average onentration of x = x =., and at various temperature distanes, plotted with respet to height (z-oordinate). 117 Conentration profiles for n-hexane in water at the vapor-liquid interfae, at a onstant temperature distane Tˆ = 1, and at various ritial or average onentrations, plotted with respet to height (z-oordinate). 119 Conentration profiles for n-hepane in ethane at the vapor-liquid interfae, at a onstant temperature distane Tˆ = 1, and at various ritial or average onentrations, plotted with respet to height (z-oordinate). 1 Normalized density profile for a polystyrene-ylohexane solution with degrees of polymerization N = 1, N = 15 and at a temperature distane of Tˆ = 1 15 viii

12 Figure 6.14 Figure 6.15 Figure 6.16 Density profiles for n-heptane in ethane at the vapor-liquid interfae, at a ritial and average onentration of x =., and at various temperatures, plotted with respet to height (zoordinate).18 Density profiles for ethane and n-hepane in ethane at the vaporliquid interfae, at a onstant temperature distane Tˆ = 1, and at various ritial onentrations, plotted with respet to height (zoordinate). 13 Density profiles for ethane and n-hepane in ethane at the vaporliquid interfae, at a onstant temperature distane ˆ ˆ T( x) = T ( µ ) 1 (without Fisher Renormalization), and at various ritial onentrations, plotted with respet to height (zoordinate) 131 ix

13 Chapter 1: Introdution The manner in whih two phases separate from one another an have a signifiant impat on many engineering appliations, whether these phases are the liquid and vapor phases of a pure (or one-omponent) fluid, or two phases of differing liquids. Diretly related to interfaial phase separation is the onept of surfae tension. Both interfaial behavior and surfae tension are important for engineering appliations when onsidering droplet formation, miroemulsions, apillary ation, nuleation, wettability, and fluid flow through miropores. 1.1 The Smooth Interfae In the study of interfaes, Gibbs 1 introdued the onept of the dividing surfae, whih is an important baseline for establishing the loation of where the surfae of tension ats in a droplet.,3 In some fluids and on larger sales, the loation of the dividing surfae an be the same as the thikness or width of the interfae between two fluids, where the interfae is only on the order of the size of a moleule of fluid. This ase is known as the sharp interfae, as the transition from one fluid phase to the other is very abrupt. In situations where the moleular size of a fluid is signifiantly large or the physial fluid properties are near the ritial point of separation, the transition between two liquid phases beomes muh more gradual, or smooth, shown in Fig In this figure, the darker olor represents the gradient of the transition from one phase to another and the light olor (or absene of olor) indiates eah of the bulk phases. This smooth interfae illustrated in Fig. 1.1 is 1

14 present in soft matter (i.e. polymer solutions and miroemulsions) and in fluids near the ritial point of separation. 4-9 Figure 1.1. Simulated urved smooth interfae of vapor-liquid separation in a droplet. The interfae was simulated using toolboxes in MATLAB developed by Adomaitis 1 and Adomaitis and Chen. 11 The darker olor indiates the steepest gradient of the interfae of separation and the lightest olor indiates the bulk phases on either side of the interfae. 1. Surfae Tension and Curved Smooth Interfaes Many hemial engineering appliations involve the formation or use of small droplets. For example, some targeted drug delivery systems utilize miro- or nanodroplets that form a flexible, biodegradable and hollow sphere into whih a pharmaeutial an be inserted. 1,13 These tiny hollow spheres an be reated dissolving a polymer in an immisible solvent; this solution is then emulsified to form small droplets. 14 After the droplets are formed, the solvent is then removed from the polymer droplet through solvent evaporation, leaving behind hollow mirospheres. 14 Another example is the formation of aerosol miro- or nanodroplets for use in oatings. Cerami oatings, due to their insulation and thermal properties, are of interest in many engineering appliations. In order to reate very thin erami films,

15 small droplets must be used to reate an ultra-fine spray. In one speifi study, onduted by Balahandran, Miao and Xiao, 15 eletrostati atomization was used to generate 4-5 µm droplets of suspended ZrO and SiC to reate a homogeneous thin erami film on alloy substrates. Although there are ountless other appliations, given these engineering examples, it is important to reognize and aount for the presene and behavior of smooth interfaes, espeially in very small droplets. Determining the surfae tension at the separation boundary of small droplets that exhibit a smooth interfae is not as straightforward as it might seem. In partiular, the effets of urvature are typially onsidered negligible for surfae tension on the marosale. However, the urvature orretion beomes inreasingly more signifiant as appliations and proesses are saled down, as has been the trend in many fields of engineering. As a result of downsaling, different physial effets and properties beome more pronouned. One suh effet is the urvature-orretion to the surfae tension. In priniple, the surfae tension of a urved interfae will behave differently than that of a planar interfae,,3,16-31 but this urvature orretion is typially ignored in engineering pratie. This is mainly due to differing opinions regarding the net effet of urvature on the surfae of tension, and as a result, the magnitude of this orretion has been the subjet of ontroversy for several deades.,1,18,,,4-8,3,3-43 In addition, when the mean-field approximation is used, the orretion to the surfae tension is insignifiant, even when onsidering a nano-sized droplet. 44 However, more reent studies have shown onvining evidene to the ontrary, indiating that this urvature orretion an signifiantly alter surfae tension alulations. 45 3

16 1.3 Asymmetry and Its Effet on Interfaial Profiles Another important onept in engineering appliations is the impat of fluid asymmetry on the behavior of an interfae. Whether or not a fluid is symmetri or asymmetri simply refers to the shape of its oexistene urve. If the phase oexistene is a symmetri parabola, the fluid is symmetri. Conversely, if the phase oexistene does not exhibit this symmetry, it is onsidered asymmetri. A simple illustration of an asymmetri phase diagram an be seen in Fig. 1.. If the phase oexistene was ompletely symmetri in this figure, the oexistene would be a symmetri parabola about the ritial point, denoted by the blue dot and ritial density, ρ. Figure 1.. Shemati of an asymmetri vapor-liquid oexistene urve. The retilinear (atual) diameter is given by the dashed green line. Fluid asymmetry an be inherent both in pure (or one-omponent) fluids as well as in fluid mixtures. The addition of a small amount of solute to a oneomponent fluid with a vastly different moleular volume and hemial harateristis an result in new fluid asymmetry whih an signifiantly impat the behavior of the mixture s interfaial profile. 4

17 Simple models, suh as the lattie-gas model offer means to model symmetri fluids and fluid mixtures, but do not aount for asymmetry. Equations of state for a partiular substane an aount for fluid asymmetry, but may not aurately aount for flutuations near the ritial point and an be partiularly ompliated if a signifiant amount of auray is desired. Therefore, we must look to a method that aounts for both asymmetry and flutuations near the ritial point of separation. 1.4 Methodology and Overview In order to develop expressions to determine both the urvature orretion to the surfae tension and interfaial behavior of an asymmetri fluid, we must utilize a method that aounts for both flutuations near the ritial point of separation as well as fluid asymmetry. A method that has been shown to aurately model this behavior is known as omplete saling, developed by Fisher and Orkulas 45 and Kim et al. 5 Prior to exploring the results of omplete saling, the simple behavior of mean-field interfaial profiles will be addressed. The expressions for the mean-field interfaial profile and surfae tension will be developed in Chapter through a mesosopi thermodynami approah that utilizes the Landau-Ginsburg loal free energy funtional. As mean-field theories do not aount for the effet of ritial flutuations, we must look to omplete saling to both aount for these flutuations and fluid asymmetry. The development of expressions desribing one-omponent and binary fluid behavior using omplete saling will be shown in Chapter 3. The urvature orretion to the surfae tension, or Tolman s length, and its impliations 5

18 will be disussed in Chapter 4. The near-ritial and general thermodynami expressions for Tolman s length will be presented as well. One the omplete saling approah has been presented, the behavior of real fluids an be approximated. As fluid asymmetry has been shown to be related to the Tolman length, 45,51 we will look to highly asymmetri fluids, suh as polymer solutions, to further explore the magnitude of this orretion in Chapter 5. Also in this hapter, the urvature orretion will be addressed both near the ritial point of separation as well as away from the ritial region. These results will be used to develop a rossover expression for the Tolman length in polymer solutions. As an alternative to the lattie-gas model, Chapter 6 provides the appliation of omplete saling to approximate the density and onentration profile behavior at the interfae of asymmetri one-omponent fluids and dilute mixtures. Based on a good understanding of symmetri interfaes, as presented in Chapters and 3, asymmetry speifi to a fluid or fluid mixture will be applied through saling oeffiients used in omplete saling. Lastly, onlusions and potential future work will be presented in Chapter 7. 6

19 Chapter : Mean-Field Approximation of the Smooth Interfae The interfae of phase separation provides insight as to how the surfae of tension, a onept important to engineering alulations, will behave as it is highly dependent on intermoleular interations. As a result, not all interfaes are the same. A smooth interfae exists when there is a gradual hange in onentration at the phase boundary between liquid-vapor or liquid-liquid separation, and is frequently observed in soft matter. Conversely, a sharp interfae exists when the hange between two phases is very abrupt, muh like a step funtion. Smooth interfaes are present in near-ritial liquid-vapor and liquid-liquid interfaes, in polymer solutions and polymer blends as well as liquid membranes and vesiles. 4-6,9,5,53 These interfaes an be represented by an interfaial density (or onentration) profile, whih will be used to determine a harateristi length sale or interfaial thikness. In order to determine this length sale, priniples of mesosopi thermodynamis will be utilized. For initial estimations of surfae tension, density and onentration profiles, the effets of flutuations near the ritial point of phase separation will be ignored and mean-field or van der Waals theory will be used to approximate a symmetri smooth interfae..1 Determining the Loal Helmholtz-Energy Density While onventional statistial mehanis has been the longstanding methodology and basis for mesosopi thermodynamis, other approahes that 7

20 emphasize universality an be suessfully applied to desribe ritial phenomena and interfaial density profiles. One suh example of this approah is use of the Landau-Ginzburg funtional. As flutuations are prevalent in mesosopi thermodynamis, we an onsider a loal or oordinate-dependent density of an appropriate thermodynami potential in order to smooth these inhomogeneities. 5-5 Beause we are ultimately interested in alulating the surfae tension, σ, we will start with the basi definition: d(surfae free energy) σ =. (.1.1) d(area) We an hoose an appropriate thermodynami potential, whih in this ase is the Helmholtz energy, beause it is related to the property of interest. By using the total Helmholtz energy, A, the following expression an be developed from Eq. (.1.1): where dxdy + d( A) σ = = ρkt B [ loal free energy density] dz dxdy, (.1.) = dσ for the surfae, ρ is the ritial density, k B is the Boltzmann onstant, and T is the temperature. The remaining unknown is the expression for the loal free-energy density, and in our ase, that is the loal Helmholtz-energy density..1.1 Landau Expansion and the Landau-Ginsburg Funtional To determine the loal free energy density, we will review some basi definitions. The total Helmholtz-energy density, ρ A, an be written as the integral taken over the total volume of the system: 55 ρa= ( ρa) dv. (.1.3) 8

21 When integrated, the Helmholtz energy density an be onsidered as the sum of three terms, as indiated by the works of others: 57,58 ρa= ( ρa) + ρµ ( T, ρ = ρ ) + A ( T, ρ = ρ ) (.1.4) where the terms ρµ and A are two bakground terms that relate to the alori properties and pressure, respetively. The last two terms on the right hand side of this equation are not affeted by density flutuations and are analyti funtions of temperature and density. However, the first term, ( ρ A), is affeted by loal inhomogeneities, and, hene, density flutuations and is therefore the funtion of interest. Another important and related onept that addresses these inhomogeneities is the order parameter, ϕ. This position-dependent loal order parameter is defined by an appropriate field-dependent thermodynami potential. In the one-omponent fluid, density is the property of interest as it exhibits strong loal flutuations. Our dimensionless order parameter based on loal density of the fluid is then defined as v v ρ( r) ρ ϕ (.1.5) ( r ), ρ where ρ( r v ) is the loal density in three-dimensional spae and ρ is the ritial density. The loal Helmholtz-energy density an be expanded in powers of the order parameter and powers of gradient of the order parameter; 8 1 ˆ r 1 r 4 1 ( ρa) = ρkbt a T[ ϕ( )] + u[ ϕ( )] + ( ϕ) dxdydz, 4 (.1.6) where the dimensionless temperature distane to the ritial point, ˆT, is defined as 9

