Copyright. David Samuel Simmons

Coyright by David Samuel Simmons 2009

The Dissertation Committee for David Samuel Simmons certifies that this is the aroved version of the following dissertation: Phase and Conformational Behavior of LCST-Driven Stimuli Resonsive Polymers Committee: Isaac Sanchez, Suervisor Nicholas Peas Krishnendu Roy Venat Ganesan Thomas Trusett

Phase and Conformational Behavior of LCST-Driven Stimuli Resonsive Polymers by David Samuel Simmons, B.S. Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosohy The University of Texas at Austin December, 2009

To my grandfather, who made me an engineer before I new the word and to my wife, Carey, for being my artner on my good days and bad.

Acnowledgements I am extraordinarily fortunate in the suort I have received on the ath to this accomlishment. My adviser, Dr. Isaac Sanchez, has made this ublication ossible with his advice, suort, and willingness to field my ideas at random times in the afternoon; he has my dee areciation for his outstanding guidance. My thans also go to the members of my Ph.D. committee for their valuable feedbac in imroving my research and exloring new directions. I am liewise grateful to the other members of Dr. Sanchez research grou Xiaoyan Wang, Yingying Jiang, Xiaochu Wang, and Fran Willmore who have shared their ideas and rovided valuable sounding boards for my mine. I would articularly lie to exress areciation for Fran s donation of his own ost-graduation time in assisting my research. I d furthermore lie to than Dr. Gabriel Luna-Barcenas, with whom I had many educational conversations during his time visiting our grou. I also have the great fortune of having received receiving invaluable mentorshi in my undergraduate years and before. My great thans go to Dr. Anuj Chauhan for rearing me for graduate school and beyond, and for teaching me the imortance of scientific rigor. I am liewise deely grateful to Dr. Leonard Pinchu, whose mentorshi began in my high school years and who oened many doors that led me to a career in engineering research. My final thans go to the Cocrell School of Engineering and Mr. John W. Bartholow, Jr. for their generous financial suort through the Thrust 2000 Endowed Graduate Fellowshi. v

Gratitude cannot begin to exress my areciation for the suort of my family, which has made all things ossible. My arents rofound faith in me has given me the confidence to ursue my dreams. My sister s certain belief in my character has driven me to live u to what she sees in me. My wife s daily encouragement, trust, and examle have heled me grow into the erson I had always hoed to become. Moreover, her graceful atience with the glow of a lato at odd hours of the night has been essential to the timely comletion of this wor. vi

Phase and Conformational Behavior of LCST-Driven Stimuli Resonsive Polymers David Samuel Simmons, Ph.D. The University of Texas at Austin, 2009 Suervisor: Isaac C. Sanchez Several analytical mean field models are resented for the class of stimuli resonsive olymers that are driven by the lower critical solution temerature (LCST) transition. For solutions above the olymer crossover concentration, a hybrid model combines lattice-fluid excluded volume and van-der-waals interactions with a combinatorial aroach for the statistics of hydrogen bonding, hydration, and ionic bonding. This aroach yields models for the LCST of both neutral olymers and lightly charged olyelectrolytes in aqueous salt solution. The results are shown to be in semiquantitative agreement with exerimental data for the cloud oint of olyethylene oxide (PEO) in aqueous solution with various salts, and some asects of the lyotroic series are reroduced. Results for lightly charged olyelectrolytes are comared to and shown to be in qualitative agreement with asects of exerimentally observed behavior. Finally, a framewor is established for extension of these models to further asects of the lyotroic series and olyelectrolyte behavior. At the nanoscale, lattice fluid (LF) and scaled article theory (SPT) aroaches are emloyed to model the LCST-related coil-globule-transition (CGT) of isolated vii

olymer chains in highly dilute solution. The redicted CGT behavior semiquantitatively correlates with exerimental results for several olymer-solvent systems and over a range of ressures. Both the LF and SPT models exhibit a heating induced coil-to-globule transition (HCGT) temerature that increases with ressure until it merges with a cooling induced coil-to-globule transition (CCGT). The oint at which the CCGT and HCGT meet is a hyercritical oint that also corresonds to a merging of the lower critical and uer critical solution temeratures. Theoretical results are discussed in terms of a generalized olymer/solvent hase diagram that ossesses three hyercritical oints. Within the lattice model, a dimensionless transition temerature Θ is given for a long chain simly by the equation B B Θ = η ( Θ) (, where Θ ) is the bul solvent 1 s occuied volume fraction at the transition temerature. Furthermore, there is a critical value of the ratio of olymer to solvent S-L characteristic temerature below which no HCGT transition is redicted for an infinite chain. η s viii

Table of Contents List of Tables... xi List of Figures... xii Chater 1. Introduction... 1 1.1. Polymer Solution Phase Behavior... 1 1.1.1. Characteristics... 1 1.1.2. Physics of the LCST... 7 1.2. The Coil-Globule Transition... 10 1.3. Charges in Solution... 14 1.3.1. Effects on stability: the lyotroic series... 14 1.3.2. The Debye-Hucel model... 17 1.3.3. Polyelectrolytes... 18 1.4. Aroaches to the Chemical Potential... 23 Chater 2. The Lower Critical Solution Temerature... 26 2.1. Review... 27 2.1.1. Lattice fluid model... 27 2.1.2. Hydrogen bonding lattice fluid model... 30 2.2. Theory... 35 2.2.1. Aqueous olymers in the resence of free salt... 35 2.2.2. Polyelectrolytes... 42 2.2.3. Solution Stability... 54 2.3. Alications, Results, and Discussion... 60 2.3.1. Neutral olymer in aqueous salt solution... 60 2.3.2. Polyelectrolyte... 67 2.4. Conclusions... 73 Chater 3. Single Chain Conformational Behavior... 75 3.1. Theory... 76 3.1.1. General model... 76 ix

3.1.2. Lattice model... 79 3.1.3. Scaled article theory... 84 3.2. Results... 89 3.2.1. Overall ressure-temerature behavior... 89 3.2.2. Physics of the transition... 92 3.2.3. The high ressure hyercritical oint... 95 3.2.4. The CGT near the solvent vaor ressure... 96 3.2.5. Chain conformation through the CGT... 101 3.3. Conclusions... 104 Chater 4. Towards a Model for the Aqueous CGT... 107 4.1. CGT with Veytsman Statistics... 107 4.1.1. Physical contribution... 109 4.1.2. Hydrogen bonding contribution... 109 4.1.3. Chain gyration radius and coil-globule transition... 110 4.2. Basis for a Consistent Aroach... 112 Chater 5. Conclusions... 115 Aendix 1. Nomenclature... 116 Aendix 2. Extended Derivations... 123 A.2.1. Solution Stability... 123 A.2.2. Scaled Particle Theory Coil-Globule Transition... 124 Aendix 3. Mathematica code... 129 A.3.1. LCST in Aqueous Solution with Salt... 129 A.3.2. Lattice Fluid CGT... 160 References... 168 Vita... 179 x

List of Tables Table 1: Comarison of theoretical HCGT temeratures at solvent vaor ressure with exerimental and theoretical LCSTs for olyisobutylene in various solvents. Solvents are groued by carbon number. SPT results are unavailable for some solutions due to resent limitations on arameter availability for this model... 99 Table 2: Comarison of theoretical and exerimental results for Θ P for P = 0 various systems. For the LF model, theoretical calculations are based on olymer molecular weights chosen to match those associated with each exerimental result. For the SPT model, calculations are at infinite molecular weight, which is exected to cause little error due to the large size of the chain. SPT results are unavailable for some solutions due to resent limitations on arameter availability for this model.... 100 xi

List of Figures Figure 1: Temerature-comosition hase diagram for wealy interacting olymer solution. The stried gray area is the metastable region and the solid gray area is the unstable region... 3 Figure 2: Effect of olymer molecular weight on solution hase behavior. Curves may reresent either sinodal or binodal loci. Dashed curves corresond to increasing chain molecular weight in the direct of the dotted line.... 3 Figure 3: Polymer solution and blend hase diagrams. Behaviors include LCST only (a), UCST only (b), LCST and UCST curves with multile extrema (c and d). merged LCST and UCST (e), closed immiscibility loos (f), and combinations of LCST, UCST, and closed immiscibility loo behavior (g and h). Curves may reresent either sinodal or binodal curves... 4 Figure 4: Schematic master curve of olymer solution hase behavior in ressuretemerature sace. The curve reresents the sinodal or binodal curve of the system. The gray area denotes two-hase states whereas the white region denotes single-hase states. Dashed lines indicate hase boundaries lacing definitive exerimental confirmation.... 6 Figure 5: Qualitative schematic of mixture entroy of as a function of comosition for a system above the LCST. The dotted straight line denotes the unmixed entroy. The curved dashed line denotes the mixture entroy neglecting equation of state effects. The solid curve reresents the actual mixture entroy, with equation of state effects... 9 Figure 6: Schematic roosed master curve for ressure- temerature behavior of the CGT. The white region corresonds to a coil state, the gray to a globule state, and the curve to the CGT itself. Points to the left of P min and to the right of P max corresond to a CCGT, while oints between P min and P max corresond to an HCGT... 12 Figure 7: Plot from wor by Florin et. al. 80 showing early exerimental results for the lyotroic series. Points denote cloud-oint measurements for PEO / water solutions with varying concentrations of salts. Triangles corresond to data with KI, circles to KBr, squares to KCl, and diamonds to KF. Curves are simly a visual aid.... 15 xii

Figure 8: Conformational diagram of a wea olyelectrolyte based on Monte- Carlo simulations 105. The x-axis, ε vdw, is the van der Waals interaction in units of B T and is thus an inverse temerature axis. Yellow sheres denote charged monomers whereas blue shere denote uncharged monomers... 20 Figure 9: Sinodal curves for aqueous PEO in the resence of salt at various concentrations. Salt roerties have been manually adjusted to aroximate exerimental results for the LCST of PEO with KI. Numbers on curves are salt occuied volume fraction corresonding to that curve. Note that the aarent meeting oint of the three curves is not truly a single oint uon closer insection.... 62 Figure 10: Cutout of isothermal ternary hase diagram for water-peo-salt system, shown in Figure 11. Curves denote sinodals at indicated temeratures. The vertical axis corresonds to anion occuied volume fraction while the horizontal axis corresonds to olymer occuied volume fraction. Salt roerties have been manually adjusted to aroximate exerimental results for the LCST of PEO with KBr. Numbers on curves denote the system temerature for that curve. Dashed line denotes an isothermal salt-induced LCST transition at 370 K. 63 Figure 11: The ternary hase diagram for the resent system. The white region corresonds to the region shown in Figure 10. Results have not been obtained for the grey region, and the resent model would liely be inaroriate for treatment of the region corresonding to higher salt concentration due to breadown of the Debye-Hucel model.... 64 Figure 12: Schematic of ossible associations in PEO-water-free salt system. Heavy dashed lined denote hydrogen bond tyes, which are resent in the absence of free ions. Light dashed lines denote ion-diole bond tyes. Bonds that share a locus comete with one another for use of that tye of site. The heavy curve denotes a PEO chain, with a articular oxygen site note. C+ and A- denote free cation and anion, resectively.... 65 Figure 13: Exerimental cloud oints 80 (oints) and manual LCST fits (curves) based on the above theory, for aqueous PEO as a function of salt concentration for various salts. Blac curve and oints corresond to the LCST in the resence of KI, green to KBr, blue to NaCl, and red to KF. The urle line demonstrates a tyical salting-in case as redicted by this model for a hyothetical salt characterized by a strong favorable xiii