22 T T Tˆ T. (.1.7) The remaining unknowns a and u are system-dependent oeffiients, where u reflets the energy of interation between flutuations, 59 and is a onstant related to the range of intermoleular interations. 8 These onstants are also important in determining other thermodynami quantities, and will be disussed in Setion.5. Note that the result in Eq. (.1.7) takes into aount that at the onditions that define the ritial point, µ / ρ = µ / ρ =, are met. In Eq. (.1.6), the first terms within the brakets are a result of Landau expansion in terms of ϕ, and the last term is the result of expansion in powers of the gradient ( dϕ / dr) v. The entire braketed expression is known as the Landau-Ginzburg funtional for the loal Helmholtz-energy density. 8 It should be noted that the order parameter symmetri, as is the Landau-Ginzburg funtional. In this speifi ase, the interest lies with the height-dependent oordinate, z, whih gravity selets as the diretion normal to the planar surfae. Sine only the braketed portion of Eq. (.1.6) an be re-written in terms of z, we an write the loal Helmholtz-energy density, f as 1 ˆ dϕ = + + f[ ϕ( z)] a T[ ϕ( z)] u [ ϕ( z)]. 4 dz (.1.8) Comparing this expression for the loal Helmholtz-energy density to that in Eq. (.1.6), you an see that this is merely the integrand in brakets in Eq. (.1.6). In the previous equation, the definition of the order parameter (Eq. (.1.5)) in terms of z only beomes 1

23 ϕ ρ( z) ρ (.1.9) ρ ( z). The expression for the loal Helmholtz-energy density in Eq. (.1.8) an be simplified by arefully hoosing values for the onstants with physial meaning and by analyzing the derivatives, or onditions, of the order parameter and what it represents in terms of real physial properties. As with other thermodynami potentials, the first derivative of the loal Helmholtz energy determines the onditions for equilibrium, or hemial potential, when equal to zero: df / dϕ = µ / kbt =. The seond and third derivatives, when equal to zero, determine the spinodal (or region of instability) and ritial point onditions, respetively. In the ase of the loal Helmholtz-energy density, only the portion of Eq. (.1.8) from the expansion of powers of the gradient (i.e. the first and seond terms) will be onsidered: f = a Tˆ[ ϕ ( z )] + u [ ϕ ( z )]. (.1.1) 4 The equilibrium, spinodal and ritial point onditions are then: ϕ 1/ a ( Tˆ ) df at = = u dϕ (.1.11) 1/ a( Tˆ ) d f sp 3u dϕ ϕ = at = (.1.1) and 3 d f CP 3 ϕ = at dϕ =, (.1.13) 11

24 respetively. By applying the expression for the equilibrium ondition in Eq. (.1.11) to the loal Helmholtz-energy density in Eq. (.1.1), we an omplete the square to simplify the equation so that f = u( ϕ ϕ) uϕ. (.1.14) 4 4 The plot of this equation is given in Fig..1, where u = 9/16. This value of u was hosen to math the equilibrium result in Eq. (.1.11) along the mean-field oexistene urve for a = 9/4 and failitate ompleting the square in Eq. (.1.14). This value is similar to the results of Anisimov and Sengers 6 and Agayan, 58 given the prefators for eah term in Eq. (.1.8). This extra seond term in Eq. (.1.14) subtrated from the squared term represents the referene value of loal Helmholtz-energy density. As this term is independent of density, it will not affet the overall shape or behavior of a dimensionless density profile. Therefore we will onsider only the first term in Eq. (.1.14), re-writing the expression as 1 f = u( ϕ ϕ). (.1.15) 4 The plot of Eq. (.1.15) is given in Fig... As expeted, there is overlap at ϕ = for sets of ± T ˆ values when the referene value of loal Helmholtz-energy density is inluded. In omparing these figures, it is important to note that the minima on the axis of the order parameter and shape of in eah of the plots for a given temperature distane remain the same. 1

25 Tˆ =.1 Tˆ =.5 Tˆ = Tˆ =.5 Tˆ =.1 Figure.1. Loal Helmholtzenergy density from Landau expansion (in terms of the order parameter) given in Eq. (.1.14) for u = 9/16 and varying temperature distanes, ˆT. ϕ Tˆ =.1 Tˆ =.5 Tˆ = Tˆ =.5 Tˆ =.1 Figure.. Loal Helmholtzenergy density from Landau expansion (in terms of the order parameter) given in Eq. (.1.15) whih inludes the referene value for the loal free energy density for u = 9/16 and varying temperature distanes, ˆT. ϕ Now that the first term in the loal Helmholtz-energy density in Eq. (.1.8) has reasonably been simplified to that in Eq. (.1.15), the new full expression for the loal Helmholtz-energy density beomes 13

26 1 1 dϕ = [ ϕ ϕ] + f u 4 dz. (.1.16) Then the omplete expression for the surfae of tension (Eq. (.1.)) an be re-written as d ( A + ) 1 1 kt B u[ ] d ϕ σ = = ρ ϕ ϕ dz dxdy + 4 dz, (.1.17) whih will assist in simplifying future alulations.. Approximating the Density Profile near the Interfae Now that the expression for the surfae of tension has been established, we an determine the interfaial or density profile for a symmetri fluid or fluid mixture. How an this interfae be determined? The surfae tension an be found by dividing the surfae or exess Helmholtz-energy between two bulk phases, either vaporliquid or liquid-liquid phases. 56 The inhomogeneous fluid maintains stability at the interfae beause the interfaial tension ompensates for the unfavorable gain in Helmholtz energy between the two stable densities, explained by the two terms in the Landau-Ginzburg funtional in brakets in Eq. (.1.17). To maintain stability, the generalized loal hemial potential must be onstant along the interfaial profile, meaning µ ( z) df d uϕ( ϕ ϕ ) ϕ kt B dϕ dz = = + =. (..1) As a result, the quadrati and gradient terms must be equal to one another when onsidering the fluid harateristis below the ritial point: 14

27 1 1 dϕ u[ ϕ ϕ] = 4 dz. (..) This shows that to minimize the surfae tension and attain a finite interfae, a balane between a sharp interfae and an infinitely diffuse interfae between the two bulkphase densities must be ahieved. 56 To determine the interfaial or density profile, Eq. (..) is simply solved in terms of the order parameter, or dimensionless density, ϕ : ϕ ϕ Eq. (..3) an also be re-written as u = tanh zϕ 1/. (..3) ϕ ϕ = tanh z a ( T ˆ) 1/. (..4) This dimensionless profile an also be re-written in terms of the harateristi mirosopi length sale related to the thikness of the interfae, or ξ, whih for the mean-field approximation is 1/ 1/ ξ = ( T ) a ˆ = ξ ( T) ˆ. (..5) Here, ξ is the mean-field amplitude of the orrelation length below the ritial point, 1/ ξ = a, (..6) 15

28 and the bar denotes that this is the result for the mean-field approximation. Now the dimensionless interfaial profile an be written in terms of a dimensionless harateristi length sale, z /ξ : ϕ z = tanh ϕ ξ. (..7) The behavior of this mean-field profile an be seen in Fig..3. Notie that the phase with the higher density, or larger value of ϕ / ϕ, has the lesser height on the z /ξ axis. This represents the separation of two fluids with two differing densities. It should be noted that for the purposes of this analysis, the effet of gravity on an infinite planar interfae, insofar that the height of the dividing surfae along its horizontal axis will flutuate as a result of apillary waves, 56,61 will be ignored..3 Determining the Width of the Interfae As disussed previously, the harateristi length, ξ, was hosen as the length sale to determine the thikness or width of the interfae. There are methods to approximate the interfae thikness; however, we an opt to hoose multiples of this harateristi length as a unit to represent this width to model interfaial behavior, as the interfaial thikness may our at a multiple 56 of ξ. The best and most aurate approximation of this thikness from the mean-field approximation should be determined from the gradient of the interfaial profile, whih will allow for better auray in alulating interfaial properties. 16

29 (a) ϕ ϕ Phase I Phase II (b) d( ϕ / ϕ) d( z/ ξ ) Figure.3. Mean-field dimensionless density profile (a) and gradient (b) as a funtion of height. The dimensionless density profile is given in Eq. (..7). The gradient, or the probability of the loation of the interfae, an be seen by alulating at the slope of this density profile with respet to dimensionless height: d( ϕ / ϕ) z = se h dz ( / ξ) ξ. (.3.1) 17

30 By looking at this gradient, the onept of the smooth, diffuse, or fuzzy interfae an be better visualized. As shown in Fig..4, we see a smooth, gradual transition between both phases at their interfae. Also, in the ase of the mean-field smooth interfae in Fig..4, the steepest gradient, and therefore the majority of the interfae, lies between ± ξ, yielding an approximated interfaial thikness of ξ. When looking at Fig..3(b), it an be seen that nearly 95% of the gradient s behavior is aounted for between ± ξ. Due to this differene, it is important to onsider the behavior of the gradient when hoosing the interfae width in order to make the most aurate alulation. Figure.4. Contour plot of the interfae between two fluid phases as predited by mean-field theory. The shading indiates a larger slope of the density profile. The dotted red lines bound ± ξ and the solid blue line indiates z /ξ =. 18

31 .4 Position and Density Dependene of the Loal Helmholtz-Energy Density We have disussed the equation for the symmetri smooth interfaial profile; however, it is important to remember that this determination was based on the loal Helmholtz-energy density. Therefore, the behavior of the loal Helmholtz-energy density itself should be noted. Considering the dependene on density and height, we an obtain a better understanding of the ontribution of these individual effets. Starting with the full equation for the loal Helmholtz-energy density in Eq. (.1.16), using the same approximation that the quadrati and gradient terms are equal as before, and inserting the expression for the interfaial profile in Eq. (..7), we obtain f ϕ 4 1 z = u ξ tanh 1. (.4.1) This result is a universal dimensionless expression for the Helmholtz energy density. This is a universal equation as it is independent of temperature due to the arrangement of dimensionless groups. Its three-dimensional graphial representation, illustrating the individual dependenes on density and height an be seen in Fig..5. Looking more losely at this figure, one an observe the individual affets of height and density on the loal Helmholtz-energy density (given by the broken blue line), eah ontribute to the resultant mean-field interfaial profile, represented by the solid red line. The dotted gray and solid light blue lines show the individual effets of density and height, respetively, on the loal Helmholtz energy density. 19

32 f ϕ 4 z /ξ ϕ / ϕ Figure.5. Three-dimensional universal plot of the dimensionless loal Helmholtz-energy 4 density ( f / ϕ ) and its dependene on height and density. The dimensionless loal Helmholtzenergy density is given in Eq. (.4.1). The solid red line shows the density profile in Eq. (..7) and the dashed blue line indiates the dependene of f on both vertial position and density. The gray line with irles and the solid light blue line indiate the dependene of f on density and vertial position, respetively..5 Mean-Field Estimation of Interfaial Tension As we are ultimately interested in the behavior of the surfae tension, there are several ways in whih we an approximate the surfae of tension based on the interfaial profile itself. For now, mean-field theory will be used to estimate this important engineering property. Sine we have a mean-field expression for the smooth interfae, we an now develop expressions to estimate the interfaial tension based on the thermodynami

33 properties speifi to the fluid. Referening Eq. (.1.17), the mean-field approximation for the dimensionless surfae tension is σ ρ kt B = dϕ u[ ϕ ϕ] dz 4 dz. (.5.1) In Setion., it was disussed that the energy is minimized when the two terms within the integral were equal to one another. Therefore, we an safely make this approximation by doubling either the first or the seond term in the braketed expression as follows: σ ρ kt B dϕ = u[ ϕ ϕ] dz = dz dz. (.5.) For the sake of simpliity, we will use the quadrati term to evaluate Eq.(.5.1): σ ρ kt B + 1 [ ] u = ϕ ϕ dz. (.5.3) Sine the integral will be evaluated in terms of ϕ, the seond expression in Eq. (.5.) ontaining the gradient will be used to perform a hange of variables, whih results in dz = 1 u dϕ 1/ ( ϕ ϕ ) It is important to note that while performing this hange of variables, the limits of integration hange sign beause dϕ / dz is negative (i.e. when height inreases, the density of the fluid ( ϕ ) dereases). Substituting this information into Eq. (.5.3) and integrating the expression for the dimensionless surfae tension we obtain:. 1

34 σ ρ kt B ( ) 1/ 3 = u ϕ. (.5.4) 3 As mentioned in Setion.1, these mean-field onstants an be used to desribe important thermodynami properties that an be measured experimentally. The mean-field orrelation length amplitude, given in Eq. (..6) is one of those values. Another important definition is that of phase oexistene or equilibrium, as given by Eq. (.1.11) for the simple mean-field approximation. Using this relation shown earlier, the mean-field ondition x 1/ ϕ = B Tˆ, (.5.5) where the mean-field amplitude of the phase boundary parabola, B, is B a = u 1/. (.5.6) The last important property is the isohori heat apaity, lassially defined as S C T T A V = = T ρ T ρ. (.5.7) At the ritial point, the isohori heat apaity exhibits a finite jump at the ritial temperature, whih an be represented in the mean-field approximation by a CV =, (.5.8) u for the parameters in this work, similar to the methodologies use by others. 57,58,6,6 Substituting the thermodynami properties in Eqs. (..6), (.5.6) and (.5.8) into Eq. (.5.4), the expression for the mean-field estimation of surfae tension beomes: ρ σ kt 8 ξ ˆ 3/ = CV ( T). (.5.9) B 3

35 This result in Eq. (.5.9) is the result for the mean-field approximation as determined by both Landau and Lifshitz 63 and Rowlinson and Widom. 56 The density gradient from Landau and Lifshitz s 63 work orresponds term by term with the expression in Eq. (..). The work of Rowlinson and Widom 56 defines the density of a fluid near the ritial point, and the expression for the hemial potential of this fluid orresponds term by term to Eq. (..1). These omparisons to both Landau and Lifshitz and Rowlinson and Widom are given in greater detail in Appendix A. This mean-field result for the approximation of the surfae tension will be ompared to other methods that take into aount the effet of flutuations near the ritial point in Chapter 3. 3

36 Chapter 3: Aounting for Flutuations near the Critial Point 3.1 Effets of Critial Flutuations and Fluid Asymmetry In the traditional approah of using the mean-field approximation, interpartile interations are assumed to be idential and independent of oordinate. 64 However, near the ritial point of phase separation, the orrelation length of flutuations exeeds the range of intermoleular interations, and an signifiantly modify the physial properties of the system. The result is that the interfaial profile beomes very smooth and therefore infinitely thik as the ritial point is approahed. 56 This also has impliations for the surfae tension of a smooth interfae as it beomes low as a result of strong flutuations. 7 Mean-field approahes to the thermodynamis of surfaes, 1,4,65-67 as presented in Chapter, ignore the effets of these loal flutuations. A shemati illustrating the effets of these flutuations near the ritial point an be seen in Fig The atual, or singular, diameter (indiated by the broken green line in Fig. 3.1) is urved near the ritial point, signifiantly departing from the mean-field diameter (indiated by the dotted red line). In this figure, you an see that the effet of ritial flutuations auses the atual diameter to be different from the retilinear diameter, or arithmeti mean between vapor and liquid densities, and not only hanges the shape of the oexistene urve near the ritial point, but lowers the ritial point as well. 68 As our desire is to make aurate alulations and approximations in the ritial region, methods that aount for this effet must be applied. 4

37 Figure 3.1. Shemati of an asymmetri vapor-liquid oexistene urve omparing mean-field (red lines) and saling (blue and green lines) approximations near the ritial point. The ritial points are eah represented by a irle. The dotted red line biseting the urve represents the retilinear (mean-field) diameter, the urved broken green line represents the atual (singular) diameter and the broken blak line represents the atual ritial density. Another property that must be taken into aount is fluid asymmetry, as it an impat the behavior of a fluid both near and far away from the ritial point. This asymmetry is aused by the differene in interations between phases of a single fluid or the differene in interations between two or more kinds of moleules; the larger the differenes, the larger the asymmetry. In a symmetri system, suh as the lattie-gas model with respet to density, the arithmeti mean of the oexisting densities, or diameter of the oexistene urve, is a straight vertial line, indiating a symmetri parabola. In an asymmetri system, this diameter is often urved near the ritial point and beomes a sloped straight line. 68 The slope of this line an vary dependent on the shape of the oexistene urve, and an illustration of this onept an also be seen in Fig Note that for a symmetri system, the line of ritial density (indiated by the broken blak line in this figure) would also represent the arithmeti mean, whereas in an 5

38 asymmetri system, the arithmeti mean (indiated by the green dashed line) would be sloped, similar to what is shown in Fig Saling Theory One method of aounting for the effets of flutuations as well as fluid asymmetry is known as saling theory. 69 Saling theory onnets lassial thermodynamis to experimental observations (thermodynami spae to experimental spae). This theory aounts for the fat that near the ritial point, all physial properties follow simple power laws, also referred to as saling laws, 6 and the universal exponents used within these saling laws are known as ritial exponents. Although various fluids and fluid mixtures display a wide range of behaviors, these ritial exponents for analogous properties remain universal. Saling theory establishes universal relationships between power laws, and a theory known as renormalization group theory provides a method to alulate these universal ritial exponents. 69 The basis for both renormalization group and saling theories is the divergene of the orrelation length of the flutuations in the order parameter near the ritial point. The expression for the near-ritial orrelation length has been determined to be 56,64,7 ξ = ξ Tˆ ν (..1) and Fisher and Zinn 7 and Pelisetto and Viari. 71 have established ν.63. In this setion and throughout this doument, means approximately equal and means asymptotially equal. 6

39 3..1 General Saling Equations The ritial behavior of real fluids and their mixtures an be asymptotially desribed by saling theory; 7 in other words, a system s atual behavior an be related to that of a symmetri system. This behavior is desribed in terms of two independent fields, h 1 and h, and one dependent field, h 3. The ordering or strong field the field that exhibits strong flutuations is represented by h 1 ; the thermal or weakly flutuating field is represented by h. These fields are all funtions of physial fields in real spae and an be ombined into a form that an be used in a general ase for fluids, using thermodynami potentials expressed through onjugate order parameters. The general form for an equation of state for any fluid in terms of thermodynami potentials is: h α ± 1 h3 h f, (..) α β h where f ± is a saling funtion. 51, 73 Within this funtion, the supersript ± refers to the sign of the independent field h, where h > referenes the region above the ritial point and h < referenes that below the ritial point. The ritial exponents α and β are universal in terms of ritial point universality; these exponents only depend on the type of saling or modeling used and are independent of the speifi fluid. The Ising values of α.19 and β.36 have been wellestablished and verified experimentally. 7,71,74-77 Related to the two independent fields ( h 1 and h ) are two onjugate saling densities, ϕ 1 and ϕ, where ϕ 1 represents the strongly flutuating order parameter 7

40 and ϕ represents the weakly flutuating saling density. These saling densities are defined as: ϕ dh 3 1 = dh1 h (..3) and ϕ dh 3 = dh h 1. (..4) These fields and densities an be expressed in a differential form, 51,73 dh = ϕ dh + ϕ dh. (..5) There are also three saling suseptibilities, or seond derivatives defined in terms of saling densities: χ 1, the strong suseptibility assoiated with strong flutuations, similar to ompressibility; χ, the weak suseptibility assoiated with weak flutuations, similar to heat apaity; and χ 1, the ross suseptibility. These are defined by the following equations: χ dϕ1 dh3 1 = = dh1 dh h 1 h (..6) χ dϕ dh3 = = dh dh h1 h1 (..7) and χ dϕ dh = = dh dh1h h1 h1. (..8) Sine we are ultimately interested in modeling the equilibrium onditions of the interfae, the simplifiation for zero ordering field (h 1 = ) an be applied to the 8

41 general saling equations. Zero ordering field exists along the ritial isohore and oexistene urve as shown in Fig. 3.. T CP Figure 3.. Shemati of a T-ρ diagram indiating the loation of zero ordering field (h 1 = ). The solid line represents the oexistene urve and the dashed line represents the ritial isohore; both lines together represent h 1 =. ρ The expressions for the saling densities (Eqs. (..3) and (..4)) and saling suseptibilities (Eqs. (..6), (..7), and (..8)) at zero order field beome ϕ dh 3 β β 1 = f () h ± B h dh1 h ˆ (..9) ϕ dh ± 3 α α = ( α) f() h h h h dh 1 α h1 A (..1) χ dϕ 1 β+ α ± β+ α 1 = f () h Γ h dh1 h ˆ (..11) dϕ χ α α α ± α = (1 )( ) f() h A h dh h1 ˆ (..1) and dϕ h 1 ˆ h χ1 = β f () βb (for h < ). (..13) dh h h h1 β β 9

42 Here, ˆB, the ritial amplitude derived from the order parameter; A ±, the heat apaity amplitude; and Γ ˆ ±, the suseptibility amplitude; are all system-dependent. As before, the ± supersript refer to h > and h <, respetively. The ± sign in Eq. (..9) refers to the two branhes of the order parameter, h 1 > and h 1 <, respetively, as they approah the limit h 1 = at the phase boundary. These approximations in Eqs. (..9)-(..13) were verified against the work of Agayan Complete Saling for a One-Component Fluid Now that the general saling equations have been established, we an use the onept of omplete saling, as developed by Fisher 49 and Kim, Fisher and Orkoulas 5 to model the behavior of a pure fluid. Compared to the symmetri lattiegas model, omplete saling has the ability to model the asymmetri behavior in real fluids though saling oeffiients. As our interest lies in the region near the vaporliquid ritial point, the independent theoretial saling fields h 1 and h an be written as linear ombinations of physial fields. Therefore, in linear approximation, the saling fields beome h = a ˆ µ + a Tˆ+ a Pˆ, (..14) h = b Tˆ+ b ˆ µ + b Pˆ, (..15) 1 3 h = Pˆ+ ˆ µ + Tˆ, (..16) where the sets of oeffiients a i, b i and i represent the effets of asymmetry dependent on the type of fluid. The dimensionless physial fields in the previous equations are defined as 3

43 µ ˆ µ ˆ µ kt µ µ kt (..17) B B ˆ T ˆ T T T T T T (..18) ˆ P ˆ P P P ρ kt ρ kt P. (..19) B B Although they may appear to be ompliated, the expressions in Eqs. (..14)- (..16) an be further simplified. Careful hoie of the value of the dimensionless ritial entropy, Ŝ, where 51,68 ˆ ˆ S P dp S ( ) 1 = = kbρ = k ˆ B T h 1=, dt x, (..) allows the simplifiation of ˆ 3 = S. Using the value of the ritial entropy in Eq. (..), the saling oeffiients an be further redued. From thermodynamis and in terms of these dimensionless variables used in this setion, dpˆ = ˆ ρ d ˆ µ + ˆ ρsdt ˆ ˆ. (..1) Considering this expression in zero ordering field ( h 1 = ), we find that dpˆ d ˆ µ Sˆ dtˆ = dtˆ +, h 1=, h1 =, therefore d ˆ µ dtˆ h1 =, =. (..) Also in zero ordering field lose to the ritial point, dh ˆ ˆ ˆ 1 dµ dt dp = = a a a dt dtˆ + + dtˆ dtˆ 1 3 h1 =, h 1=, h1=, h1=,. 31

44 Therefore, based on the results in Eq. (..) a dpˆ = a dt ˆ = 3 h1,. (..3) Sine only two of the physial fields are independent in eah of the equations for the thermodynami fields, only two of the oeffiients in eah equation are independent. Eq. (..14) an be signifiantly simplified 51,68 as the oeffiients a 1 and b 1 an be absorbed by the saling amplitudes in f ±, allowing a 1 = b 1 = 1. Therefore, the simplified saling equations for a one-omponent fluid beome ˆ ˆ dp h ˆ ˆ 1 = µ + a3 P T dtˆ x, (..4) h = Tˆ+ b ˆ µ + b Pˆ (..5) 3 and ˆ ˆ dp h ˆ ˆ 3 = P µ T dtˆ. (..6) x, In many ases, the saling oeffiients a 3 and either b or the sum b + b 3 an be approximated or determined experimentally. We an use the definition in Eq. (..) and the result in Eq. (..) to rewrite the temperature derivative of h in Eq. (..15), whih beomes dh ˆ dp = 1 + b. dtˆ ˆ 3, dt, h1 = h1 = (..7) Therefore, by integration of Eq. (..7) and inorporating the results in Eq. (..3), the results for zero ordering field are 3