cation-water hydration interaction and wea anion-water and cationolymer interactions.... 66 Figure 14: Effect of fraction of electrolytic monomers on PEO LCST, for monomers with a very low K 0 corresonding to a strong electrolyte, in 0.05 molar aqueous salt solution. The roerties of the salt do not corresond exactly to any real salt but fall within the range of lyotroic salting-out behavior.... 70 Figure 15: LCST of aqueous PEO with electrolytic subunits substituted for 0.1% of its monomers, as a function of the K 0 of the electrolyte subunits. The vertical dashed line indicates the C of the solution (the negative base ten logarithm of the counterion concentration) in solution. The free ions do not corresond to any real salt, but are hyothetical ions constructed to fall into the tyical range of salting out behavior described above... 71 Figure 16: LCST of PEO with strong electrolytic subunits substituted for 0.1% of its monomers, as a function of anion volume fraction. The equivalent range of salt molarity is zero to about 0.7... 72 Figure 17: Plot of dimensionless CGT ressure versus temerature as redicted by equation (3.32) for an infinite chain in solution with r s = 10 and * * * ζ = T T = 2.0 Θ Θ 2T = T 2, the scaling of the. Because ( ) P S α = 1 temerature axis is such that the temerature value given for any oint on the curve corresonds to the value of Θ ~ at that ressure. The solid square denotes the critical oint of the solvent. Values to the left of the hyercritical oint Pmax corresond to an HCGT while values to the right corresond to a CCGT.... 90 Figure 18: Plot of CGT of olyisobutylene in n-entane. The solid line is the lattice fluid rediction, while the blue line is the scaled article theory rediction.... 91 Figure 19: P-T lot of the HCGT of olyisobutylene (M w = 1.66 10 6 g/mol) in various solvents. Solid lines corresond to redictions based uon equation (3.32) for the lattice fluid model for an infinite chain. Dashed lines corresond to redictions based uon equation (3.54) for the SPT model for an infinite chain. Points corresond to exerimental data 11... 92 Figure 20: Qualitative contributions by excluded volume and olymer selfinteraction energy to the RHS of equation (3.22) as a function of temerature for a system in which the olymer S-L characteristic xiv

temerature is greater than the solvent S-L characteristic temerature. Coil-globule transitions occur when the sum of these contributions is zero. 93 Figure 21: LCST and CGT data for olyisobutylene in various solvents as a function of solvent density... 94 Figure 22: Contour lot of temerature Θ Pmax at the high ressure hyercritical oint as a function of interaction ratio ζ and solvent size r s, for the LF model in the limit of infinite chain legnth. The numbered lines indicate the value of Θ Pmax along that contour. White oints indicate the osition of various olymer / solvent systems. Each vertical trilet of oints corresonds, from to to bottom, to olystyrene, olyisobutylene, and PDMS in the labeled solvent... 96 Figure 23: Quantitative lot of dimensionless transition temerature vs. the ratio of the S-L characteristic temeratures of the olymer and solvent for infinite chain length. The vertical axis is the dimensionless transition temerature Θ. Figure 23a is for infinite r s while Figure 23a is for r s equal to 10. The oint mared in red on each lot is the liquid-vaor critical oint of the solvent. The green curve denotes the critical oint of the solvent as a function of ζ. Each branch from the critical oint corresonds to a coil globule transition for a chain in a different solvent hase, as labeled on the lots. Note that in the limit of infinite r s (Figure 23b) both the gas and suercritical hases are at zero solvent density. Sub-critical data is at the solvent saturated vaor ressure while suercritical data is at the solvent critical ressure... 98 Figure 24: Plot of theoretical vs exerimental values of Θ P for systems P = 0 shown in Table 2. Blac diamonds denote LF-CGT results while blue squares denote SPT-CGT results. The 45 degree line indicates the locus of oints along which theoretical and exerimental values would agree... 101 Figure 25: Exansion factor as a function of temerature for various molecular weights of olyisobutylene in n-entane near the HCGT. The red line corresonds to a molecular weight of 10 6, the blue to a molecular weight of 10 7, and the blac to a molecular weight of 10 8.... 103 Figure 26: Polymer volume fraction as a function of temerature for various molecular weights of olyisobutylene in n-entane near the HCGT. The red line corresonds to a molecular weight of 10 6, the blue to a molecular weight of 10 7, and the blac to a molecular weight of 10 8.... 103 xv

Figure 27: Tyical lot of chain exansion factor through the ressure induced globule-to-coil transition as redicted by this model, shown as calculated for the system olyisobutylene / n-entane. Curves corresond to different temeratures. From leftmost to rightmost curve, corresonding temeratures are 353 K, 354 K, 355 K, 356 K, and 357 K. 104 xvi

Chater 1. Introduction The lower critical solution temerature (LCST) transition is a ubiquitous and fundamental mechanism by which many stimuli resonsive olymers react to environmental changes. This mechanism has drawn great interest in the develoment of synthetic smart olymers, and it is believed to contribute to the function of many bioolymers. LCST hysics drive two qualitatively distinct but related behaviors. For solutions of uncrosslined olymers above the chain overla concentration, the LCST resents as a macro-scale hase searation. On the other hand, in macromolecular systems that are either highly dilute or are covalently constrained from hase searation, the LCST taes the form of a swelling transition at the scale of the molecule or covalently constrained networ. These behaviors exist over a wide range of olymer solution roerties, from nonolar olymers in organic solvents to olyelectrolytes in aqueous solution. Desite this extensive alicability, a redictive and even qualitative theoretical understanding of these henomena is lacing in many cases. Accordingly, this wor resents a new set of models extending theoretical understanding and semi-quantitative rediction of LCST henomena to a range of reviously unaddressed systems. 1.1. Polymer Solution Phase Behavior 1.1.1. Characteristics It has long been understood that olymer solutions exhibit rich hase behavior that is qualitatively distinct from that of small molecule mixtures. Whereas small molecule solutions exhibit thermally induced mixing, as early as 1960 Freeman and Rowlinson reorted thermally induced hase searation in olymer solutions 1. The modern icture of binary olymer solution hase behavior is of a variety of ossible hase diagrams. The tyical temerature-comosition hase diagram for a wealy interacting olymer solution includes two hase boundaries, the lower corresonding to thermally induced mixing and the uer corresonding to thermally induced demixing, as 1

shown in Figure 1 2-4. For the lower boundary, the maximum temerature at which thermally induced mixing occurs is nown as an uer critical solution temerature (UCST). For the uer boundary, the minimum temerature at which thermally induced demixing occurs is nown as a lower critical solution temerature (LCST). Alternatively, these terms are sometimes used to refer to the infinite molecular weight limit of these extrema or to the entire corresonding sinodal or binodal curve. The asymmetry in the critical curves in Figure 1 originates from the asymmetry in comonent size; in general, the more dissimilar the molecular size of the comonents, the greater the asymmetry in the hase diagram. As shown in Figure 2, as the olymer molecular weight goes to infinity and the solvent molecular weight is held constant, the LCST and UCST move to zero comosition at a condition corresonding to the Flory theta oint. Beyond this tyical behavior for wealy interacting olymers, many other T-x hase diagrams exist for olymer solutions in general 2, 3, 5, 6, articularly in the resence of strong intermolecular interactions. Solutions may exhibit LCST or UCST behavior only (Figure 3 a, b), and these critical curves may ossess multile extrema 7 (Figure 3 c, d). Merging of an LCST and a UCST curve may result in a neced immiscible region, as in Figure 3e. Closed immiscibility loos are also observed (Figure 3f), as are such loos in the resence of other tyical or atyical LCST and UCST curves (Figure 3 g, h). Furthermore, the system ressure and comonent molecular weights may control which such behavior is observed. 4, 8. 2

Temerature LCST UCST Binodal Sinodal Comosition Figure 1: Temerature-comosition hase diagram for wealy interacting olymer solution. The stried gray area is the metastable region and the solid gray area is the unstable region. Temerature Comosition Figure 2: Effect of olymer molecular weight on solution hase behavior. Curves may reresent either sinodal or binodal loci. Dashed curves corresond to increasing chain molecular weight in the direct of the dotted line. 3

Temerature a b c d e f g Comosition h Figure 3: Polymer solution and blend hase diagrams. Behaviors include LCST only (a), UCST only (b), LCST and UCST curves with multile extrema (c and d). merged LCST and UCST (e), closed immiscibility loos (f), and combinations of LCST, UCST, and closed immiscibility loo behavior (g and h). Curves may reresent either sinodal or binodal curves. The ressure-temerature behavior of these hase boundaries in olymer solutions has liewise been the subject of great interest. For wealy interacting olymer solutions where the T-x hase behavior is generally of the form of Figure 1, the LCST is observed to have a universally ositive sloe with ressure. The UCST, on the other hand, may have a ositive or negative sloe, or both in different ressure ranges. Furthermore, at low ressure the UCST and LCST may meet at a so-called hyercritical oint 3. In a seminal wor 9, Konyenburg and Scott in 1968 (reublished in 1980) 4

resented a theoretical develoment of the ossible critical behaviors of binary van der Waals mixtures. This wor has been alied 3, 10 for classifying the ressure-temerature hase behaviors of various wealy interacting olymer solutions. More recently, Imre and associates have roosed 3, 4, 8, 10 that a single master curve may combine the behavior of the various Konyenburg and Scott classifications for wealy interacting olymer solutions. In articular, they argue that aarently different classifications aear only because the liquid-liquid hase boundary sometimes extends into a metastable region with resect to liquid-vaor or liquid-solid hase stability. The roosed master curve, dislayed in Figure 4, includes three hyercritical oints : one at a minimum temerature T min, one at a minimum ressure P min, and one at a local ressure maximum P max. Points on the curve to the left of P min corresond to the UCST in Figure 1, while oints between P min and P max corresond to the LCST. The high ressure hyercritical oint P max, and the UCST to its right have not been exerimentally confirmed, liely due to olymer degradation at elevated temeratures 3, 11. However, evidence for such a maximum and ensuing UCST exists in several forms. From a theoretical standoint, the ossibility of a high temerature closed immiscibility loo has been shown based on the Sanchez-Lacombe (S-L) lattice fluid model 5. Exerimentally, results in olymer-solvent systems have exhibited distinct negative curvature in the LCST, indicating the ossibility of a maximum 11, 12. In addition, the hase lines of several small molecule systems 13 as well as systems of hydrocarbons in CO 14 2 have been exerimentally shown to exhibit a maximum in ressure. Differences in the ositions of the hyercritical oints of Figure 4 in different systems then exlain the aearance of qualitatively different behaviors in revious studies. When the metastable liquid state found at ressures below the vaor-liquid equilibrium ressure (including at negative ressures) is not considered, systems with P min below this curve aear are incorrectly taen to have searate and non-contacting LCST and UCST hase boundaries. Similarly, when T min is at negative ressure, the UCST would aear to have a urely ositive sloe, and when it is at extremely high 5

ressure the system would aear to have a urely negative sloe. Furthermore, T min may be exerimentally inaccessible if it falls below the freezing temerature of the solvent. In suort of this ersective, Imre and associates have demonstrated exerimental results in which the LCST and UCST cloud oint curves continue into the negative ressure domain and meet at a reviously unnown low ressure hyercritical oint 8. P 1 Phase T min P max 0 P min 2 Phase T Figure 4: Schematic master curve of olymer solution hase behavior in ressuretemerature sace. The curve reresents the sinodal or binodal curve of the system. The gray area denotes two-hase states whereas the white region denotes single-hase states. Dashed lines indicate hase boundaries lacing definitive exerimental confirmation. 6

1.1.2. Physics of the LCST The LCST can be viewed suerficially as an inverse of the UCST in that it is characterized by a chain collase or hase searation with increasing temerature. However, the hysical basis of the LCST transition is quite different from that of the UCST. Whereas UCST hase searation is driven by attractive enthalic considerations, the LCST hase searation is driven by entroy. In articular, above the LCST, the searation is actually entroically favorable. This reversal of the usual role of entroy can be understood, deending on the system, as stemming either from so-called equation of state (EOS) effects or from secific interactions such as hydrogen bonding. The entroically driven nature of the LCST was rigorously demonstrated for a two comonent mixture by Sanchez in the following way 15. The limit of stability for a two hase mixture is defined by the condition where g is intensive free energy, and g = 0, (1.1) xx g xx 2 g 2 x P, x, (1.2) and where x is a comosition fraction conjugate to the definition of the intensive free energy. Since the two-comonent condition for stability is g xx > 0, (1.3) g xx is negative within the sinodal and ositive without it. By definition of the LCST and UCST we then have that g Alying the definition of entroy, xx T Px, > 0 at a UCST =. (1.4) < 0 at an LCST g s = T P, x, (1.5) 7

and inverting the order of differentiation in equation (1.4) reveals that s xx is always ositive at an LCST and negative at a UCST. At the sinodal gxx = hxx Tsxx = 0, (1.6) where h is intensive enthaly. Consequently, at the LCST s xx are h xx are both ositive; thus for the LCST entroics are universally destabilizing and the hase searation is urely entroically driven. This is in contrast to the UCST, at which entroics are universally stabilizing and the hase searation is driven urely and universally by enthalics. More recently, Sanchez also demonstrated formally that solution comressibility is always destabilizing (ie. a comressible solution is always less stable than the corresonding incomressible solution) and that this effect is a central element of the LCST transition 15. This fact is shown simly by searating the free energy into incomressible and comressible arts: g = a vκ a, (1.7) x 2 xx xx T v where v is intensive volume, a is the intensive Helmholtz free energy, and κ T is the isothermal comressibility. The first term is the constant volume contribution to the stability, and the second term is the comressible contribution. Since the second term is always ositive, and remembering equation (1.3) for binary stability, solution comressibility always detracts from system hase stability. This is the thermodynamic origin of the so-called equation of state effects. The hysics of the above arguments are shown at a thermodynamic level in Figure 5 for a system above the LCST. In articular, the inclusion of equation of state effects can reveal a local minimum in the mixture entroy. Since the curvature of the entroy will become ositive in this region, this tyically yields an unstable region of negative curvature in the free energy, leading to hase searation. Put another way, the maximum system entroy will be obtained by hase searation into two artially demixed states. 8