45 h = 1 ˆ a h = T 1 b3 a 3 h = Pˆ ˆ µ Sˆ Tˆ. 3 (..8) It is important to note that the expression for h in Eq. (..8) an be further simplified by onsidering basi thermodynami definitions in terms of the dimensionless variables established thus far: ρ dpˆ ˆ ˆ and ˆ ˆ ρs dp ρ = = ρs = = ρ ˆ µ ρ ˆ d k dt Tˆ B ˆ µ. (..9) Considering Eqs. (..5) and (..), the differential expression for h 3 in Eq. (..8) an be solved in terms of the dimensionless entropy density in Eq. (..9): ( ˆ ) dh = dpˆ d ˆ µ Sˆ dtˆ = + ϕ 1+ b S dtˆ 3 3 dpˆ d ˆ µ ˆ ˆ ˆ dt ( 1 ˆ dt S = ϕ + b S ) dt dt dt dt 3 ˆ ˆ ˆ ˆ ˆ µ ˆ µ ˆ µ ˆ µ ˆ ˆ dp ˆ ρs = = Sˆ + ϕ 1+ b Sˆ ˆ dt ˆ µ ( ) 3 This follows that by the definition of the dimensionless entropy density, ( ˆ ρs ˆ ) ˆ ρs ˆ ˆ ρ ˆ ˆ ˆ ˆ S ρs S = =. (..3) the result in Eq. (..3) yields: ( ˆ ρsˆ) ϕ( 1 b ˆ 3S) = + (..31) indiating that the entropy density is proportional to the weakly flutuating saling density, ϕ. As shown here, and as disussed by Wang and Anisimov, 68 the 33

46 oeffiient b 3 does not appear to play a signifiant role in affeting fluid asymmetry and was assumed to be zero. However, later works by Wang et al. 51 onsider and inlude the effets of b 3 in an effetive onstant, beff suh that b eff b + b 1 Sb ˆ 3 =. (..3) Complete Saling for a Dilute Binary Mixture Another ase that an be modeled by omplete saling is a dilute binary fluid mixture. As two omponents are now being onsidered, an additional field is onsidered in the linear ombination of the saling fields near the ritial point, namely the exhange hemial potential ( ˆµ 1 ), suh that Here, ˆ µ ( µ µ ) 1 = 1 1 B where 1 and ˆ µ ( µ µ ) / kt 1 = 1 1 B, where 1 h = ˆ µ + a Tˆ+ a Pˆ+ a ˆ µ, (..33) h = Tˆ+ b ˆ µ + b Pˆ+ b ˆ µ, (..34) h = Pˆ+ ˆ µ + Tˆ+ ˆ µ. (..35) / kt µ is the hemial potential of the pure solvent µ is the differene in hemial potential between the solvent and solute, dependent on mole fration. Not shown are oeffiients a 1 and b 1, as only three of the saling oeffiients in eah saling field are independent, as disussed in the previous subsetion. It should be noted that all saling oeffiients and the ritial parameters depend on the loation of the ritial lous, whih an be speified by any of the four ritial parameters: 73 T, P, µ 1 and µ 1. 34

47 These saling equations for a dilute mixture an then be written in terms of onentration and density. Given that from thermodynamis dpˆ = SdT ˆ ˆ+ ˆ ρ d ˆ µ + ˆ ρ xd ˆ µ, 1 1 where x represents the mole fration of solute, the expression for onentration of solute in terms of saling fields and densities an be written as 51 x = x + a4ϕ1+ b4ϕ. (..36) 1+ ϕ + b ϕ 1 The density and entropy density an be written as 51 ˆ ρ 1+ ϕ + b ϕ 1 aϕ bϕ 1 = (..37) and ˆ ρsˆ Sˆ 1 + a ϕ + ϕ aϕ bϕ 1 =, (..38) respetively. It should be noted that the full expressions in both Eq. (..37) and (..38) are idential to those for one-omponent fluid. 51,68 Also, similar to a oneomponent fluid as disussed previously, 1 = 1, = 1 and 3 = S, while ˆ = x. 4 As given by the expression for onentration in Eq. (..36), asymmetry in the mixture must now be aounted for in the remaining oeffiients. As we are onsidering the dilute solution, where a very small amount of solute added to a pure fluid or binary fluid mixture, the oeffiient b will be approximated to be the same as for a pure fluid and unaffeted by a small addition of solute. However, the oeffiients a 4 and b 4 must be alulated from properties of the mixture itself. 35

48 Using a method developed by Anisimov et al., 78 a 4 and b 4 an be approximated using the derivatives of h 1 and h with respet to the mixture term, µ 1. Along the ritial line, h 1 and h vanish, therefore a 4 an be determined from Eq. (..33): h = = ˆ µ + a Tˆ+ a Pˆ+ a ˆ µ d dt dp d = µ + a + a + a µ dµ dµ dµ dµ To solve for a 4 in terms of dimensionless groups, we obtain: dµ 1 dt 1 dp a4 + akb + a3 (..39) dµ 1 dµ 1 ρ dµ 1 Note that also along the ritial line, µ 1 = kt B ln x, therefore: d dx kt x µ 1 = B. (..4) The unknown temperature and pressure derivatives in Eq. (..39) an then be determined as dt dt dx dt x = = (..41) dµ dx dµ dx k T 1 1 B and dp dp dx dp x = = dµ dx dµ dx k T 1 1 B dp x dpˆ = x. (..4) dx k T dx ρ B The unknown derivative dµ 1 / dµ 1 an be found from the Gibbs-Duhem relation: 36

49 N dµ = SdT + VdP i i dµ = SdT + VdP xdµ. 1 1 Therefore, at ritial onditions dµ = S dt + V dp x dµ (..43) 1 1 Taking the derivative of Eq. (..43) with respet to dµ 1 the expression beomes dµ dt dx dp dx = S + V x. (..44) dµ dx dµ dx dµ As the referene value of entropy of the fluid is ompletely arbitrary (as only the value of the entropy differene is of onern), the ritial value of entropy an be hosen in suh a way that is simplifies the overall expression. In this ase, the definition in Eq. (..) will be used, where S kb( P/ T) h1 = ˆ ˆ. Given this =, relation and noting that V = 1/ ρ, Eq. (..44) beomes dµ 1 x P dt dp = + ρkt B. (..45) dµ 1 ρ kbt T h 1=, dx dx Noting that the Krihevskii parameter is defined as K dp P dt, (..46) dx T dx h 1=, Eq. (..45) an be re-written as dµ x dµ ρ ( K ρ k T ) 1 = B. (..47) 1 kbt The Krihevskii parameter represents is the differene in slope between the ritial line and saturation urve and applies the physial harateristis of the mixture and its dependene on onentration of solute. 37

50 Now that the derivatives have been determined in terms of known and measurable values, the expression in Eq. (..3) an be re-written with dimensions as a 1 P = a 3 ρ kb T h 1=,. Given this expression, Eq. (..39) an be simplified as ( ) a4 x ˆ 1+ a3 K 1, (..48) noting that the dimensionless Krihevskii parameter is defined as Kˆ K / ρkbt. Similar to the method used previously for the saling oeffiient a 4, a relationship an be established between the saling oeffiient b 4 with respet to oeffiients b 1, b and b 3 along the ritial line, where h =, using the method developed by Anisimov et al., 78 this time in terms of dimensionless variables: b k dt + b dµ + b 1 dp 1 4 B 3 dµ 1 dµ 1 ρ dµ 1. (..49) Using the derivatives previously determined in Eqs. (..4), (..4) and (..47), the expression for the saling oeffiient b 4 an be written as dtˆ b x + b ( Kˆ 1) + b dx 4 3 dpˆ. (..5) dx Depending on how the effets of b and b 3 are to be onsidered, there are two possible treatments of their approximation. As they annot be determined independently by experiment, they an be onsidered in two ases: one approximation 38

51 in whih b = b 3, and in the other approximation, where b 3 =. In the ase of the approximation were b = b3, Eq. (..5) beomes dtˆ ˆ ˆ dp b4 x + b K 1 + (for b1 = 1 and b = b3). dx dx (..51) Alternately, in the ase of the approximation where b 3 =, Eq. (..5) beomes dtˆ 4 ( ˆ b x + b K 1 ) (for b1 = 1 and b3 = ). dx (..5) These expressions an now be used to model the behavior of real dilute solutions, aounting for asymmetry. Similar to the effet of solute onentration on the saling oeffiients a 4 and b 4, the saling oeffiients a 3 and b an also be impated by onentration. Cerdeiriña et al. 79 and Wang and Anisimov 68 have shown that these saling oeffiients show a strong dependene on moleular volume. Therefore, when there is a large differene in moleular volume between solvent and solute, espeially when a large amount of solute is present, the alulation of fluid density is affeted. Based on the findings of Wang and Anisimov, 68 a potential solution to this issue is to approximate the dependene of a 3 on the hange in ritial volume with a hange in ritial onentration of a fluid mixture as x dv a ( x ) a +, (..53) 3 3 V dx where a 3 is the saling oeffiient of the pure solvent. Note that the orretion fator is added to the pure omponent. This is beause a 3 should inrease if the ritial 39

52 density dereases when the onentration of solute is inreased (i.e. dv / dx is positive). Conversely, an inreased onentration of solute with a smaller moleular volume and larger ritial density (i.e. dv / dx is negative) will derease the asymmetry of the solution. In a similar manner, we an approximate that the oeffiient b is also dependent on onentration as, x dv b ( x ) b + k (..54) V dx where b is the saling oeffiient of the pure solvent and k is an empirial onstant approximated to be k ~1/3 in this work, based upon the findings of Wang and Anisimov. 68 Sine the values of b tend to be muh smaller in magnitude 68 relative to a 3 and due to a redution by an empirial onstant k, the oeffiient b may not have a signifiant impat on fluid asymmetry. 3.3 The Smooth Interfae Sine the modeling of real fluids, whih now takes into aount the affet of ritial flutuations, the same should also be onsidered for the behavior of the smooth interfae. The mean-field approximation of the smooth interfae was disussed in Chapter ; however, the effet of ritial flutuations is not onsidered in this method. To aount for these flutuations, and their impat on the smooth interfae, the results of renormalization group theory an be applied. In general terms, renormalization group theory utilizes the Ising model, aounting for flutuations near the ritial point by inluding the effet of diretional spin 4

53 densities. 56 This theory shows that the universal expression for the density profile an be written in the form, ϕ =Ψ ( z / ξ ), (.3.1) ϕ similar to what was expressed in Chapter. Ohta and Kawasaki 8 suggested that the profile for the surfae an be written as follows to more aurately fit the atual density profile for a given substane: a Ψ ( z/ ξ) = tanh( z/ ξ) 1+ seh ( z/ ξ) 3+ a 1/ (.3.) where 3 a= πε ε = 4 d, 6 and d is the number of dimensions. The same oordinate system as before, where the more dense fluid lies in the z diretion, is applied to Ohta and Kawasaki s result in Eq. (.3.). Note that when ε = (for d = 4 ), the mean-field approximation result is obtained as in Eq. (..7). A omparison of these results an be seen in Fig. 3.3, where the profile gradient is smoothed further when aounting for the effets of ritial flutuations. The interfae thikness an be approximated in terms of ξ by assessing the magnitude of the profile gradient of the renormalization group theory results shown in Fig This figure illustrates the dimensionless density gradient based on Eq. (.3.). The shading indiates where the magnitude of the gradient is the greatest, from whih we an estimate the interfae thikness to be ± ξ, or ξ. This an be 41

54 onfirmed by a visual inspetion of Fig 3.3, where the largest gradient of the density profile resides between ± ξ. ϕ ϕ Figure 3.3. Comparison of mean-field and renormalization group theory density profiles as a funtion of height. The mean-field theory profile (based on Landau expansion of the loal Helmholz energy density, shown in Chapter ) in Eq. (..7) is represented by the dashed red line. The results renormalization group (RG) theory 76 in Eq. (.3.) are represented by the solid blue line. Figure 3.4. Contour plot of the interfae between liquid and vapor phases as predited by renormalization group theory. 8 The shading indiates a larger slope of the density profile. The solid blue lines bound ± ξ and the two solid red lines bound ± ξ. 4