S S 2 S 1 x Figure 5: Qualitative schematic of mixture entroy of as a function of comosition for a system above the LCST. The dotted straight line denotes the unmixed entroy. The curved dashed line denotes the mixture entroy neglecting equation of state effects. The solid curve reresents the actual mixture entroy, with equation of state effects. From a molecular standoint, equation of state effects emerge from differences in size and acing between solution comonents. In essence, for mixtures of comonents that differ greatly in interaction and/or size, there can be a densification on mixing that is entroically unfavorable. A simle examle of this effect can be seen in a system of two tyes of hard sheres of greatly differing diameters. In this case, the smaller sheres can ac easily in the saces between the larger sheres, yielding a densification and overall loss of free volume. This, in turn, yields an unfavorable contribution to the free energy of mixing. In addition to the comressibility-related origin of the LCST, the LCST can also emerge from strong directional interactions such as hydrogen bonding. As in the comressibility origin, this mechanism is entroically driven. In this case, the entroic loss emerges from the reduction in degrees of freedom with hydrogen bond formation. 9

Above some temerature, it becomes more favorable for the system to hase searate in order to reduce this enalty. This second class of LCST is articularly relevant in the aqueous smart olymer systems that are of great interest in biological settings. For examle, a hydrogen-bonding-based LCST transition near hysiological temeratures drives the stimuli-resonsive behavior of Poly(N-isoroylacrylamide) (PNIPAAM), maing it an excellent candidate for use in biological systems. The above distinctions between the LCST and UCST initially emerged from the inability of the Flory-Huggins lattice fluid model to cature LCST behavior. Because this model included no vacancies, it did not allow for variable density and hence had no equation of state and could not redict the LCST. In resonse to this limitation, Flory and associates develoed a simle mean field theory that qualitatively redicted the resence of an LCST via the introduction of equation of state effects in the form of variable density 16-18. Some years later, Sanchez and Lacombe incororated equation of state effects into the original Flory-Huggins lattice framewor by introducing vacant lattice sites 5, 19, 20. The Sanchez-Lacombe lattice fluid model semi-quantitatively redicted LCST transitions for a wide range of olymer solutions and became the gold standard for this urose. This model accordingly rovides the basis for much of the theoretical develoment in the resent study. However, although this model effectively addresses comressibility driven LCST henomenon, it does not in its original form reroduce the hydrogen bonding-driven LCST. Some more recent extensions of this model will thus be invoed in order to allow consideration of aqueous systems. 1.2. The Coil-Globule Transition A arallel issue to that of olymer solution hase behavior is that of olymer satial conformation. In articular, how does the conformation of an isolated chain in solution below the chain crossover concentration reflect hase transitions in the analogous semi-dilute or concentrated solution? Flory began to address this question from a theoretical standoint with the observation that there should be a theta condition 10

at which the solvent quality is such that the olymer s attractive and excluded volume interactions exactly cancel out. Flory argued that in this state the chain would assume the ideal Gaussian Configuration of a random flight 21, 22, with its radius scaling as the root of chain length 23. Under solvent conditions better than this theta condition, he argued that the chain would assume a more extended configuration in which its radius scales as the chain length to the three fifths ower 23. Stocmayer later noted 24 that all chains should assume a collased conformation when their effective self interaction becomes strong enough; such a collase occurs for solvent conditions significantly oorer than the theta condition, with the collased globule radius scaling as chain length to the one third ower. It has since been shown that although the second virial coefficient vanishes as exected at the theta oint, the chain conformation is erturbed by the retention of a nonzero third virial coefficient 25. It has liewise been argued that ternary interactions cannot be neglected at the theta oint and that it thus does not strictly corresond to the ideal chain state 26. Nevertheless, for many uroses the chain configuration can be treated as essentially ideal in the dilute theta state. Indeed, this essential concet that emerged from Flory s wor a coil-to-globule transition (CGT) as the solvent quality dros through the Theta oint forms the basis of the resent investigations regarding single-chain conformational behavior near the LCST. The above icture of the CGT immediately suggests a strong lin with hase transition behavior; in articular, both the UCST and associated cooling-induced CGT (CCGT) occur at or near the Flory theta condition. Similarly, in 1979, Sanchez ointed out that there should be a heating induced CGT (HCGT) closely related to the LCST 27. Based on this close corresondence, the qualitative form of the master hase diagram shown in Figure 4 for wea olymers can liely be alied to the CGT as well, as shown in Figure 6. In this concetion, an HCGT locus lies between high ressure and low ressure hyercritical oints P max and P min, while CCGT loci are found at temeratures outside of these oints. In fact, simulation 28, 29 and exerimental 30 studies of oligomers in 11

suercritical solvents that have suggested the resence of a high-temerature CCGT at temeratures above the HCGT, and drawing this arallel between conformation and hase behavior thus lends further suort for the existence of a high temerature UCST in olymers. P Coil State T min P max 0 P min Globule State T Figure 6: Schematic roosed master curve for ressure- temerature behavior of the CGT. The white region corresonds to a coil state, the gray to a globule state, and the curve to the CGT itself. Points to the left of P min and to the right of P max corresond to a CCGT, while oints between P min and P max corresond to an HCGT. Possibly as a result of its relative tractability, the CCGT to the left of P min has long been the focus of theory 31-35, exeriment 36-44, and simulation 45, 46. However, the HCGT has recently received increased attention, articularly due to its connection with the functionality of biological macromolecules. Early studies in this area indicated that CGTs are of relevance in the functionality of DNA 47, 48. More recent results have 12

confirmed the resence of a coil-globule transition in DNA 49-51 and have demonstrated a relationshi between rotein coil-globule transitions and folding 52-55. Numerous studies have also documented high ressure denaturation or conformational changes in roteins 56-58 that can be understood as a ressure induced CGT. Comuter simulation 28, 29, 59-62 and exeriment 63-65 have confirmed the existence of this inverse collase transition, and further studies have suggested that it may be the dominant mechanism in many alications. Indeed, Urry argued that the LCST transition rovides a fundamental mechanism whereby roteins fold and function and whereby the energy conversions that sustain living organisms can occur at constant temerature. 66 Within the context of many dilute biological systems, this crucial LCST mechanism must tae the form of an HCGT. In addition to direct alications in single molecule systems, the CGT has useful arallels with the swelling behavior of olymer networs. Flory commented that, desite quantitative differences between the two, the single chain case may quite roerly be regarded as a submicroscoic rototye of olymer networs, and that qualitatively, the two situations are striingly similar. In articular, both tyes of system are subject to the essential forces that would otherwise drive hase searation, but they are covalently restricted from doing so. Many roosed alications for synthetic stimuli resonsive olymers see to harness these underlying hysics. The LCST-driven behavior of PNIPAAM, for examle, is commonly alied as a swelling transition that can be triggered ay hysiological temeratures 67-71. Via this mechanism, such materials have been roosed for use in controlled drug delivery due to their ability to release an absorbed drug in resonse to hysiological triggers 72. Their utility has liewise been demonstrated as rototye sensors 73, actuators, and micro-scale valves 74 for use in microfluidic systems. Recently, a limited model for an HCGT has been develoed for the case of a symmetric solvent one in which the solvent-olymer interaction, the olymer self interaction, and the solvent self interaction are all equal. 75 However, to our nowledge 13

no general model exists that redicts HCGT temerature and behavior for arbitrary combinations of olymer and solvent. A more general model, however, is available for the CCGT. Sanchez, in 1979, showed that a lattice based model for a olymer chain in vacuum could redict the gyration radius of the chain given the exerimental CCGT temerature 27. The resent wor on dilute chain conformational behavior centers on extensions of this model to the HCGT. 1.3. Charges in Solution 1.3.1. Effects on stability: the lyotroic series The resence of free salt has been shown to strongly affect 76-80 and even induce 81-84 hase and swelling transitions both in biological 85 and synthetic olymers. To comlicate matters, differing salts, even among those with equal valencies, have been shown to roduce qualitatively different affects on the solubility of macromolecules. Certain salts have a monotonic salting out effect, whereas other salts exhibit a salting in effect at low concentrations and a salting out effect at high concentrations. The lyotroic (or Hofmeister s) series rans ions in terms of relative salting-out effect. For examle, the lyotroic series for the anions fluorine, chlorine, bromine, and iodine is tyically F > Cl > Br > I 86. This series is demonstrated by Figure 7 from early wor by Florin and associates on the cloud oint of PEO in the resence of various salts 80. A central observation regarding this behavior is that in most cases the lyotroic series for anions contains far more variation of behavior than that for cations; ut another way, the effect of a salt on solution stability is tyically controlled much more strongly by it s anion than cation. This exerimental result has formed the basis for much of the investigation of the underlying hysics of the lyotroic series. 14

Figure 7: Plot from wor by Florin et. al. 80 showing early exerimental results for the lyotroic series. Points denote cloud-oint measurements for PEO / water solutions with varying concentrations of salts. Triangles corresond to data with KI, circles to KBr, squares to KCl, and diamonds to KF. Curves are simly a visual aid. At the simlest level, the salting out behavior observed in the lyotroic series could be ascribed to occuation by ions of water sites needed for macromolecule hydration. In this vein, Par and Hoffman have argued 81, based on exeriments with PNIPAAM, that direct interactions between the cation and the olymer chain drive the lyotroic series in at least some systems. Such exlanations are recommended by their simlicity in that they consider only binary interactions. However, they do not clearly 15

exlain the salting-in effect of some salts, and it has been argued that they also do not exlain the dominance of anion identity in determining salt effects 86. An alternative set of models have focused on ions effects on hydrohobic hydration of the macromolecule. Melander and Horvath have develoed a highly cited such theory based on salt-induced changes in solvent surface tension and on electrostatic interactions, which they argue naturally yields a theoretical lyotroic series 85. Alternatively, Inomata and associates have noted an excellent correlation between the B coefficient of viscosity for the anion and the temerature deression of the LCST for a small range of selected salts, and on this basis they and others 78 argue that varying effects of anions on the icelie hydration structure of water are resonsible 77. The B coefficient of viscosity is a fitting arameter for highly dilute salt solutions that is a constant of the solute molecule and is understood to relate to the ion-water interaction 78, 87, 88. In articular, anions with a ositive B coefficient (tyically small and/or olyvalent ions) are understood to augment icelie water structure formation and therefore strengthen the hydrohobic interaction between the olymer and itself. On the other hand, those with negative B coefficients (tyically large and/or monovalent ions) are understood to interfere with such structure formation and stabilize hydrohobic hydration. In a third aroach, Satoh and associates have argued that, for at least some systems, salt effects are determined by their role in hydrogen bonding hydration rather than in hydrohobic hydration of olymers 86. In articular, they argue that the ability of water molecules to hydrogen bond to the olymer is modified by their concurrent hydration of ions. In this mechanism, hydration of an anion is taen to decrease the ositive charge on the water s hydrogen atoms and increase the negative charge on the water s oxygen. This, in turn, increases water s ability to act as a hydrogen bond roton accetor and decreases its ability to act as a hydrogen bond roton donor. Hydration of a cation is taen to have the oosite effect. As a consequence, the hydrohilic hydration of a olymer that is a roton accetor will be weaened by anions and strengthened by 16