55 The signifiane of the interfaial thikness is highlighted by the example of a sub-miron droplet of water near the vapor-liquid ritial point. A simple simulation of this droplet was reated using toolboxes in MATLAB developed by Adomaitis and Chen 1,11 and is shown in Fig Based on this simulation, the interfaial thikness ξ.r, therefore, for a droplet nm in radius, the interfaial width would be 4 nm. Taking into aount Eq. (..1) for the orrelation length of flutuations in density, approximating the orrelation length amplitude as moleular size ( ξ.1 nm ), and the ritial temperature of water ( T 647 K ), this simulation represents a temperature ~. K from the ritial point of water. Figure 3.5. Simulation (applying toolboxes in MATLAB developed by Adomaitis and Chen 1,11 ) of a smooth interfae for a droplet near the ritial point of fluid-fluid separation using RG theory. 76 The intensity of shadowing represents the square of the density/onentration profile in Eq. (.3.) The thikness of the interfae, given by a harateristi deay-length sale of the profile, is indiated by ξ. It should be noted that sine the thikness of the smooth interfae is mesosopi in sale and an extend from nanometers to mirons, the droplet size annot be defined smaller than this harateristi thikness, as there would be no 43

56 distintion between the behaviors of the bulk phases. This thikness is dependent on the properties of the fluid, but in the ase of the symmetri interfae given in Figures 3.3 and 3.4, the properties of the bulk phases are not apparent until ± 6ξ. 3.4 Estimations of Surfae Tension Similar to the alulation of surfae tension for the mean-field approximation given in Chapter, definitions of orrelation length and amplitudes will be employed. However, in these alulations, the effets of ritial flutuations will be aounted for. The methodologies disussed will inlude the results from renormalization group theory, a loal approximation of the slope of the renormalization group theory profile and dimensional saling analysis Renormalization Group Theory Results Not only does renormalization group theory provide the results for an interfaial profile, but this theory an also assist in the estimation of surfae tension. This simple alulation an be made by using two definitions from this theory. The surfae energy of an area of interation, obtained from the Ising lattie model and renormalization group theory has been determined by Zinn, 81 ( ) σ ξ + (.4.1) kt B.377 where the + sign indiates the value above the ritial point. This equation an be divided by the two-sale universality fator, 48,7 ( ξ ) Aρ.171, (.4.) 44

57 whih yields the following expression for the oeffiient of surfae tension, based on the renormalization group theory, for amplitudes above the ritial point: σ. ρ kta + ξ. (.4.3) + B Similarly, we an alulate the surfae tension above the ritial point by using the ratio of orrelation lengths above and below the ritial point, 8 + ξ 1.96 ξ, (.4.4) and the ratio of the ritial value of the heat apaity above and below the ritial point, 7 + A.53, (.4.5) A where ± indiates the value above and below the ritial point, respetively. Applying the ratios in Eqs. (.4.4) and (.4.5) to Eq. (.4.3), the oeffiient of surfae tension for amplitudes below the ritial point is σ.6 ρ kta ξ. (.4.6) B 3.4. Dimensional Saling Analysis We an also develop a more general result for the surfae tension using dimensional saling by estimating the surfae tension as the produt of the density of the Helmholtz-energy density and the width of the vapor-liquid interfae. Beause we are only onsidering the behavior below the ritial point along the oexistene urve, we only need to onsider only one variable in this ase, temperature as 45

58 density along the oexistene urve is a funtion of temperature. The Helmholtz energy per unit volume sales as 8 A A T V ( α)(1 α) α ˆ. (.4.7) Therefore taking into aount this relation, the expression for the orrelation length in Eq. (..1) and assuming the interfae thikness is ξ as before, the expression for the dimensionless surfae tension an be approximated as σ A A ξ = ρ kt V ( α)(1 α) B ξ ˆ T α ν. (.4.8) Noting that the universal exponents α ν = ( d 1) ν = ϑ. (.4.9) Therefore, the relation in Eq. (.4.8) an be written in terms of the general exponent in Eq. (.4.9): σ A ρ kt ( α)(1 α) B ϑ ξ ˆ T. (.4.1) Note that van der Waals theory, 8,54,56 whih neglets ritial flutuations, predits an exponent of ϑ = 3/. The expression for the dimensionless surfae tension oeffiient is σ 1.19 ρ kta ξ, (.4.11) B where in the ase of d = 3, Tˆ ν σ = σ. (.4.1) Above the ritial point, taking into aount Eqs. (.4.4) and (.4.5), this expression for the surfae tension oeffiient beomes 46

59 σ 1.16 ρ kta + ξ (.4.13) + B If the surfae tension is alulated for an interfae thikness of ± ξ, or 4ξ, these oeffiients would be σ.37 ρ kta ξ (.4.14) B and σ.3 ρ kta + ξ (.4.15) + B for values below and above the ritial point, respetively. ϕ ϕ Figure 3.6. Representation of the interfae thikness onsidered for the dimensional saling analysis alulation of surfae tension in Eqs. (.4.11) and (.4.13)-(.4.15). The profile plotted is the result from RG theory in Eq. (.3.). The red box highlights the portion of the density profile used to alulate the surfae tension in the dimensional saling analysis between ± ξ. The green box highlights the portion of the density profile between ± ξ. 47

60 A graphial representation of the estimation of the interfaial thikness for the alulations in Eqs. (.4.11) and (.4.13)-(.4.15) an be seen in Fig Note that a large portion of the profile slope is negleted by the small box, whereas the large box is more inlusive of the behavior of the gradient. Table 3.1. Comparison of mean-field and saling methods for the estimation of surfae tension oeffiients. Method Coeffiients for Amplitudes below the Critial Point Equation Mean-field Approximation Renormalization Group Theory Dimensional Saling Analysis ( ξ ) Dimensional Saling Analysis ( 4ξ ) σ 8 ρ ktc V ξ = (.5.9) 3 B σ.6 ρ kta ξ (.4.6) B σ 1.19 ρ kta ξ (.4.11) B σ.37 ρ kta ξ (.4.14) B σ Experimental Results a ρ kta ξ N/A a Referene 83 B Comparison of Methodologies This setion and Setion.5 both address methods to approximate the surfae tension at an interfae. A summary of these approximations is ompared to experimental results in Table 3.1. Note that the result of renormalization group theory is well within the experimental range of values for the amplitude oeffiients. The dimensional saling analysis with the large interfae thikness ( 4ξ ) also 48

61 provides an estimate lose to experimental values as ompared to the mean-field approximation and the results of the dimensional analysis with the small interfae thikness ( ξ ). These results highlight both the need to take into aount the effets of ritial flutuations as well as a areful hoie of the interfaial width. 49

62 Chapter 4: Curved Interfaes and Surfae Tension As disussed previously, surfae tension is ruial for many engineering appliations inluding droplet formation, miroemulsions, nuleation, flow through miropores and apillary ation. Usually, the effets of urvature are ignored as they are onsidered negligible on the marosale. However, as proesses are saled down to onsider small droplets, suh as aerosol prodution or modeling of biologial ells, surfae urvature an play a more signifiant role on the alulation of surfae tension. 4.1 Tolman s Length: Curvature Corretion to the Surfae Tension Young-Laplae Approximation of Surfae Tension A well-known starting point for the alulation for the surfae tension is the Young-Laplae equation, P P = σ α β +. (4.1.1) R1 R Here, P α and P β represent the pressures of the phases on the onave side (or inside) and outside, respetively, and the radii R 1 and R represent the radii of urvature. The surfae tension, σ, is onsidered to be that of an infinite plane without urvature. The Young-Laplae equation an be simplified for spherial bubbles and droplets when all radii of urvature are equivalent ( R= R1 = R), therefore 5

63 σ Pα P β =. (4.1.) R As before, the pressures P α and P β represent the pressures of the phases on the inside and outside, respetively, of the bubble or droplet. This is illustrated in Fig α R β Figure 4.1. Simple shemati illustrating the properties of the variables used in the Young-Laplae relation for surfae tension at the interfae between two phases, α and β Expression for Tolman s Length It is ommonly believed that the effet of urvature on surfae tension is negligible for alulations on the marosale, therefore the surfae tension of an infinite plane an be used for alulations on this sale as seen in Eq. (4.1.1). Some have refuted the laim that there is any effet of urvature on the surfae tension; however, many believe that for very small droplets and bubbles the surfae tension is affeted by strong urvature.,16,-3,6,7,44,88,89 In 1949, Tolman onsidered the effet of droplet size on the alulation of the surfae of tension as previously desribed by Young and Laplae and determined there should be a orretion parameter. This parameter, known as Tolman s length 51

64 (δ ), represents the differene in radius between the equimolar surfae ( R e ) and surfae of tension ( R ) in a droplet of liquid,,16,19,56 δ = R R, (4.1.3) e and an be applied to the expansion of the planar surfae tension as δ σ( R) = σ R. (4.1.4) This is ommonly known as the Laplae-Tolman equation. Note that Eq. (4.1.3) establishes the sign of the Tolman length as negative for droplets and positive for bubbles, whih has been verified by other studies. 89 An illustration of this onept for a droplet of liquid an be seen in Fig. 4.. As it an be seen from Eq. (4.1.4), the smaller the radius R, the more signifiant Tolman s length beomes. δ Figure 4.. Sketh of the onept of the Tolman length ( δ ) for a droplet of liquid, where R is the radius of the surfae of tension and Re is the radius of the equimolar surfae. R e R 5

65 A more general equation for the surfae tension was developed later by Helfrih, 89 whih inludes a seond-order orretion due to urvature in the expansion of surfae tension, δ k σ( R) = σ R R, (4.1.5) where k is a onstant that aounts for the rigidity of the interfae. However, this equation is not widely aepted as studies indiate that higher order terms will not likely provide a more aurate result. 37 Therefore, the fous of this study will be on the Laplae-Tolman expression in Eq. (4.1.4) Preditions of the Behavior of Tolman s Length Studies of the Tolman length have yielded various results. For sharp interfaes (as mentioned in Chapter ), where the vapor-liquid or liquid-liquid interfae thikness is on the order of the size of a moleule, Tolman s length is expeted to be very small. This is one of the reasons that a signifiant urvatureorretion has only been believed to affet nano-sized droplets. The sign of Tolman s orretion to the surfae tension has been a subjet of debate as well. Some studies have onfirmed that the Tolman s length is negative for droplets, 16,,6,9,31 while moleular dynamis simulations have indiated that it may be positive. 3,9,91 Other studies have found that the sign of Tolman s length annot be de determined with muh ertainty, while Lei et al. 4 and Guermer et al. 34 ite that the sign is only negative for droplets that are not lose to moleular size and, in the ase of very small droplets, the surfae tension atually vanishes. 53

66 There are also varying onlusions regarding the behavior of the Tolman s length for a smooth interfae near the ritial point. Square gradient theories have provided results onsistent with mean-field theory, 43,44 as disussed in Chapter. It should also be noted that symmetri systems, suh as the lattie-gas and regular solution models, do not exhibit an effet on Tolman s length, partially as a result of the values of the mean-field ritial exponents. Gránásy 6 and Moody and Attard 9 predited that the Tolman length would beome inreasingly negative when approahing the ritial temperature. Moody and Attard 9 also indiate that the Tolman length may even hange sign very lose to or at the ritial temperature. These findings further support the ambiguity of the Tolman length behavior near the ritial point due to the effet of ritial flutuations. 43 Studies that aknowledge the existene of Tolman s length have given many different results: Tolman s length is finite, logarithmially divergent, or algebraially divergent with varying exponents. Phillips and Mohanty 9 suggested that the Tolman length diverges with the same exponent as the orrelation length of ritial flutuations( ν.63), while others made different preditions. Fisher and Wortis 93 predited two terms: one weakly diverging term with an exponent of.65 and the other, more strongly diverging term with an exponent of.13. Using the penetrable-sphere model developed by Widom and Rowlinson, 94 and Rowlinson 19,35 determined that the exponent for the Tolman length showed a weak divergene (.65). These results are ommonly aepted, as they are supported by an exat result using saling obtained by Fisher and Wortis