cations, with the reverse holding true for a roton donating olymer. The balance of these effects is then said to control the effect of the salt on solution stability. At this time, none of these aroaches aears to have been strongly demonstrated to fully account for the exerimental lyotroic effect. It seems liely that all of these mechanisms may lay a role in various systems: straightforward occuation of hydrogen bonding hydration sites by ions; modification of hydrogen-bonding affinities as a consequence of ion hydration; and modification of hydrohobic hydration induced by ion-driven changes in the ice-lie structure of solvating water. The first mechanism follows simly from binary interactions, whereas the latter two would seem to require the consideration of ternary or higher order interactions. 1.3.2. The Debye-Hucel model At a far simler level than the above, the most essential behavior of ions in electrolyte solution was catured by the early and still highly useful Debye-Hucel model for charge screening 89. At its core, the model constitutes a first order series exansion of the sherically symmetric Poisson-Boltzmann equation for an ion in a sea of charges. The central hysical conclusion of the model is that any such ion will be surrounded by a relative scarcity of coions and relative glut of counterions, eaing at a screening length 1 κ, given by 2 κ 8π IlB = (1.8) where I is ionic strength and l B is the Bjerrum length, corresonding to the charge searation at which electrostatic energy equals thermal energy. This is given by l B 2 βq =, (1.9) 4πε ε 0 r where β = 1 T B, q is the elementary charge, ε 0 is the ermittivity of free sace, ε r is the dielectric constant of the medium, B is Boltzmann s constant, and T is temerature. The screened electrostatic otential U ij through which any two charges of tye i and j in 17

solution will interact at distances greater than the screening length is then shown to be given by ex( κ r) βu = lbzizj, (1.10) r where z is the valency of ion and r is the sacing between the ions. The electrostatic free energy u i of any articular free ion of tye i is liewise shown to be given by βu z l 2 1 i = i κ B I 1+ κσ i, (1.11) I where σ i is the ionic diameter of an ion of secies i. Equation (1.11) may then be summed over all ions to yield the total electrostatic energy of the solution, and the activity of each ion secies may liewise be obtained. The Debye-Hucel model is effective only within significant limitations. First, it is restricted to fairly low charge densities, above which a first order solution of the Poisson-Boltzmann equation becomes inadequate. This limit can be stretched by using a higher order solution at the cost of significantly increased comlexity. On the other hand, the aroach fundamentally fails to address henomena related to strong correlations between ions such as ion bridging; similarly, it fails to roerly address multivalent ion behavior. Several studies have indicated that such henomena can lay a strong role in olyelectrolyte hase and conformational behavior 90-92. Finally, it can be inadequate in the resence of closely fixed charges as such arrangements render invalid the assumtion of sherical symmetry on which the above solution is based. This last limitation in articular has led to the develoment of ion distribution models for the secial case of olyelectrolyte systems, to be discussed in the following section. 1.3.3. Polyelectrolytes Polyelectrolyte hase and conformational behavior resent unique challenges relative to that of uncharged olymers. The underlying origin of these challenges is the resence of charges, both free in solution and fixed on the olymer chains themselves. 18

The long range nature of the Coulombic interactions between these charges yields behavior qualitatively different from that of olymer solutions without charges. In an excellent review aer 93, Dobynin and Rubinstein have given a modern view of many of the relevant issues. Phase and conformational behavior The introduction of charges to the chain bacbone introduces a rich array of conformational behavior not seen in uncharged olymers. Extensive theoretical attention has been focused over the course of several decades on the electrostatic ersistence length and modifications of chain stiffness with addition of charges to a chain 94-100. Moreover, it has long been understood that Coulombic self-reulsion can cause olyelectrolyte chains to assume a more exanded conformation than the equivalent neutral chain 101. As early as 1952 Hill further noted that such interactions may in fact distort the chain from a sherical shae 102. More recently, Dobrynin, Rubinstein, and Obuhov have shown that sufficiently charged olyelectrolyte chains in oor solvent will assume a bead neclace conformation consisting of multile sherical globules attached by strings of olymer 103, 104. Both conformations are essentially adatations to maintain the local collased globular conformation demanded by the solvent quality while rogressively increasing the average distance between lie charges on the olymer bacbone so as to reduce the associated unfavorable interaction energy. A recent simulation study 105 by Ulrich and coworers has demonstrated these and other conformational behaviors, as shown in Figure 8. At high effective charges, the chain assumes a highly extended conformation for good solvents and a bead neclace conformation for oor solvents. At intermediate charge fractions, a cigar conformation is found. Finally, as exected, at low effective charge fractions the standard uncharged globule and coil conformations are observed. However, it is imortant to note that Figure 8 omits an exected high temerature transition bac to a collased globule as the LCST is encountered at low ionization fraction. 19

Figure 8: Conformational diagram of a wea olyelectrolyte based on Monte-Carlo simulations 105. The x-axis, ε vdw, is the van der Waals interaction in units of B T and is thus an inverse temerature axis. Yellow sheres denote charged monomers whereas blue shere denote uncharged monomers. As suggested by Figure 8, the inclusion of charges on the olymer causes its hase and conformational behavior to become sensitive to qualitatively new environmental stimuli. For examle, dissociable subunits such as acrylic and methacrylic acid have been coolymerized with PNIPAAm in order to yield a H sensitive LCST 106-108. This mechanism has been roosed for use in controlled H triggered drug delivery 72 as well as in H based sensors 73. For olyelectrolytes that are olyacids or olybases, the mechanism of this sensitivity is actually counterion dissociation equilibrium. As a searate issue, solution ionic strength affects olyelectrolyte behavior due to charge screening. Furthermore, electrical fields have also been shown to induce conformational changes in olyelectrolytes 109, 110, and such effects have been roosed for use in drug delivery 111, artificial ums and muscles 112, and so on. Relative to an uncharged olymer, the inclusion of charges in the chain may either stabilize or destabilize the solution, and this effect is determined by the balance of several mechanisms. Perhas the most intuitive such mechanism is that of Coulombic reulsion between lie charges on the chain. Such reulsions are tyically exected to lead to 20

enhanced solution stability as the olymer imbibes more solvent in order to increase the searation between charges. With increasingly unfavorable electrostatics, this effect is exected to result in more exanded chain conformations exhibiting the emergence of non-isometry and longer range order. However, this mechanism may be mitigated by charge screening, and it may not be the dominant mechanism at low olymer charge densities or high screening. Furthermore, at high counterion concentrations, the bacbone charges may be effectively neutralized, eliminating this effect entirely. A second mechanism for modulation of solution stability is then altered chain hydration. In articular, fixed ions on the charge bacbone are exected to be hydrohilic and to increase the net interaction between the chain and water. However, it has been shown that this effect can be reversed if the relevant counterion concentration is sufficiently high so as to lead to a low dissociation fraction of ionizable grous on the chain 113, 114. For examle, in random coolymers of N-isoroylacrylamide and acrylic acid, the solution is destabilized relative to the neutral olymer at low H 115. This effect is attributed to intrachain hydrogen bonding 113 or attractive interactions between the roton accetor sites on the isoroylacrylamide subunits and the non-dissociated acidic sites on the acrylic acid subunits 115. Furthermore, this exlanation is qualitatively consistent with theoretical 116 and simulation 117 results inointing counterion condensation and ensuing intramolecular interactions as an origin of chain collase. Free ion distribution A ey element of the behavior of charges in solution is the balance between dissociated and associated ions, characterized by the ionization fraction. For wea electrolytes, this balance is given at infinite charge dilution by the commonly tabulated dissociation constant K 0. However, at finite charge concentrations, which are resent almost by default in the case of a olyelectrolyte wherein dissociable grous are covalently connected, the actual ionization fraction α is altered by electrostatic 105, 118-120 interactions, such that 21

K 0 1 α 1 gel = H + log +, (1.12) α T ln10 α where g el is the intensive electrostatic free energy of the system and where K 0 is called the intrinsic dissociation constant, corresonding to the value of the dissociation constant in the limit of zero charge density. In general, H here is the negative base ten logarithm of the relevant counterion concentration rather than simly of the hydrogen cation concentration. Equation (1.12) can be understood as a modified Henderson-Hasselbalch equation that accounts for long-range electrostatic interaction between charge. In rincile, this equation is all that is needed to rigorously calculate the ionization fraction of charges in a olyelectrolyte. However, determining g el f exactly can be quite challenging, articularly in the case where the Debye-Hucel aroach is inadequate. A number of alternative aroaches to determining ion distribution vis-à-vis a olyelectrolyte chain have thus been develoed. One of the earliest and most influential such attemts was the Manning condensation model 121. As in the Debye-Hucel case, this model emloys the Poisson- Boltzmann equation, but it relaces the sherical symmetry of that earlier aroach with a cylindrical geometry centered on a charged rod-lie chain. Its ey conclusion is that there is a critical linear charge density uon a rodlie olymer above which counterions condense into the immediately surrounding region to neutralize some of its effective charge. For monovalent counterions and monovalent dissociable grous uon the chain, this limit is at the oint where the Bjerrum length equals the mean distance between charges. The ractical effect of this henomenon is to render the effective charge of highly ionized olymers significantly less than would be redicted from intrinsic dissociation constants alone. Recently, more advanced models for ion distribution about the chain have been roosed. A three domain extension of the Manning model has been devised which establishes three hases of counterion behavior 122. The rincile of this model is that a Manning-lie cylindrical region around a stiff olymer chain is further embedded within 22

a much larger sherical region. Desite such imrovements, however, the alication of a model based on a rodlie olymer to flexible chains clearly resents certain limitations. In order to address this roblem, Muthuumar has more recently roosed a model for counterion distribution around flexible olyelectrolytes 123, and has shown that the behavior in this case is qualitatively different than that redicted by the Manning model. Furthermore, he and Kundagrami more recently demonstrated that the dielectric mismatch between the bul solvent and the domain immediately surrounding the olyelectrolyte may have a strong effect in biasing counterions toward the condensation on the chain 124. Several studies have also indicated the imortance of counterion valency in counterion and olyelectrolyte behavior, with evidence of ion bridging by multivalent salts 91, 92, 124, 125. 1.4. Aroaches to the Chemical Potential There are two general statistical mechanical aroaches to obtaining the chemical otential of a comonent in solution. The most common method is to begin by calculating the Gibbs artition function Ω or Gibbs free energy G of the system. The chemical otential of a comonent is then the derivative of the free energy with resect to the number of molecules of : G μ = N T, P, N j where T is temerature, P is ressure, and { j }, (1.13) { } N denotes the number of molecules of all secies other than. A second aroach to the chemical otential, initially develoed by Benjamin Widom in 1963 126, is the insertion method. The ey to this aroach is the searation of the contribution of one article within the configurational artition function. For a onecomonent fluid of N articles, the configuration artition function may be written as QN = ex( βwn) dτ 1 dτ N, (1.14) V V 23

where WN is the interaction energy of the N articles within volume V, and β is the inverse of the roduct of Boltzmann s constant and temerature. Searating out the N th article gives Q N N 1 ( ) ex( W ) = ex βψ β dτ dτ V V ( βψ ) = Q V ex N 1 1 N, (1.15) where ψ is the interaction energy of the one selected article with all the remaining N-1 articles as a function of their osition. The bracets denote an average over all ossible ositions of this article. The activity a i of the article of secies i is then given by ρi ai =, (1.16) ex( βψ ) where ρ i is the number density of secies i. The chemical otential can then be written as βμ = i i ρ λ B, (1.17) 3 ln i i i where λ i is the thermal wavelength and B i is called the insertion arameter and is given by B = ex( βψ ). (1.18) i i For systems with a hard core reulsion, the above may be simlified. For any inserted osition at which the article overlas with another, the interaction energy ψ will be infinite and the contribution of these configurations thus will be zero. It follows that equation (1.18) can be written for hard sheres as This may be rewritten exactly as B = P ex( βψ ). (1.19) i i i ( βψ ) ex β( ψ ψ ) B i = P iex i i i..(1.20) Within a mean field aroximation, the latter factor in equation (1.20) is neglected, obtaining i 24

( βψ ) B = P ex (1.21) i i i for a hard shere fluid within a mean field model. Furthermore, Sanchez, Trusett, and in t Veld showed 127 in 1999 that equation (1.21) may be alied even to non-hard shere fluids under most circumstances by taing accetable insertions to be only those resulting in a negative interaction energy. Although in rincile both of these aroaches will yield valid comonent roerties in any system, in ractice one or the other often offers considerable simlification. Furthermore, they often yield different and even contradictory results as a consequence of differences in the way concetually equivalent aroximations lay out in each aroach. For examle, the mean field aroximation used in the insertion aroach may yield quantitatively and even qualitatively different results than a mean field aroximation used in directly calculating the system artition function. As a result, the choice of aroach can be quite imortant, and attemts to mix results stemming from the two aroaches may be roblematic. 25

Chater 2. The Lower Critical Solution Temerature A model for the LCST of charge-containing aqueous olymer solutions is needed in order to facilitate understanding and design of such systems in a variety of alications. Such a model must encomass several hysical interactions: excluded volume interactions; wea van der Waals interactions including disersion forces and fixed diole interactions; hydrogen bonding; ion-diole interactions, and ion-ion interactions. Furthermore, it should account for cometition and cooeration between the ion-ion, ion-diole, and hydrogen bonding interactions, so as to reroduce and elucidate the hysics of the lyotroic series of salts as described in section 1.3.1. The hydrogen bonding lattice fluid model 128 for the LCST incororates the excluded volume, van der Waals, and hydrogen bonding interactions from the above list. The resent develoment extends this model to interactions involving charge, subject to several limitations. The first such limitation is that, with one excetion, only binary interactions will be considered. As discussed in section 1.3.1, this may omit some roosed mechanisms for the lyotroic effect that rely uon ternary or higher interactions. Secondly, the extension will focus on monovalent salts and will not be directly amenable to treating many of the henomena, such as ion bridging, that are believed to occur with higher valency salts. In fact, these two limitations are related, and a framewor develoed for treatment of ternary interactions could ossibly be modified to achieve better treatment of multivalent ions. For the resent, however, the rimarily binary aroach is recommended by its relative simlicity, and it should facilitate a useful investigation of the extent to which binary interactions lay a role in establishing lyotroic behavior. An extension to olyelectrolytes is offered that addresses the basic hysics by which incororation of charges into the olymer bacbone introduces new sensitivities of the LCST to environmental roerties. Consideration is made of the unfavorable Coulombic interactions between olymer charges that can stabilize the system at 26

sufficiently high olymer charge density and low screening. Enhanced hydration of olymer chains with addition of charges is exlicitly addressed. Furthermore, a framewor is rovided for treating interactions between non-dissociated charges on the olymer with each other and with the chain bacbone; such interactions are osited to drive destabilization of olyelectrolyte solutions at high counterion concentration. 2.1. Review Two existing models rovide many of the underinnings for the current 5, 19, 20, 128 develoment of the LCST. The Sanchez-Lacombe (SL) lattice fluid (LF) model rovides the basic mean field aroach for redicting the LCST in wealy interacting olymers. The hydrogen bonding lattice fluid (HBLF) model extends this aroach to treat hydrogen bonds via Veytsmann statistics 129. A brief recaitulation of these models follows. 2.1.1. Lattice fluid model Consider a mixture consisting of t comonents with N molecules of each comonent, at temerature T and ressure P. Further consider the system to be divided into a lattice of N r sites. Each molecule of secies occuies r such sites, with each site occuying * v volume in the ure state. The total number of sites occuied is t Nr Nr, (2.1) leaving N 0 sites unoccuied such that the total number of sites is r = 1 The fraction of occuied sites is then given by N = Nr+ N. (2.2) 0 0 Nr ρ =. (2.3) Nr + N The volume fraction occuied by secies in the mixture is then defined as 27

rn rn r r φ = x, (2.4) where x N N is the mole fraction of secies. Similarly, comonent surface fractions may be defined as t θ = φ s φ s = φ s s, (2.5) j j j= 1 where s is a surface to volume ratio characteristic of the molecule, equal to the number of contact sites er segment of molecule. This ratio was treated as unity in the original develoment of the lattice fluid model; however, it is included here in the interest of consistency with recent wor and results in the literature. The average interaction energy of a site of secies in its ure state is s =, (2.6) 2 * ε ε where ε is the interaction energy between two adjacent sites of secies. The following mixing rules are alied: v * t = φ v ; (2.7) = 1 = 1 l= 1 * t t * s ε = θθε l l ; (2.8) 2 and a Berthelot-tye rule is alied for the cross-interaction terms, given by ( ) 12 ε = ξ ε ε, (2.9) l l ll where ξ l is a dimensionless arameter exected to have value close to one. The combination of equations (2.8) and (2.9) indicates that only ratios rather than absolute values of s s are imortant. Furthermore, both s s and ξ l s have been taen to be unity successfully in a number of alications 15, 19, 20, 127. The system volume is V * = rnv v, (2.10) 28

where v is the reduced volume, which is the inverse of the reduced density ρ. Similarly, the total energetic contribution from hysical interactions is given by EP * = rnρε. (2.11) The hysical artition function is then given by 128 t N (, ) ( ) ( ) ( 0 N N Q * P T N0, N 1 ρ ρ ω φ ex βrn ρε ) =, (2.12) where ω is the number of configurations available to a chain of secies in the close aced state. This is treated as a constant of the molecule and will dro out in calculations of hase stability. The Gibbs artition function is given by = 1 ( T, P, { N} ) QP( T, N0, { N} ) ex( β PV) Ψ = N0 = 0. (2.13) The Gibbs free energy is related to the Gibbs artition function by G= Tln Ψ. (2.14) The maximum term aroximation may be alied to the Gibbs artition function as usual; the equivalent minimization condition on the free energy is ( G v) { } The free energy of the system is then given by =. (2.15) 0 T, P, N where and t 1 φ φ βg = rn ρ T + P ρt ( 1 1 ρ) ln 1 ρ+ ln ρ+ ln, (2.16) r = 1 r ω T BT T = =, (2.17) * * T ε * P v P P = =. (2.18) P * ε * From equations (2.15) and (2.16), the equation of state (EOS) of the system is 29

2 ρ P T ln ( 1 ρ) ρ 1 1 + + + = 0. (2.19) r The chemical otential of comonent from equation (2.16) is r ρ P 1 1 βμ = lnφ + 1 + r + 1 ln( 1 ρ) ln r T Tρ ρ + ρ r, (2.20) t t j 1 s + r ρ θixi θθ i j Xij i= 1 j= 1 i= 1 si where X ij is the Flory χ arameter modified to account for the surface to volume ratio arameters included in this version of the lattice-fluid model: X 12 s s = β ε + ε 2 ε. (2.21) * i * i * ij i j ij s j s j 2.1.2. Hydrogen bonding lattice fluid model Consider a mixture consisting of t comonents with N molecules of each comonent, at temerature T and ressure P. The system contains m d tyes of roton donors and m a tyes of roton accetors. Each molecule of secies contains d i such donor sites of tye i and a such accetor sites of tye j. The total number of donors of j tye i is then N i d t = N, (2.22) = 1 while the total number of accetors of tye j is d i N j a t = N = 1 a j. (2.23) The configurational artition function of the system is then assumed to be factorable. One factor, factor, Q HB, considers only hydrogen bonding interactions. A second Q P, considers only hysical interactions such as excluded volume, induced 30

diole, and wea olar interactions. Each factor exlicitly ignores the resence of the interactions catured in the other factors, although they are lined imlicitly. The canonical artition function can thus be written as Q= Q Q. (2.24) This decouling of interactions is clearly an aroximation; however, its success has been demonstrated in a number of aers 128-131. In fact, a recent wor has further decouled the above aroach in the charge free case by slitting the hysical contribution into random and nonrandom contributions in order to facilitate a quasichemical aroach to the hysical interactions 130. The hysical contribution to the artition function of this system is given by the lattice fluid model, above. The hydrogen bonding contribution is given as follows. The system will contain a number of bonds between donors of tye i and accetors of tye j equal to N ij HB P. The total number of unbonded donors of tye i and unbonded accetors of tye j, resectively, are then given by and i0 i d m N = N N (2.25) j= 1 ij n j = a ij i= 1 The total number of hydrogen bonds in the system is N0 j N N. (2.26) N HB n m = Nij i= 1 j= 1. (2.27) The hydrogen bonding artition function will have several contributions. The first, Q HB, C, is an entroic combinatorial factor accounting for the number of ossible ways of forming N ij bonds for all i-j airs. This has been shown via the Veytsman aroach, which is based on straightforward combinatorial considerations 128, to be given by Q HB, C i j Nd! Na! 1 N! N! N md ma md ma =! i= 1 i0 j= 1 0 j i j ij. (2.28) 31

The second contribution to the hydrogen bonding artition function, Q HB, G, is a geometric robability factor accounting for the robability that each of the airs considered in the combinatoric factor Q HB, C are actually satially roximate to one another. Equivalently, this term can be understood to account for the loss of translational entroy with bond formation. For any articular air, this contribution will scale as the ratio of the volume of an ion to the volume of an entire system; there is one such factor for each association air in the system. In terms of lattice fluid arameters, this factor is thus Q HB, G ρ = rn N ij. (2.29) The third contribution to the hydrogen bonding artition function, Q HB, S, will be an entroic loss factor accounting for the loss of rotational degrees of freedom with bond formation and for local steric considerations. Alternatively, it may be understood as the robability that all airs are correctly oriented and aligned to form hydrogen bonds. It may be written in terms of an entroy change of hydrogen bond formation Q HB, S 0 ( βtsij ) N ij 0 S ij =. (2.30) The final contribution, Q HB, E, will be an energetic factor accounting for the energy of formation of all associative bonds in the system. It may be written in terms of the energy of hydrogen bond formation 0 E ij as Q HB, E 0 ( β Eij ) Combining equations (2.28) through (2.31) yields where Q HB N ij =. (2.31) m 0 d i ma j md ma N!! ( Fij ) d N β a ρ = N! N! N! rn i= 1 i0 j= 1 0 j i j ij F = E TS 0 0 0 ij ij ij Nij Nij as, (2.32). (2.33) 32

The Gibbs free energy is obtained as in the lattice fluid model, with the modification that the Gibbs artition function is now given by ( { } ) QP( T, N0, { N} ) ( β ) Ψ T, P, N = ex PV N0 = 0 QHB ( T, N0, { N},{ Nij} ) and the system volume is now md ma * 0 V = rnv v + NijVij i= 1 j= 1 HB, (2.34). (2.35) It follows that the Gibbs free energy may also be artitioned into contributions from each of the above grous: G= G + G. (2.36) P The hysical contribution to free energy is given by equation (2.16) from the lattice fluid model. The hydrogen bonding contribution is: βg rn v G v v v v 0 0 where G = F + PV md ma md ma 0 ij i i0 j 0 j HB = ij 1+ β ij + ln + d ln + a ln i j i= 1 j= 1 ρvi0v 0 j i= 1 vd j= 1 va 0 ij ij ij, and v i d i Nd Ni0 =, vi 0 =, v rn rn N ij ij =, and so on. rn v, (2.37) The free energy minimization condition on the density is essentially the same as that in the lattice model: ( G v) TP { N}{ Nij} =. (2.38),,, 0 However, an additional set of free energy minimization conditions now constrain the hydrogen bonding numbers: ( G Nij ) T, P, v,{ N},{ Nlu ij} = 0 (2.39) for all i and j. From equation (2.38), the density equation of state is now given by where 2 ρ P T ln ( 1 ρ) ρ 1 1 + + + = 0, (2.40) r 33

md ma 1 1 v. (2.41) ij i 1 j 1 r r = = Note that equation (2.40) is of the same functional form as the density EOS for the ure lattice fluid EOS, given by equation (2.19). The effect of hydrogen bonding on the density equation of state is simly to modify the effective average molecular size. However, there is now an additional set of equations of state on the hydrogen bond numbers, deriving from equations (2.39): β G 0 ij v ij + ln = 0. (2.42) ρvv i0 0j The chemical otential of comonent is in general given by G G v μ = + N v T, P, { N },,{ } TP,,{ N}{, Nij} N l v N ij T, P, { Nl }..(2.43) md ma G Nij + i= 1 j= 1 N ij N T, P, N { } { l } T, P, N, v However, by alying the minimization conditions of equations (2.38) and(2.39), equation (2.43) may be reduced to G P G HB μ = + N N T, P, { Nl }, v, { Nij} T, P, { Nl }, v,{ Nij}. (2.44) = μ + μ P, HB, The hysical contribution to chemical otential is given by the lattice fluid result of equation (2.20). The hydrogen bonding contribution is given by βμ v n m n m i0 0 j HB, = r vij+ di ln + jln i a. (2.45) j i= 1 j= 1 i= 1 vd j= 1 va v 34