67 There have been more reent developments in saling that more properly aount for fluid asymmetry. Fisher and Orkoulas 49 reognized the orretion needed to model this behavior and developed an approah known as omplete saling, whih was elaborated by Kim et al. 5 Anisimov and Wang verified this approah and further developed it for onvenient use in pratial appliations, 68,95 and others have applied these results to binary fluids. 79 Anisimov 45 reently indiated that a omplete saling approah to alulating the near-ritial Tolman length yields a muh stronger algebrai divergene than shown in previous works, with an exponent of β ν.34. This effet is the result of ritial flutuations and its amplitude is a funtion of fluid asymmetry; therefore this behavior does not exist in the mean-field regime. This result showing a strongly divergent behavior indiates that the magnitude of the Tolman length may result in a signifiant urvature orretion alulation in highly asymmetri fluids and fluid mixtures. 4. Tolman s Length Near the Critial Point As fluid asymmetry, in addition to urvature, impats the Tolman s length 19,43,93 the method of omplete saling an be used to aount for both fluid asymmetry and ritial flutuations as disussed in Chapter 3. For a general derivation of the Tolman length using saling theory, Anisimov 45 related the pressure differene in the Laplae-Tolman equation to the suseptibility with respet to flutuations in density. The pressure differene between the inside of a droplet of liquid and its surrounding vapor is derived from two parts: a symmetri part whih is independent 55

68 of speifi fluid harateristis and an asymmetri part, whih is speifi to the fluid, ( ) ( ) P P = P P + P P. (4..1) α β α β sym α β asym As the Laplae-Tolman equation (Eq. (4.1.5)) aounts for both urvature and asymmetry in the Tolman length, δ, this expression an be used to estimate the total pressure differene as σ δ + 1 sym asym R R. (4..) ( Pα Pβ ) ( Pα Pβ) Sine the Tolman length is attributed to asymmetry, 45 we an assume that symmetri ontribution to this pressure differene is related to the planar surfae tension desribed in Eq. (4.1.) ( Pα Pβ ) sym σ =. (4..3) R Therefore, ombining this equation with Eq. (4..), the ratio of Tolman s length with respet to the radius of surfae tension beomes 45 ( α β) ( ) asym α β δ P P R P P sym. (4..4) Anisimov 45 suggested that this same pressure differene an be approximated using the ritial part of the grand-anonial potential per unit volume ( Ω / V = P) as Pr, multiplied by the orrelation length, as Pr is related to the surfae tension. Sine R is the length sale in Laplae s pressure differene equation, the orrelation length, ξ, is the orresponding length sale for the pressure differene Pr. Following the saling approximation, 56

69 ( P P ) R~ ( Pr ) α β ξ sym sym and ( P P ) R ~ ( Pr ). α β ξ asym asym Therefore, Anisimov s 45 result is obtained for Eq. (4..4) as ( r ) ( P ) δ P δ ξ r asym sym (4..5) where δ is a universal onstant, estimated as.6.7 from saling theory. Close to the ritial point, Pr an be approximated as 45 r ( ) P χ µ, (4..6) where the total suseptibility with respet to density flutuations is defined as χ ρ P = = µ µ T T. (4..7) Just as with the pressure differenes, the total suseptibility an be written in terms of a symmetri and asymmetri part, analogous to the pressure in Eq. (4..) and the ritial part of the grand-anonial potential in Eq. (4..5) so that χ = χ + χ. (4..8) sym asym Therefore Eq. (4..5) for the Tolman length-orrelation length ratio an be re-written in terms of the suseptibility in Eq. (4..6) as δ ξ χ asym δ. (4..9) χsym In order to use omplete saling as disussed in Chapter 3, Eq. (4..9) must be re-written in terms of saling variables. The total suseptibility in Eq. (4..8) is also a 57

70 ombination of the strong, weak and ross saling suseptibilities in Eqs. (..11)- (..13). Based on the results developed by Kim 96 and Kim et al. 5 for omplete saling, keeping only the lower order terms ( ϕ 1, ϕ and ϕ 1 ) and eliminating any produt terms (e.g. ϕϕ 1 ), the following result for the dimensionless total suseptibility an be obtained: ˆ χ (1 + a ) (1+ 3 aϕ ) 5 bϕ χ + ( b + b ) χ + (1 + a )( b + b ) χ (4..1) To further simplify this equation, the strength of the divergene of eah term will be ompared, and only the most dominant terms will be kept. In the ase of the one-omponent fluid, in the first ε -approximation where γ = ν 1.6, the temperature diverges as β γ ϕχ ~ T ~ T ϕχ ~ T ~ T 1 α χ ~ T ~ T 1 α γ (4..11) χ β ~ T ~ T. Given these results, the total suseptibility in Eq. (4..1) an be simplified to ˆ χ (1 + a ) (1 + 3 aϕ ) χ + (1 + a )( b + b ) χ. (4..1) By applying the results of saling for a one-omponent fluid in zero ordering field ( h 1 = ) where ˆ = a ˆ, (4..13) h T 1 b3 T a3 58

71 the order parameter and strong and ross saling suseptibilities given in Chapter 3 in Eqs. (..9), (..11) and (..13) an be approximated as ϕ Bˆ 1 T β χ Γˆ ± 1 T γ (4..14) χ β 1 ˆ 1 βb T. Substituting these relations in Eq. (4..14) into the expression for the total suseptibility in Eq. (4..1) results in ˆ χ (1 + a ) (1+ 3 a Bˆ T ) Γˆ T + (1 + a )( b + b ) Bˆ T. (4..15) β ± γ β β In terms of the approximation where b 3 =, Wang and Anisimov 64 determined the amplitudes ˆB and Γ ˆ ± to be Bˆ B = 1 + a ˆ Γ Γ =. 3 ± ± ( 1+ a ) 3 (4..16) This provides the result for the total suseptibility in terms of variables with dimensions: ˆ 3a ˆ χ 1 ˆ Γ + ˆ ± γ 3 β β 1 T B T bβb T 1+ a3 where the symmetri and asymmetri parts were determined to be 45. (4..17) ˆ χ Γˆ ± sym T γ 3 1 ˆ a β ˆ χ Γ B ˆ T b B ˆ T. ± 3 β γ asym β 1+ a3 (4..18) 59

72 Substituting these results into the expression for the Tolman length-orrelation length ratio in Eq. (4..9) yields the new expression δ 3a ˆ ˆ β B δ B T b T ξ 1+ Γ 3 β β + 1 γ a ˆ ± 3. (4..19) Applying the expression for the orrelation length in Eq. (..1), the expression for the Tolman length beomes 45 3 ˆ β ν 1 ˆ β B α β ν δ m δξ aeffb T b T ˆ ±, (4..) Γ where the prefator m refers to a bubble of vapor or a drop of liquid, respetively and a eff a 1 a 3. (4..1) + 3 This expression indiates that there are two ontributions to the Tolman length near the ritial point: one that weakly diverges as 1 α β ν.65 Tˆ = Tˆ (as shown by others 19,35,94 ) and a new term from omplete saling that diverges more strongly, as β ν.34 Tˆ = Tˆ. 4.3 General Thermodynami Expression for Tolman s Length Anisimov 45 and Wang et al. 51 also suggested a general thermodynami expression for the behavior of the Tolman length both near and far beyond the ritial region. Based on fluid asymmetry, the Tolman length with respet to the thikness of the interfae is suggested to be 45,51 δ ˆd ρ ξ m ˆ ρ ˆ ρ, (4.3.1) 6

73 where ˆρ and ˆρ are the densities in the liquid and vapor phases, respetively. In this expression, the prefator m refers to a droplet of liquid and bubble of vapor, respetively based on the definitions of oexisting phases. The diameter of phase oexistene is represented by ρ ρ ˆ ρd = (4.3.) ρ where ρ and ρ represent the liquid and vapor branhes, respetively. A simple shemati of this onept an be seen in Fig 4.3. ˆT ρ d ρ ρ Figure 4.3. Sketh of the onept of using the oexistene diameter to approximate the ratio of the Tolman length to the interfae thikness in Eq. (4.3.1). The green line represents the atual diameter, the blak dotted line is the ritial density and the blue line is the oexistene urve. Equation (4.3.1) an also be verified in the ritial regime using the leading and dominant term from Wang and Anisimov s 68 expression for the oexistene diameter ˆ ˆ d ρ ˆ ˆ d 1 aeffb T β ρ = (4.3.3) 61

74 and the definition given by Wang and Anisimov 68 for the differene in density along the two sides of the phase boundary ρ ρ Bˆ ˆ T β. (4.3.4) ρ Substituting Eqs. (4.3.3) and (4.3.4) into Eq. (4.3.1) and approximating δ / 3 yields the leading term in Eq. (4..19). Additionally, Eq. (4.3.1) has also been verified for use in the mean-field regime for a van der Waals fluid. 45 Sine these relations have shown to be universal, they will be applied to asymmetri solutions in following hapters. 6

75 Chapter 5: Tolman s Length for Polymer Solutions and Other Dilute Binary Mixtures 5.1 Polymer Solutions: Theory and Bakground Semidilute polymer solutions tend to beome highly asymmetri as the degree of polymerization or number or monomer units interonneted inreases. This is beause as the length of a polymer hain inreases, the polymer has less preferene for the solvent, and therefore solubility dereases Phase Coexistene There are two distint regimes in polymer solutions: the ritial regime and the polymer regime. The harateristi variable, w, first introdued by Widom, 97 separates these two regimes and is defined as: 1 1/ ˆ w N T, (5.1.1) where N is the degree of polymerization of a polymer hain. Widom s variable provides an indiation of where a partiular polymer solution lies on a phase diagram. When w << 1, it indiates a near-ritial polymer solution, when the degree of polymerization is small and the distane to the distane to the ritial temperature is also very small; this is termed the ritial regime. Conversely, when w >> 1 (and N ), it is only dependent on the degree of polymerization and is termed the polymer regime. 63

76 The overall behavior of polymer solutions in general an be visualized by a simple phase diagram. This behavior will be modeled in terms of known limiting behaviors and established relations in both the polymer and ritial regimes. Based on Flory theory, Povodyrev et al. 98 indiated that at lower volume frations of polymer, the relation between temperature and volume fration beomes linear, and the polymer regime in the limiting phase behavior approahes T = φ + 1, (5.1.) θ 3 where φ is the volume fration of polymer in solution and T is the solution temperature. The theta temperature, θ, is the ritial temperature of phase separation for an infinitely dilute polymer solution with an infinitely long polymer hain. This temperature is speifi to the properties of a speifi polymer and solvent ombination. In terms of our temperature distane variable, T ˆ, Eq. (5.1.) an be re-written for the volume fration of the solution: T T φ = + 3 ˆ T 1 θ θ. (5.1.3) This equation an be written in a more general form, sine the expression T / θ an be found in terms of the degree of polymerization from Flory 99 theory as T θ = + 1 ( 1 1/ N ). (5.1.4) Here, it an be seen that in the limiting ase as the degree of polymerization inreases ( N ), T θ. 64

77 In the ritial regime, analogous to the differene in density along the two sides of the phase boundary, Eq. (4.3.4) an be re-written in terms of volume fration as φ φ Bˆ ˆ T β (5.1.5) φ where φ is the polymer-rih phase and φ is the solvent-rih phase. This result is similar to that presented by Widom 97 and the experimental results presented by Dobashi et al. 1 The amplitude ˆB an be determined for polymer solutions of moderate to high degrees of polymerization, aounting for the results of Anisimov et al. 11 suh that: Bˆ B N β /, N = 1, (5.1.6) where B ˆ, N = 1 1 is determined from the three-dimensional Ising model, whih takes into aount ritial flutuations. 5 The development and N-dependene of the ritial amplitude in Eq. (5.1.6) will be disussed in greater detail in Setion 5.. Then, in terms of the Widom variable, w, Eq. (5.1.5) beomes: φ φ ( w) β. (ritial regime) (5.1.7) φ In the polymer regime, based on the results from Anisimov et al., 51 the two sides of the phase boundary an be approximated as: φ φ φ 3. (polymer regime) φ φ w (5.1.8) Combining what is known for both the ritial and polymer regimes in Eqs. (5.1.7) and (5.1.8), respetively, a simple rossover expression an be developed to model 65

78 the behavior in both regimes as well as approximate the behavior in between. This simple rossover expression for the phase boundary is then β 3 ( w) 1+ w φ φ. (rossover) β 1 φ ( + w) (5.1.9) Lastly, based on the works of Widom 97 and Anisimov et al., 11 the solvent-rih branhes in the ritial and polymer regimes are given by: ( w N) 1/ φ 6 / ( w ) φ. (5.1.1) φ 1/ N ( w ) Therefore, a simple rossover expression an be developed to onnet the polymer and ritial regimes for the solvent-rih phase: φ φ 6w N + ( 1 6w). (rossover) (5.1.11) The general behavior of a polymer solution with respet to temperature distane to the ritial point and the volume fration of polymer (φ ) from Eqs. (5.1.3), (5.1.9) and (5.1.11) is presented in Fig In this figure, the phase separation urve is represented by the solid blue line and the arithmeti mean of phase oexistene (or diameter, φ d ) is represented by the dotted green line. Similar to the oexistene urve for an asymmetri fluid shown in Fig. 4.4 in Chapter 4, asymmetry is evident from the slope of the diameter. The red line represents the phase-limiting behavior at an infinite degree of polymerization ( N ), whih terminates at the theta temperature, θ, as predited by Flory theory

79 Also, similar to the fluids desribed previously the same urved diameter is observed near the ritial point, seen in the inset in Fig As before, this urvature is a result of flutuations as the ritial point is approahed. ˆT Figure 5.1. Phase diagram for a polymer solution ( N = 1 ) near the ritial point. The solid blue line represents the phase separation urve. The dotted red line represents the phase-limiting behavior at an infinite degree of polymerization, whih originates at the theta point (or theta temperature) as predited by Flory theory. 99 The oexistene diameter ( φ d ) is illustrated by the green dotted line. φ and φ represent the volume frations of polymer-rih and solvent-rih phases, respetively. The inset shows a region very near the ritial point Flutuations Semidilute polymer solutions exhibit flutuations in two different order parameters that beome oupled and ontribute equally: one is a salar parameter 67

80 assoiated with the polymer volume fration, and the other is a vetor parameter assoiated with the behavior of long polymer hains. 1 The former is an observable quantity, while the latter an be modeled by de Gennes radius of gyration 13 for a polymer hain, R g, Rg = a N, (5.1.1) where a is the step size in a random walk. The radius of gyration will be the parameter representing the orrelation length of flutuations for the polymer hain order parameter, whih is assoiated with the onentration of hain ends, or probability that the hain ends will meet one another. 13 Asymptotially lose to the ritial point of phase separation, the orrelation length of onentration flutuations will dominate and beome larger than the polymer hain s radius of gyration, R g. In this regime, a polymer solution behaves similarly to an ordinary fluid near its ritial point. As the solution approahes the theta point, the radius of gyration will beome larger than the ritial flutuations, and the solution will exhibit theta-point triritiality. 11 This behavior is very lose to mean-field behavior, with logarithmi orretions required as a result of flutuations. 97 These saling onepts, along with rossover between ritial fluid behavior and triritial theta-point behavior, will be applied to alulate the interfaial thikness and Tolman s length in polymer solutions in this hapter. The limiting behaviors that bound this analysis are low and very high degrees of polymerization as well as temperatures very near and far away from the ritial point. For the infinite polymer hain, the polymer volume fration tends to zero as the orrelation length, or radius of gyration, diverges. This ondition, when present 68

81 with a temperature above the theta temperature (θ ) where the seond virial oeffiient is zero 99 is analogous to a ritial state. By appropriate tuning of the seond virial oeffiient, this ritial state an be modified to a first-order phase transition where the polymer hain ollapses onto itself and phase separation ours. This separation would result in the introdution of a new onentration-based order parameter. 5. Tolman s Length in Near-Critial Polymer Solutions Complete saling is a universal approah to modeling fluid behavior, and individual harateristis are taken into aount by the saling oeffiients. Therefore, this method will be used to model the behavior of a polymer solution and determine the Tolman length near the ritial point of separation. In the ritial regime, where w << 1, the Tolman length for a polymer solution will be determined from Anisimov s 41 result given in Chapter 4 (Eq. (4..)): 3 ˆ β ν 1 ˆ β B α β ν δ m δξ aeffb T b T ˆ ±. (5..1) Γ 5..1 Approximation of Saling Coeffiients As the degree of polymerization and volume fration affets the asymmetry in a semidilute polymer solution, these are the parameters that will be onsidered in determining the saling oeffiients in omplete saling. To approximate these values, Flory theory 99 of polymer solutions will be used, where the Gibbs energy of mixing, mix G, is mix G kt B φ = (1 φ) ln(1 φ) + ln φ+ χφ(1 φ). (5..) N 69

82 and the polymer-solvent interation parameter, χ, as presented in this hapter, is ω χ =. (5..3) RT The variable ω represents the moleular interations between solvent moleules, polymer moleules and eah with themselves, T is the solution temperature and R is the universal gas onstant. Assuming a high degree of polymerization where N, the theta temperature an be expressed as 99 θ = ω / R, therefore the polymer-solvent interation parameter an be expressed as θ χ =. (5..4) T Sine the dimensionless hemial potential of a fluid mixture is an indiator of its asymmetry, Wang and Anisimov 7 found the saling oeffiients a eff (from Eq. (4..1)) and b to be a eff a ˆ µ 1 ˆ µ = = 1+ a 3 ˆ µ 5 ˆ µ (5..5) and b 1 ˆ µ ˆ 1 1 µ 4 ˆ µ ˆ ˆ. (5..6) 11 µ 11 5 µ 3 The notation ˆ i + µ = j ˆ µ / ˆ φ i Tˆ j in Eqs. (5..5) and (5..6) represent the ij derivatives of the hemial potential taken at the ritial point, where the volume fration is analogous to density. The Flory theory expression in Eq. (5..) an also be written in terms of the hemial potential by taking the first derivative of the Gibbs energy of mixing with respet to solvent volume fration: 7

83 mix µ ( G / kbt) T ˆ µ = = kt B φ T T 1 1 ˆ N θ µ = lnφ ln(1 φ) ( 1 φ) T + + N N T. (5..7) Noting that the dimensionless volume fration is φ φˆ, (5..8) φ the derivatives needed to solve for the saling oeffiients in Eqs. (5..5) and (5..6), evaluated at the ritial point beome ˆ µ ˆ µ T φ φ θφ ˆ µ = = φ = + 1 ( φ / φ) φ T 1 φ T T N T ˆ µ ˆ µ T φ 1 ˆ µ = = φ = ( φ / φ) T φ T T ( 1 ) N φ ˆ µ 3 ˆ µ T φ 1 ˆ µ 3 = φ 3 = 3 = + 3 ( φ / φ) φ T T T ( 1 φ ) N ˆ µ ˆ µ 4 ˆ µ T φ 1 = 4 = φ 6 4 = 4 ( φ / φ) φ T T T ( 1 φ ) N ˆ µ ˆ µ φ 1 ˆ µ φ T T T N 11 = = T = + ( φ / φ) ( / ) φ ( 1 φ) ˆ µ ˆ µ φ 1 ˆ µ = = φ =. T T T N T ( φ / φ) ( / ) φ ( 1 φ ) (5..9) The derivatives in Eq. (5..9) an also be written in terms of known quantities for polymer solutions, suh as the degree of polymerization, N. Aording to Flory 71

84 theory, 99 the ritial temperature and ritial volume fration of a polymer solution are T = θ ( 1+ 1/ N ) (5..1) and 1 φ =, (5..11) 1 + N respetively. Applying these relations to the derivatives in Eq. (5..9) yields the following: T(1+ 1 N) N + 1 ˆ µ 3 = θ N N (1+ 1 ) 1 4 T N N ˆ µ = 6 θ N ˆ µ 11 N + N = N N (5..1) ˆ µ =. 1 Finally, substituting these relations into Eqs. (5..5) and (5..6) gives the result for the saling oeffiients a eff and b in terms of the degree of polymerization: a eff 3 N 1 = 5 N + N (5..13) b a eff N N + N N, (5..14) whih is the result of Anisimov and St. Pierre. 14 7

85 Plots of Eqs. (5..13) and (5..14) an be seen in Fig. 5.. As shown in Fig. 5., the saling oeffiient a eff approahes 3/5 as the degree of polymerization inreases. Also, the oeffiient b follows a power law funtion as it approahes an infinite degree of polymerization. Figure 5.. Plots of aeff and b against inverse degree of polymerization (1/N) for the range N = 1 to 1 5. Plots for aeff and b are based on Eqs. (5..13) and (5..14), respetively. 73

86 5.. Approximation of Critial Amplitudes The neessary ritial amplitudes to alulate the Tolman length an also be determined for polymer solutions. The amplitudes, not surprisingly, are also dependent on the degree of polymerization. Anisimov et al. 11 and Anisimov and Sengers 15 determined that the ritial amplitudes (as the average moleular volume beomes infinitely large) beome 1 (1 γ ) 1 N Γ = a g B bλ N = 1/ 3 1/4 1/ 6(6 v ) ( β 1)/ (5..15) 1/ N ξ r (1 ν )/ where a and u are system-dependent onstants from Landau expansion disussed in Chapter, λ is a onstant related to the seond virial oeffiient, v is the moleular volume, is a system-dependent onstant related to the Ginsburg number, r is the bare interation range for N = 1, and g is a universal numerial prefator. The ritial exponent γ follows the Ising relation α + β + γ = ; therefore, γ Taking into aount the definition of dimensionless suseptibility for polymer solutions, ( φ / φ) ˆ χ ˆ µ Tˆ, (5..16) the expressions for the ritial amplitudes an be made dimensionless and simplified to fous only on the dependene on the degree of polymerization, suh that 74

87 ( 1+ N ) N ( 1+ 1/ N ) (1 γ )/ Γ kt B, N = 1 φ Γ ˆ = =Γ, N = 1 φ ( ) ˆ B B = = B 1+ N N ( β 1)/ (5..17) ξ ξ N r N (1 ν)/ (1 ν)/, N = 1. Based on the approximation of for Flory theory, 11 we an assume that the oeffiient is of the order of monomer or fragment size where ξ, N = 1 r. The oeffiients in Eq. (5..17) an also be approximated from different theories. Based on the Ising model for the mean-field approximation ( N = 1), the amplitudes are Γ, N = 1 1 and B, = ,98 However, as disussed previously, the N value of B, N = 1 dereases to unity ( B, N = 1 1) due to ritial flutuations in the threedimensional Ising model for short-range interations, 5 therefore this value will be used Tolman s Length near the Critial Point Sine the ritial amplitudes and saling oeffiients an all be approximated in terms of the degree of polymerization as shown in Eqs. (5..13), (5..14) and (5..17), we an determine the Tolman length for a polymer solution based on the degree of polymerization, temperature distane to the ritial point using Eq. (5..1). This equation an be made dimensionless and universal for polymer solutions by dividing the Tolman length by the moleular size, r, suh that m. (5..18) δ (1 ν )/ 3 β ν β B 1 α β ν N δ aeff B T b T r Γ 75

88 A three-dimensional graphial representation of the dimensionless Tolman length for a droplet of polymer in solution, based on these approximations and the assumption that δ / 3, is given in Fig Notie that in this figure there is a dramati inrease in Tolman s length lose to the ritial point and at a relatively high 4 degree of polymerization ( N = 1 ). The magnitude of the Tolman length also appears to be more strongly dependent on the degree of polymerization, and thus asymmetry, relative to temperature distane. δ r 1/ N Tˆ Figure 5.3. Three-dimensional plot of the dimensionless near-ritial Tolman length for a droplet of polymer in solution (from Eq. (5..18)). It is assumed δ /3 and ξ r., N = 1 76

89 Tˆ = 1 4 δ r Tˆ = 1 3 Tˆ = 1 Figure 5.4. Plots of the two terms from the dimensionless Tolman length for a droplet of polymer in solution at three temperatures. Shown here are Eqs. (5..18) and (5..19) for the full and partial expressions, respetively Comparison of Terms Anisimov s 45 expression for the Tolman length given in Eq. (5..18) an also be further simplified for a more simple approximation. By omparing the ontribution of eah term in this equation where 77