2.2. Theory 2.2.1. Aqueous olymers in the resence of free salt Model descrition As discussed in the introduction, it is well nown that the LCST of aqueous olymers may be considerably altered by the resence of modest concentrations of free salt. A significant limitation of the hydrogen bonding lattice fluid model is its inability to account for this effect. The resent model remedies this deficiency through a simle aroach. The electrostatic interactions characteristic of ions are divided into two tyes: short range ion-diole interactions (exe. ion hydration), and long range ion-ion interactions. The long range interactions will be addressed by including an electrostatic factor in the artition function based on the Debye-Hucel aroximation. The short range interactions will be addressed by alying Veytsman 129 statistics to the combined networ of ion-diole and hydrogen bonds. Note that ionic bonding is not considered in this model; hence it is not alicable to solutions containing wea electrolytes. The later model for olyelectrolytes resented in section 2.2.2 secifically considers ionic bonding and could be alied in a simlified form to non-olyelectrolytic systems containing wea free salts. As in the hydrogen bonding lattice fluid model, the system artition function is treated as factorable. A factor second factor, Q, will relace A Q P will account for hysical interactions as before. A Q HB and will account for secific associating interactions including hydrogen bonding and ion-diole interactions. A final factor, accounts for long range electrostatic interactions between ions. The canonical artition function can thus be written as E A P Q E, Q= Q Q Q. (2.46) 35

Physical artition function In general, the hysical artition function may be based on any number of theories. As noted above, for examle, a quasi-chemical aroach has recently been alied for hysical interactions in the charge-free case 130. However, as in the hydrogen bonding lattice fluid model 128 and in the interest of simlicity, the resent develoment is based uon the lattice fluid model, above. The hysical artition function is then given as before by equation (2.16). Associating artition function The associating artition function follows the same develoment as that for the hydrogen bonding lattice fluid model, albeit with alterations to account for ion-diole interactions. Namely, hydrogen and ion-diole bonds are lumed together as association bonds. As such, m d and m a exress the number of association donors and accetors rather than simly the number of tyes of hydrogen bond donors and accetors. Each molecule of secies then contains such association accetor sites of tye j. d i such association donor sites of tye i and Other than these changes of definition, the derivation for the hydrogen bonding artition function alies exactly. Equations (2.22) and (2.23) aly for the total number of association donors and accetors, resectively. The number of association bonds between donors of tye i and accetors of tye j is equal to interactions are now included in the given by N ij, where ion-diole { N ij }. The total number of association bonds is then N A md ma ij. (2.47) i= 1 j= 1 = N The associating artition function is of the same form as that of the hydrogen bonding contribution in the rior model: a j 36

Q A m 0 d i ma j md ma N!! ( Fij ) d N β a ρ = N! N! N! rn i= 1 i0 j= 1 0 j i= 1 j= 1 ij Nij Nij, (2.48) where F = E TS 0 0 0 ij ij ij. (2.49) Electrostatic artition function terms by The dimensionless ionic strength of the resent system is given in lattice fluid 1 1 I ρ φ vi= Nz = z, (2.50) t t * 2 2 ρ 2 rn = 1 2 = 1 r where z is the charge valency of secies and I is dimensional ionic strength. For sufficiently low ionic strengths, the interaction energy u l of any selected free ion of tye l with the surrounding cloud of free ions is given by the Debye-Hucel theory 89 : βu l 3 2 κ zl = I 8π I 1+ κσ l, (2.51) I where σ l is the diameter of the ion and κ is a dimensionless Debye-Hucel inverse screening length, given in lattice-fluid terms by 3 32 3 * 3 κ v κ 8π B* 12 ( I) l =, (2.52) v where κ is the usual Debye-Hucel inverse screening length, l is the Bjerrum length, q is the electron charge, ε 0 is the vacuum ermittivity, andε r is the dielectric constant of I the medium. The ion s diameter σ l may be based uon its van der Waals radius or, as an aroximation, uon its ure state lattice sacing constant is a function of comosition, density, and temerature: { } B *1 3 v l. In general, the dielectric εr = εr ρ, φ, T. (2.53) 37

However, this develoment will neglect the density and comosition deendencies of the dielectric constant. The electrostatic otential energy is simly the sum of the er-ion electrostatic energy given in equation (2.51) over all ions in the system, halving to revent doublecounting: E E t 1 = Nu. (2.54) 2 Combining equations (2.51) and (2.54) gives the contribution to otential energy from electrostatic interactions: β E E = 1 3 κ I κ = rn, (2.55) 8πρ I where I κ can be understood as a screening-adjusted ionic strength, given by 1 z I φ κ ρ. (2.56) 2 1 t 2 I = 1 r + κσ It then follows from equation (2.55) that the electrostatic artition function is given by Q E 3 κ I κ = ex rn. (2.57) 8πρ I Note also that the comonent mole numbers are constrained by the requirement of electrical neutrality: m C Nz = 0. (2.58) l= 1 Free energy The Gibbs free energy is obtained as in equation (2.14). The Gibbs artition function is now given by ( T, P, { N} ) Ψ = ( ) (, 0, { }), 0, { },{ ij} (, 0, { } ) ex( β ) Q T N N Q T N N N P A, (2.59) N0 = 0 QE T N N PV 38

where V is given by equation (2.35), with the modification that 0 V ij now alies to formation of ion-diole bonds as well as hydrogen bonds. As in the HBLF model, it follows that the Gibbs free energy may also be artitioned into contributions from each of the above grous: G GP GA G E = + +. (2.60) The hysical contribution to free energy is given by equations (2.16) from the SL model. The associating contribution is of the same form as equation (2.37) from the HBLF model, albeit with the altered variable definitions described above. The electrostatic contribution is given, from equation (2.57), by βg E 3 κ I κ = rn. (2.61) 8πρ I Equations of state The minimization conditions on free energy, as in the HBLF model, are given by equations (2.38) and (2.39). The associating equations of state are given by equation (2.42) from the HBLF model. The density equation of state is now where ( I κ I 2κ) 3 2 1 κ ρ + P + T ln ( 1 ρ) + ρ 1 + = 0, (2.62) r 16π I 1 I φ κσ 2 1 t I 2 2κ = ρ *1 3 2 = 1 r v z I *1 3 ( + κσ v ), (2.63) and where r is given by equation (2.41) as in the HBLF model. Note that equation (2.62) is of the same functional form as that of the lattice fluid and hydrogen bonding lattice fluid models, albeit with an additional term for Coulombic interactions between free ions. Furthermore, the electrostatic term will be negligibly small at most charge concentrations of interest and can thus usually be neglected. This result emerges from the κ 3 scaling of this interaction, which in essence is the cube of the ratio of the lattice sacing to the Debye-Hucel screening length. As the lattice sacing is of the length 39

scale of atoms and the screening length is considerably larger for modest charge concentrations, the order of this term is much less than one. Chemical otential As before, the chemical otential may be divided into contributions from the various artition function factors: G μ = N T, P, { Nl }, v,{ Nij}. (2.64) = μ + μ + μ P, A, E, The hysical and associating contributions are given by equations (2.20) and (2.45) from the LF and HBLF models. The electrostatic contribution is given by βμ κ 1 1 I I ( I κ I κ) 3 * 2 κ 2κ E, = z + 2 * 16πI 1+ κa 2 I v ρ rv. (2.65) Note that for uncharged secies, the electrostatic contribution to chemical otential reduces to βμ κ 16πI ( I κ I κ) rv ρv 3 * E, = 2 *. (2.66) Furthermore, as in the density equation of state, for most cases the contribution to chemical otential from electrostatics will be negligible due to its Heat of mixing 3 κ scaling. The heat of mixing is given by the difference in energies between the comonents in the ure states and the mixture. For covalent comonents such as solvents and olymers, the ure state energies may be calculated consistently via this model. However, for free ions for which the ure state is a crystalline ionic lattice, it is necessary to call uon some other theoretical or exerimental data in order to determine the ionic lattice energy E Li, of the ure state solid. Given this quantity, the energy of mixing is 40

md ma * 0 mix ερ ij ij i= 1 j= 1 Δ E = rn + N E + md ma * 0 0 0 rn ερ Nij Eij Ns, iel, i = covalent secies i= 1 j= 1 i= salts, (2.67) where 0 ρ is the reduced density of comonent in its ure state and 0 N ij denotes association counts in ure comonent. The sum of over covalent secies indicates a sum over comonents whose ure state is not an ionic crystal. The summation over salts is a summation over electrically neutral salt secies, such as NaCl or HCl (rather than a summation over dissociated ion secies such as Na, Cl, and H as other secies sums in this aer are). Thus, N s, i is the number of salt molecules of tye i necessary to rovide the ions resent in solution, and is related to the N s of the ion secies by stoichiometric considerations. For examle, in a simle system containing ions dissociated from only a single salt secies, N s, i would be given by N si, = = ion secies = ion secies Nn n i,, i (2.68) where n i, is the number of ions of secies er molecular unit of salt molecule i. For examle, n = for chlorine in NaCl or 2 for chlorine in MgCl 2. i, 1 Equation (2.67) for the heat of mixing rovides an aroach to obtaining associating arameters for ion hydration. Such an aroach first requires an indeendent determination of lattice fluid arameters for ions, in order to obtain a hysically meaningful searation of hysical and associating interactions. Atomic ions lattice fluid arameters may be obtained by aroximating their hysical interactions to be equal to those of a hyothetical Noble gas of equal van der Waals radius. The roerties of this hyothetical Noble gas can, in turn, be determined via interolation with resect to the van der Waals radius of real Noble gases. Once lattice fluid arameters are determined via this aroach, the only remaining arameters are those for association interactions. 41

These can then be obtained by a best fit to heat of mixing data over a range of temerature and ressure. The major hurdle in fitting ion association arameters to salt heats of solvation based on a single salt solution is that there is no basis for the establishment of indeendent arameters for the salt s constituent ions. This roblem can be surmounted by simultaneously fitting to heat of solvation for four searate water-salt systems in which each cation and anion aears exactly twice. An ideal such set of systems would contain only strong 1:1 salts. One examle reasonably satisfying these requirements would consist of the systems water-nacl, water-hcl, water-hbr and water-nabr. The effect of fitting all four ions contained in this set to all four systems at once would be to distinguish each ion from any articular ion airing. Once arameters for these ions were established, they could be used to establish those of other ions by fitting to heat of solvation data for a salt consisting of one of these ions and the new desired ion. An informative test of this model would be whether ion hydration arameters obtained from a heat of solvation otimization such as the one above are also able to yield LCSTs in agreement with exeriment. Success in such a test would indicate that the model consistently catures the hysics of both ion hydration and solution stability. 2.2.2. Polyelectrolytes Model descrition Two modifications must be made to the above model in order to treat olyelectrolytes. First, the electrostatic term must be modified to account for long range Coulombic interactions involving the charges fixed on the chain. Second, whereas the above model does not consider ionic bonding, many olyelectrolytes of interest only artially ionize at exerimental H, and it is thus necessary to incororate this interaction exlicitly. 42