90 first term: 3 β ν (1 ν )/ N a ˆ ˆ δ eff B T m β Bˆ 1 α β ν seond term: ± N b Tˆ, δ (1 ν )/ ˆ Γ (5..19) we an determine a way to simplify the omplete expression for Tolman s length. In the first term, temperature depends as ˆ.34 T, showing a muh stronger divergene as ompared to the weak divergene in the seond term,.65 Tˆ. A visual omparison of the two terms is in Eq. (5..19) is made in Fig In this figure, we an see that the ontribution of the first term dominates by far, due to the magnitude of the ritial exponents. The first term diverges strongly while the seond term is nearly independent of temperature and only indiates a slight differene at high degrees of polymerization. As a result of this omparison, a simplifiation an be made for engineering approximations whih takes only the first term into onsideration. The equation for the near-ritial Tolman length (Eq. (5..18)) an also be expressed in terms of Widom s variable w in Eq. (5.1.1). Considering Eqs. (5..13), (5..14) and (5..17) at a moderate to high degree of polymerization ( N 1 ), these values beome: 3 aeff ( N 1) (5..) 5 3 b ( N 1) N (5..1) 5 ˆ ( 1) N ( )/ Γ N Γ, 1N γ = (5..) 78

91 ˆ / ( 1), N = 1. B N B N β (5..3) The omparison of the results in Eqs. (5..)-(5..3) for a moderate to high degree of polymerization, and the results for the full N-dependene equation in Eq. (5..17) for the amplitudes Γ ˆ and ˆB an be seen in Fig Note that there is little differene in the plots for Γ ˆ and ˆB, whih indiate little differene for values of N 1. By applying the values in Eqs. (5..1)-(5..3), as well as the definitions of w and radius of gyration, the expression for Tolman s length in polymer solutions near the ritial point beomes δ 3 β ν 1 α β ν m Ra δ g eff w βw (5..4) when Γ, N = 1 1 and B, N = 1 1 using the three-dimensional Ising model, and 1 β ν α β ν 1. The seond of the two temperature-dependent terms in this equation diverges weakly, as w w 1 α β ν.65, and the leading term diverges more strongly as w β ν w.34. A omparison of the ontribution of these two terms indiates that the only term of signifiane is the leading term, espeially when approahing the ritial point as shown in Fig Therefore, the expression for the Tolman length in the ritial regime simply beomes whih assumes δ / 3. aeff Rg x β δ m ν. (ritial regime) (5..5) 79

92 ˆB Γˆ Figure 5.5. Comparison of N-dependent amplitudes ˆB and Γˆ for a moderate to high degree of polymerization. Eqs. (5..) and (5..3) represent the high N estimate and Eq. (5..17) represents the equation for all values of N. 5.3 Tolman s Length at High Degrees of Polymerization As the orrelation and Tolman s lengths at high degrees of polymerization (when w >> 1) are expeted to behave differently than in the ritial regime, a new set of equations will be developed. In the mean-field approximation, where β = ν = 1/, Eq. (5..5) for the ritial regime would be independent of temperature ( δ m aeff Rg ), a signifiant departure from the results of omplete saling. Some 8

93 studies indiate similar mean-field-like results are expeted for the polymer regime. 13,15,16 As disussed in Setion 5.1, for an infinite degree of polymerization, the limit of phase separation approahes an asymptoti line that extends from the theta temperature, shown in Fig At this limit of the infinite degree of polymerization, the radius of gyration of a polymer hain beomes signifiantly greater than the orrelation length of ritial flutuations. The orrelation length at a high degree of polymerization, ξn, then beomes dependent on the radius of gyration and is given by degennes 13 saling theory for the semi-dilute phase of a polymer solution: ω 1 ξn ~ r χ ~ r RT. (5.3.1) At a high degree of polymerization, the theta temperature beomes θ = ω /R, and Eq. (5.3.1) beomes ξ N ~r θ T T 1. (5.3.) At a high degree of polymerization when w, sine N at any given temperature, we an approximate the theta point temperature distane as θ T T Tˆ. (5.3.3) Therefore, Eq. (5.3.) for the polymer orrelation length an be expressed as 1 ˆ, (5.3.4) ξ N r T or in terms of the Widom variable (Eq. (5.1.1)) and radius of gyration (Eq. (5.1.1)), the equation beomes 14 81

94 ξ N Rw 1. (5.3.5) g Here, it an be assumed that the size of the monomer, r, is on the same order as the random walk step size in a polymer hain ( r a). This expression in Eq. (5.3.5) supports a similar result obtained by Szleifer and Widom 1 for the mean-field (Flory) approximation for semi-dilute polymer solutions. It should be mentioned that there is some disagreement over the proper temperature variable in the polymer regime. De Gennes 13 and Broseta 17 stated that the temperature variable away from the ritial regime ( w >> 1) should be defined as ( T T)/( T ) θ instead of T T T T ˆ ( )/ as shown in this work and others 97,14 beause it aounts for the temperature at whih monomer-monomer interations vanish. However, as disussed further by Widom, 97 the temperature variable T ˆ and related variable w provide omparable results to this suggested temperature variable when ompared to experimental results. As presented in Chapter 4, the asymmetry of a solution as indiated by the oexistene urve an assist in determining the Tolman length within droplets of solution. Moving away from the ritial regime toward the polymer regime, the oexistene diameter beomes linear and its slope represents the asymmetry of the solution. As previously indiated in Eq. (4.3.1), this diameter an generally be used to determine the Tolman-orrelation length ratio; in terms of volume fration, this equation beomes 14 δ φ ξ φ φ m d (5.3.6) 8

95 where φ is the volume fration in the polymer-rih phase and φ is the volume fration in the solvent-rih phase. By this definition, the prefator " " refers to the orretion for a polymer-rih droplet in solution and the " + " refers to the orretion for a droplet of solvent. Away from the ritial point as the solution diameter ( φd = φd φ) for a highly asymmetri solution approahes one half of the distane between the oexisting phases ( φ φ ), the Tolman-orrelation length ratio for polymer solutions in the polymer regime beomes 14 δ ξ N = m 1. (5.3.7) This expression for w >> 1 is quite dramati, as it indiates that the Tolman length is expeted to be as large as half of the interfae thikness, if the interfae is defined as ξ. This result is onsistent with Rowlinson s 31 and Anisimov and St. Pierre s 14 preditions that the Tolman length is expeted to be on the same order as the interfae thikness for small droplets. 5.4 Crossover Expression for Tolman s Length in Polymer Solutions Now that the behavior of Tolman s length has been determined for the limiting behavior in both the ritial and polymer regimes, the behavior between these two limits an be modeled. In order to ahieve an understanding of the Tolman length in this unknown region, simple rossover expressions will be developed based on the Widom variable, w, at the known behavior at the limit N and approahing the ritial point. As before, the defining parameters for the two regimes 83

96 are w and the size of flutuations in the polymer hain relative to the radius of gyration: Critial regime: w<< 1 and R << ξ g Polymer regime: w<< 1 and R. g Crossover Correlation Length Sine the basis of the approximation of Tolman s length is relative to the thikness of the interfae, whih is based on the size of the flutuations in a polymer solution, the first expression developed is the rossover orrelation length. As the expression for the orrelation length of ritial flutuations in the ritial regime given in Chapter 3 (Eq. (..1)), ξ = ξ Tˆ ν, (ritial regime) (5.4.1) is universal for any fluid, we an apply this same expression to polymer solutions in the ritial regime, taking in the system-dependent amplitude ξ in Eq. (5..17). In writing this expression in terms of Widom s w, it beomes 14 ν sine 1. ξ = Rw ν g, (ritial regime) (5.4.) As disussed in the previous setion, the limiting behavior in the polymer regime is given by Eq. (5.3.5). Combining what is known of this behavior in the polymer regime with that is known for the ritial regime in Eq. (5.4.), we propose as a simple rossover expression: 14 ξ Rw ν g ν ( 1+ w) 1. (rossover) (5.4.3) 84

97 A simple hek of this rossover expression yields the limiting expressions: for large values of w, ξ is the result for the polymer regime (Eq. (5.3.5)) and for small values of w, ξ is the result for the ritial regime (Eq. (5.4.)) Crossover Tolman-Correlation Length Ratio The rossover onept an be taken a step further and applied to the ratio of the Tolman length to the orrelation length to obtain a good approximation of the size of the urvature orretion relative to the interfae thikness of a droplet of polymer in solution. As before, this orretion for a droplet of polymer-rih phase would be negative, whereas the orretion to a solvent droplet would be positive. Referening Eq. (4..19) in Chapter 4, one of the universal equations for the Tolman-orrelation length ratio, δ 3a ˆ 3 β 1 ˆ β B β + γ m δ B T b T ξ 1 a ˆ ±, (5.4.4) + 3 Γ we an determine the ratio speifially for polymer solutions. As before, the harateristi variable, w and the system-dependent amplitudes and oeffiients for polymer solutions for moderate degrees of polymerization (Eqs. (5..1)-(5..3)) an be applied to this equation, therefore: δ ξ 3 β 1 α β m a δ eff w β w. (5.4.5) Given that the first term diverges more strongly and the results of the term analysis in Setion 5. show that the first term dominates by far, we an simplify the equation to: δ aeff w β, (ritial regime) ξ m (5.4.6) 85

98 where, as before, we an assume 41 δ /3. Sine the oeffiient a eff 3/5 at moderate degrees of polymerization ( N 1 ), the saling oeffiient an be approximated as aeff 1, as given by Anisimov and St. Pierre. 14 Therefore, the ratio of Tolman length relative to interfaial thikness in Eq. (5.4.6) simply beomes δ w β. (ritial regime) ξ m (5.4.7) As the Tolman-orrelation length ratio for the polymer regime has been established to be ( ) δ / ξ = m 1 in Setion 5.3, the following rossover expression N between the polymer and ritial regimes an be developed: β δ aeff w m. (5.4.8) 1/ β β (1 + aeff w) ξ As presented by Anisimov and St. Pierre 14 for a simple approximation at moderate degrees of polymerization, Eq. (5.4.8) an be simplified to β δ w m. (5.4.9) β (1 + w) ξ This equation an provide a good approximation of the Tolman s length orretion to the surfae tension relative to the orrelation length of flutuations Crossover Tolman s Length Lastly, an expression that bridges the gap between the polymer regime limit and the ritial regime limit an be developed for the Tolman s length. Combining the relationships developed in Eqs. (5.4.3) and (5.4.8), the rossover Tolman s length expression beomes: 86

99 δ a R w m. (5.4.1) β ν eff g 1/ β β 1 ν (1 + aeff w) ( 1 + w) For the sake of simpliity, Eq. (5.4.1) an be written as δ Rw m, (5.4.11) (1 + w) β ν g 1+ β ν as presented by Anisimov and St. Pierre. 14 Again, these simple rossover equations an be tested by onsidering the ritial and polymer regime limits. When w is very small, Eq. (5.4.1) redues to the expression for the Tolman length in the ritial regime in Eq. (5..5). Equation (5.4.11) redues to the simplified expression Rw β δ m ν g, (ritial regime) (5.4.1) where aeff 3/5 1. When w is very large, Eqs. (5.4.1) and (5.4.11) redue to the expression for the polymer regime (Eq. (5.3.5)). Figure 5.6 illustrates the universal Tolman s length behavior for a drop of polymer-rih phase in solution as predited by Eq. (5.4.11). As expeted, these results show a divergene of the Tolman s length with inreasing degree of polymerization and dereasing proximity to the ritial point of phase separation. Further analysis of this methodology revealed that this rossover expression is in agreement with the seleted harateristi variable, w. Fig. 5.7 illustrates the rossover temperature dependene of the Tolman s length between the ritial and polymer regimes for a given degree of polymerization. For the ase of 4 N = 1, areful examination of the intersetion of the lines tangent to the ritial and polymer regime behaviors ( β ν.34 and 1, respetively) showed the rossover 87

100 temperature to be 1.67, yielding a value of unity ( w 1 Tˆ 1 ), whih is in very lose agreement with the definition used in Widom s 97 harateristi variable. δ r 1/ N ˆ T Figure 5.6. Universal behavior of the dimensionless Tolman length with respet to degree of polymerization and temperature distane to phase separation for a polymer-rih droplet as predited by Eq. (5.4.11) Two simple examples further illustrate the effets of Tolman s length as shown in Fig Consider a polymer solution with a degree of polymerization 4 N = 1 and a monomer size (or random walk step size) of r. nm and a temperature distane from the ritial point of separation, 4 Tˆ 1. When the random walk step size is approximated as the monomer size ( a r ), a droplet of this 88