Consider a system containing t secies, in which there are N molecules of each secies. Of these secies, t are olymeric or macromolecular and t s are small molecules. All dissociable ionic secies are considered to be fully dissociated into their constituent ions for the urose of these counts. Each molecule of secies contains C l cationic sites of tye l and A anionic sites of tye u; the system contains u m C tyes of such cationic sites and of tye l is then m A tyes of such anionic sites. The total number of cationic sites N l C t = N = 1 and the total number of anionic sites of tye u is C l, (2.69) Of these, I N lu N t u A N u = 1 = A. (2.70) airs of a cationic grou of tye l and an anionic grou of tye u will be in a bound state. Each cationic site of tye l has a charge valency C a l, and each anionic site of tye u has a charge valency C zl and an ionic radius A z u and an ionic radius A a u. The system also contains md tyes of association bond donor sites and association bond accetor sites, where association bonds include hydrogen bonds and ion hydration or ion-diole tye bonds, and otionally may include some strong diolediole bonds. Association donors include hydrogen bond roton donors and hydration sites on cationic sites and on some artial ositive oles of dioles. Association accetors include hydrogen bond roton accetors and hydration sites on anionic sites ma and some on artial negative oles of dioles. Each molecule of secies resents d i association bond donors sites of tye i and a j association bond accetor sites of tye j that are not associated with an ionic site. Furthermore, each charged site is associated uniquely with a single tye of association site which by convention shall have the same index as the charged site tye. Thus each cationic site of tye i exhibits C d i association 43

donor sites of tye i, and each anionic site of tye j exhibits A a j association accetor sites of tye j. Finally, bound airs of a cation of tye l with an anion of tye u may exhibit lu d i association donors of tye i and lu a j association accetors of tye j; this reresents diolediole interactions between bound ion airs and other dioles and can often be neglected. However, note that, for bound ion airs on a olyelectrolyte, this class of interactions has been osited to drive solution destabilization with resect to the neutral olymer at high counterion concentration 113, 115. Omission of such interactions may thus omit some olyelectrolyte hysics. Furthermore, note that this contribution reresents the introduction of a limited class of ternary interactions into the model; secifically, it accounts for ion-ion-diole ternary interactions. This aroach could be further extended to account for ternary interactions more generally. For examle, water s hydrogen bonding-energy to the olymer could be modified if the water is also hydrating an ion. The ion-ion-diole ternary interaction has been selected for secial treatment in this model simly because the energy of ion airing is so high as to be exected to have a far more significant ternary effect. The total number of association donors of tye i is t mc ma i C C i I d = i + αi i C + lu = 1 l= 1 u= 1 N N d d N N d and the number of association accetors of tye j is t mc ma j A A j I a = j + α j j A + lu = 1 l= 1 u= 1 N N a a N N a lu i lu j, (2.71), (2.72) C A where α i and α j denote cation and anion ionization fractions, resectively, defined as and α C l 1 = 1 l N C m A I Nlu (2.73) u= 1 44

A 1 αu = 1 N m C I Nlu. (2.74) u A l= 1 The requirement of charge neutrality rovides a constraint uon the molecule numbers: mc ma l C u A NCzl NAzu l= 1 u= 1 + = 0. (2.75) As in revious models the artition function of this system is considered to be factorable. The first factor, Q P, accounts for hysical interactions between molecules: excluded volume interaction, disersion forces, and most diole-diole interactions. A second factor, Q I, accounts for ionic bonding. The third factor,, accounts for associating interactions such as hydrogen bonds, ion-diole bonds, and some strong diole-diole bonds. The final factor, Q, accounts for long range electrostatic interactions. The canonical artition function may thus be written as E P I A E Q A Q= Q QQ Q. (2.76) As in the revious model, the lattice fluid model result given by equation (2.12) will be used for the hysical artition function. Similarly, equation (2.48) for associating interactions will be used for the associating artition function. Ion-binding artition function The ionic bonding artition function will follow the same develoment as the associating artition function. The number of unbonded cationic grous of tye l and number of unbonded anionic grous of tye u are and m A I l C l I Nl0 = NCαl = NC Nlu (2.77) u= 1 m C I u A u I N0u = NAα u = NA Nlu. (2.78) l = 1 45

of where As in the association contribution, this leads to an ionic bonding artition function Q mc l ma u m I C ma N!! ( Flu ) C N β A ρ = N! N! N! rn I I I I l= 1 l0 u= 1 0u l= 1 u= 1 lu F = E + TS I I I lu lu lu I Nlu I Nlu, (2.79), (2.80) E lu S lu I I and where and are the energy and entroy, resectively, of air formation between a cationic grou of tye l and an anionic grou of tye u. Electrostatic artition function For simlicity, note that the total number of tyes of ionic grous in the system is given by mi mc m A +. (2.81) Then define the number of ionic grous of tye l er molecule of tye Ι l such that Ιl Cl l m C. (2.82) Al m l > m C C Similarly, α αl l m C l C A αl m l > m C C, (2.83) z l z l m z l m C l C A l mc > C, (2.84) and σ I l σ l m C l C A σ l m l > m C C, (2.85) The ionic strength of this system may now be defined in dimensionless form as 46

The contribution from free ions to I is 1 φ. (2.86) t m I * 2 I Iv = ρ αlιlzl 2 = 1 r l= 1 1 φ, (2.87) t mi * 2 I f I fv = ρ αlιl zl 2 = t + 1 r l= 1 where the sum from t + 1 to t denotes a sum over small molecules only. There will be two distinct tyes of long range electrostatic interactions. The first tye includes interactions involving free ions (those on small molecules). Due to their small size, the satial distribution of such ions is exected to be dominated by their electrostatic interactions, and these interactions are thus modeled via the Debye-Hucel theory. Within this framewor, the interaction energy of any ion with the surrounding cloud of free ions is given by 89 where βu f l 3 κ z = 8πI 1+ κσ f 2 l I l 3 32 3 3 * κ κ v 8π B* v *1 3 12, (2.88) l = ( I f ), (2.89) v and where κ is the Debye-Hucel inverse screening length and l is the Bjerrum length, defined as usual by l B 2 βq. (2.90) 4πε ε The second tye of Coulombic interaction includes only those between fixed ions (those on macromolecules). The satial distribution of such ions is taen to be dominated by considerations of the chain bacbone rather than by electrostatics, and the charges thus do not articiate in Debye-Hucel screening. They are however, still screened by the free ions, such that the interaction between two such fixed ions is given the Debye-Hucel screened air interaction energy: 0 r B 47

= l. (2.91) r bb B βu ex( κr) Integrating equation (2.91) over all sace (considering all other fixed charges) yields for the interaction energy b u l of a fixed charge of tye l with all other fixed charges z ρ β u = b l I l I f, (2.92) where ρ I is the total dimensionless charge density of fixed olymeric charges: t mi * 1 φ I I v = l l z l 2 = 1 r l= 1 ρ ρ ρ α Ι. (2.93) The total electrostatic energy of the system is the sum of equations (2.88) and (2.92) over all alicable ions, halving as necessary to revent double-counting: βg rn κ = + f 8 3 2 ( ρ ) ( I κ I κ) ρi π E I where I κ is a screened ionic strength, given by, (2.94) 1 φ ρ 2 1 α Ι t m I 2 l l l I κ I *1 3 = 1 r l= 1 + κσ l v z, (2.95) and I κ is the contribution from olymeric charges to this screened ionic strength, given by t mi 2 1 l l zl I φ α Ι κ ρ I *1 3 2 = 1 r l= 11+ κσ l v The electrostatic contribution to the artition function is then Q rn κ = ex + f 3 2 ( ρ ) ( I κ I κ) ρi 8π E I. (2.96). (2.97) For modest salt concentrations, both terms in equation (2.97) with be considerably less than one and will thus be negligible relative to the hysical, associating, and ion-binding contributions. 48

Free energy The Gibbs artition function is given by I (, 0, { }) (, 0, { },{ },{ u} ) I I ( 0 { } { }) ( 0 { } { }) QP T N N QA T N N Nij N l Ψ ( T, P, { N} ) = QI T, N, N, Nlu QE T, N, N, N lu, (2.98) N0 = 0 ex( β PV ) where the system volume V is given by md ma mc ma * 0 I I ij ij lu lu i= 1 j= 1 l= 1 u= 1 V = rnv v+ N V + N V, (2.99) and where 0 V ij is the volume change of association formation between a donor of tye i and an accetor of tye j and I V lu is the volume change of ionic bond formation between a cationic grou of tye of tye l and an anionic grou of tye u. As before, the Gibbs free energy is given by equation (2.14) and the free energy may be segregated into the same contributions as the artition function: G= GP + GA+ GI + G E. (2.100) The hysical and associating contributions to free energy are given by equations (2.16) and (2.37), resectively, as in the HBLF model. The electrostatic contribution to free energy is given by βg rn κ = + f 8 3 2 ( ρ ) ( I κ I κ) ρi π E I The ionic bonding contribution is given by v v βg rn v G v v. (2.101) mc ma I mc I ma I I I lu l l0 u 0u I = lu 1+ β lu + ln C ln Aln I I + + l u l= 1 u= 1 ρvl0v0u l= 1 vc u= 1 va v,(2.102) where G = F + PV I I I lu lu lu and v l C l I NC I Nl0 =, vl 0 =, rn rn v I lu I Nlu =, and so on. rn Extra care must be taen in determining the maximum term of the artition function due to comlications arising from interactions between the associating and 49

ionic-bonding contributions to the artition function. In articular, the maximum term is equivalent to minimizing the Gibbs free energy with resect to v and each of the N and υ, where I lu N v υ = = (2.103) I I I lu lu lu 12 12 l u l u ( NCNA) ( vcva) Minimizing with resect to a non-intensive measure of ionic bonding or to an intensive measure which is asymmetric for anions and cations results in a model that is not thermodynamically self consistent. In articular, for such a model the system free energy is not correctly recovered from the chemical otential. Accordingly, the correct minimization conditions are G v = 0, (2.104) I { }{ υlu}{ ij} TP,, N,, N G = 0, (2.105) N ij I TPv,,, { N}{, υlu}{, Nyz ij} and G 0 I =, (2.106) υlu I T, P, v,{ N}{, υyz lu}{, Nij} ij N ij I for all i and j and all l and u for which and, resectively, are not identically zero due to the system definition (i.e. airings of ions comrising a strong electrolyte). Equations of state The associating equations of state are given by equation (2.42) as in earlier models. Equation (2.104) yields the density EOS for the system: N lu where 3 1 κ + P + T ln ( 1 ) + 1 + ( I κ I 2κ + I κ I 2κ) = 0, (2.107) r 16π I f 2 ρ ρ ρ 50

and I I md ma mc ma 1 1 v v r r = = = =, (2.108) I ij lu i 1 j 1 l 1 u 1 1 φ κσ α Ι z 2 r v 1 v t mi I 2 l l l l 2κ = ρ *1 3 2 = 1 l= 1 I *1 3 ( + κσ l ) t mi I 2 1 φ κσl αlιlzl 2κ = ρ *1 3 2 = 1 l= 1 I *1 3 ( + κσ l ) 2 r v 1 v, (2.109). (2.110) Note that long range electrostatic interactions do not lay a significant role in the density equation of state of modest charge density solutions. This surrising conclusion emerges from two redictions. The first rediction, as in the revious model, is that the energy of Coulombic interactions involving free ions is negligible at most salt concentrations of interest. The second rediction is that the energy of Coulombic interactions between fixed charges is constant with resect to density. This emerges from the fact that as density dros, distances between fixed charges increase but charge screening decreases at the same rate, with the two effects cancelling exactly. Desite these conclusions, however, it is imortant to note that the resence of charges still significantly effects the density equation of state via the contribution of ionic bonds and of ion-diole associations. Accordingly, discarding the electrostatic term gives for the density equation of state: 2 ρ P T ln( 1 ρ) ρ 1 1 + + + = 0, (2.111) r which is the same as that for the hydrogen bonding lattice fluid model, albeit with an additional contribution to account for ionic bonding. Equation (2.106) yields the set of ionic bonding equations of state: 51

v C dl v v = ρv v I I I l0 0u lu l0 0u l u vd va A au ρ I s C2 s A2 C A ρi ( fcl zl + fauzu ) 2( fcl zl + fauzu ) I f I 1 I I I2 I G κ + κ κ β 2κ s C2 s A2 lu 2 ( f ) 3 Clzl + fauzu I f 2I f ex κ + 8π z z + + + + 1+ κσ l v 1+ κσ u v md ma v lu vi 0 lu 0 j di ln aj ln i j i= 1 vd j= 1 va C2 A2 l u *1 3 ( 1 fcl ) 1 f C A *1 3 ( ) Au,(2.112) where s f Cy and s f Az are the fraction of cations of tye y and anions of tye z, resectively, that are on small molecules: f s Cy C (2.113) t φ y y = t + 1 r vc and Similarly, f Cy and f φ A. (2.114) t s z Az z = t + 1 r va f Az are the fractions of cations of tye y and anions of tye z, resectively, found on olymer molecules: and f f Cy C (2.115) t φ y y = 1 r vc t φ Az Az z = 1 r va =. (2.116) Note that for most cases, (2.113) through (2.116) will have values of either zero or one as the same tye of charged grou by definition will generally not be found on both a small and large molecule. Note also that equation (2.112) may be considerably simlified in the limit that long range electrostatics may be neglected. 52

Chemical otential As in earlier models, the chemical otential may be searated into contributions from the various artition function factors: G μ = N I { y } { ij} { υlu}. (2.117) T, P, N, v, N, = μ + μ + μ + μ P, A, I, E, The hysical contribution to chemical otential is given by equation (2.20) from the lattice fluid model. The associating contribution is: βμ = r md ma v A, ij i= 1 j= 1 1 C A d d C v md + i= 1 + + + ma j= 1 The ionic bonding contribution is: ma C C I i u i + i α i i + iu i u 2 u= 1 vc va vi 0 ln i mc ma 1 v I lu C d l A u vludi l u 2 l= 1 u= 1 vc v A mc 1 A A I C A l j aj + aj α j Aj vlj l j 2 l= 1 vc v A v0 j ln j mc ma 1 a I lu C v l A u + vluaj + l u 2 l= 1 u= 1 vc va. (2.118) μ The electrostatic contribution is: 1 A C G v. (2.119) v v + + mc ma I I I u l lu lu I, = vlu r + + + ln u l I I l= 1 u= 1 2 va vc T ρvl0v0u mc I ma I l0 0u Cl ln u ln l A u l= 1 vc u= 1 va 53

βμ E, = mi ρ I αlιlzl 2H t zl H t 1 l 1 I = f ρi mc ma I Cl A u ρ I C2 s A2 s C A vlu l u ( zl fcl zu fau ) ( zl fcl zu fau ) l= 1 u= 1 vc v A 2I f m * I 2 rv αlιl I κ + I κ I 2κ I zl H t 1 2I f * 2κ l= 1 ρv 1 + 2I mc ma f 1 I Cl A u C2 s A2 s 2I + vlu l u ( zl fcl zu fau ) f 2 l 1 u 1 vc v = = A 3 mi 2 κ αlιl z l + *1 3 ( 1+ H t I ) 8π l= 1 1+ κσ l v C 2 z l ( 1+ fcl ) m C *1 3 C ma 1 1 I Cl A + κσ u l v + vlu l u A2 2 l= 1 u= 1 vc va z u ( 1+ f Au ) A *1 3 1+ κσ u v, (2.120) where H[ x ] is the Heaviside ste function, given by Note that the relation [ ] H x 1 if x 0 = 0 if x < 0 (2.121) t G = N μ (2.122) = 1 roerly recovers the system free energy, as it should. Also note that in the absence of olyelectrolytes, equation (2.120) reduces exactly to equation (2.65) from the rior, olyelectrolyte free, model. 2.2.3. Solution Stability The above models will entail systems with at least four comonents (olymer, solvent, cation, and anion), with the electroneutrality condition constraining one of the comonent mole numbers. It is thus useful to develo in a general way the sinodal of a 54

multicomonent system under additional constraints. The develoment will differ deending uon whether the constraint is on comosition fractions only, or uon mole numbers and comosition fractions, as in the resent case. For comleteness and comarison, the comosition fraction only case is addressed first. Sinodal with a constraint on the comosition fractions For a t comonent mixture, in general the system comosition will be described by t-1 indeendent comosition variables. An aroriate gradient oerator with resect to comosition may be defined in vector form as φ φ φ. (2.123) 1 φ 2 1 j 1 φ t j 2 φ j t 1 In general, the limit of stability (sinodal) for the system with resect to comosition is defined by ( g ) det = 0, (2.124) where g is the intensive system free energy and denotes the outer roduct of the gradient vector with itself. The tye of intensive free energy (molar, volumetric, etc.) should be chosen to be conjugate to the comosition variable chosen. Several comosition variables are of the form rn rn φu = = rn rn u u u u t = 1, (2.125) where r u is some secies-secific roerty of comonent u on which the comosition variable is based. For examle, for the mole fraction r u is identically one, and for the volume fraction r u is some measure of molecular size. In many cases, it is convenient to frame the sinodal in terms of chemical otentials rather than in terms of free energy, both because the chemical otential is a commonly calculated quantity and because this form sometimes allows the straightforward use of free energy minimization conditions, such as equations (2.104) 55

through (2.106), in simlifying the sinodal. The chemical otential is given at constant temerature and ressure in terms of system free energy by G μ =. (2.126) N { N j } In terms of intensive free energy and the comosition variable form given by equation (2.125), chemical otential is g μ = rg + rn N { N j } (2.127) Alying the chain rule (see aendix A.2.1 for more detail), the general exression for second order derivatives of intensive free energy with resect to comosition is then 1 μ r φl { φ j l, t} g 1 =, (2.128) φ 1 {, } 1 t l φ φ φ g j t { φ j l, t} + φ u u φl φu { φ j u, t} { φ j l, t} where all comosition derivatives are taen with all other comosition fractions excet the deendent one held constant. For a two comonent system equation (2.128) reduces to a single equation and the sinodal is simly 2 g 1 1 μ 2 φ r 1 φ φ = = 0. (2.129) However, for a ternary system equation (2.128) yields a set of three equations: g g φ 1 1 μ 1 g = φ φ φ 2 11 12 1 1 1 1 r1 2 12 22 1 1 1 1 r1 1 φ2 φ 1 1 μ 1 g = φ φ φ 2 φ1 ; (2.130) ; (2.131) 56

where g φ 1 1 μ 2 g = φ φ φ 1 22 12 1 2 1 2 r2 2 φ1 ; (2.132) g l g. (2.133) φl φ { φ j } { φ j l} This set of equations may be solved through straightforward linear algebra, yielding exlicit equations for the g l in terms of comosition derivatives of the comonent chemical otentials. These exressions may then be substituted into equation (2.124) to yield the sinodal in terms of chemical otentials. Sinodal with a constraint on the mole numbers A comlication in the above aroach emerges when the system is also subject to a constraint on the mole numbers, such as the electroneutrality condition in the resent models. Such a constraint will always also imose an associated constraint on the comosition fractions, and it is at first temting to simly aly the above develoment on this basis. However, equation (2.128) becomes incorrect if a mole number constraint is alied to the free energy before the sinodal is calculated. In this case, it is necessary to account for the altered meaning of the chemical otential, as follows. In general, the free energy is a function of all mole numbers: ({ t} ) G = G N. (2.134) However, when there is a constraint on the mole numbers, the mole number of one of the comonents (comonent ) is a deendent function of the other mole numbers such that The free energy can then be exressed as ({ }) Nt = Nt N. (2.135) ({ }, ({ })) t G = G N N N. (2.136) 57

Furthermore, via substitution of this constraint into the free energy, a new form Ĝ may be arrived at for the free energy which is equivalent to the original form but which has no N t exlicit deendence uon : ({ } ) G = Gˆ N. (2.137) Throughout this derivation, hats on variables will similarly indicate their form with the substitution of equation (2.135) having already been made. By equating mole number derivatives of equations (2.136) and (2.137), the chemical otential ˆ μ as determined based on where Ĝ is then related to the fundamental chemical otential μ based on G via N t ˆ μ = μ + μt, (2.138) N { N j, t} Gˆ ˆ μ N. (2.139) { N j } As noted above, equation (2.135) will yield a corresonding constraint on the comosition fractions: ({ }) φt = φt φ. (2.140) In general, as in the form given by equation (2.125), the comosition fractions are a function of all the mole numbers such that ({ N }) φ = φ. (2.141) However, as in the free energy alying equation (2.125) yields an alternate form that is not an exlicit function of the deendent mole number: ({ N }) φ ˆ = φ. (2.142) Equation (2.125) may be rewritten in this form as ˆ rn φ =. (2.143) rn ˆ ˆ 58

In terms of intensive free energy, the substituted form of chemical otential ˆ μ may now be written as rn ˆˆ gˆ ˆ μ ˆ = g + rn, (2.144) N N { N j } { N j } which is comarable to equation (2.127) in the former develoment. Second derivatives of intensive free energy with resect to comosition may then be show via the chain rule to be given by ˆ ˆ 1 rn ˆ gˆ 1 φ r N φ { j } l φ N { φ j u} { φ j l}. (2.145) t 1 1 ˆ μ 1 ˆˆ ˆ ˆ rn g = + φu r φl u { j l} r N φ φ { N j } l φ u { φ j u} { φ j l} Furthermore, in this case the gradient vector from equation (2.123) will exclude the derivative with resect to N. For a quaternary system, equation (2.145) yields the triad of simultaneous equations: and g g g ˆ φ rn ˆˆ 1 1 ˆ μ g = 2 1 11 12 ˆ 1 ˆ 1 2 2 ˆ 1 rn ˆ N N ˆ 1 rn ˆ r φ1 φ r 1 1 φ1 1 φ1 r1 N1 r N 1 N 1 1 N2 1 ˆˆ ˆ 1 rn 1 1 ˆ μ φ g = 2 22 1 12 ˆ 2 2 ˆ 2 2 ˆ 1 rn ˆ r N N ˆ 1 ˆ 1 rn r φ 1 φ2 1 φ2 r2 N2 r N 2 N 2 1 N1 1 1 rn ˆˆ 1 1 ˆ μ t 1 1 12 ˆ φ2 g22 = ˆ 1 1 ˆ 1 2 ˆ 1 rn ˆ u r N N ˆ 2 ˆ 1 rn r φ 1 φ1 1 φ1 r1 N1 r N 1 N 1 2 N2 ; (2.146) φ 1 φ ; (2.147) 1. (2.148) It is imortant to note that equation (2.138) must be alied in order to obtain equations in terms of standard chemical otentials. 59

2.3. Alications, Results, and Discussion Numerical solution of the above models focuses on the base system of olyethylene oxide (PEO) in water. This rovides an exerimentally relevant model system for which hydrogen-bonding arameters are already available in the literature. Furthermore, it has reviously been shown that redictions from the HBLF model rovide excellent quantitative agreement with exeriment for the ure binary water-peo system 131. All numerical results shown in the below sections are for PEO molecular weight 6 10 5 grams er mole, with lattice fluid and hydrogen bonding arameters for PEO and water taen from a recent study fitting these arameters to the water-peo binary system 131. Furthermore, for simlicity, all ions are taen to have a hydration coordination number of six. 2.3.1. Neutral olymer in aqueous salt solution System definition Consider a system of olyethylene and water containing a strong monovalent free salt. Water, PEO, cations, and anions will be considered comonents 1, 2, 3, and 4 resectively. Each molecule of water contains two association donors (the hydrogen atoms) and two association accetors (free electron airs on the oxygen atom), 1 1 corresonding to d 1 = 2 and a 1 = 2, resectively. Each olymer chain contains one accetor site (the oxygen atom) er monomer, so that to 2 a 2 is equal to the degree of olymerization of the olymer (from a hysical standoint, it is debatable whether the correct number of accetors er PEO oxygen is one or two, but one has been used successfully in the literature 131 ). Each cation has some number of hydration sites which serve as association donors, with this number given by 3 d 2. Similarly each anion has a number of accetor sites given by 4 a 3. These grous will form five tyes of association bonds: 1-1 (water-water); 1-2 (water-peo); 1-3 (water-anion); 2-1 (cation-water); and 2-2 (cation-peo). Associations between anions and cations are neglected on the basis that 60

a strong salt will be almost entirely dissociated. Finally, the ratio s 1 s 2 = 1.3424 on a hard shere model, and ξ 12 = 1.0472 has been shown to rovide a good fit with based exerimental data. All other Berthelot arameters will be taen to have a value of unity and all the other { s } will be taen to have a value equaling that of water. One density equation of state and five associating equations of state characterize the system s state. The electroneutrality constraint from equation (2.58) becomes Nz 3 3 Nz 4 4 0 and the volume fractions are liewise constrained: φ z r + =, (2.149) φ + z = 0. (2.150) 3 4 3 4 3 r4 It then follows from section 2.2.3 that the sinodal condition is given by the determinant of a two-by-two matrix, and equations (2.146) through (2.148) may be alied to write the condition in terms of chemical otentials rather than free energies. The above system is solved for the sinodal in the following way. The system of equations of state is first solved over a grid of temerature, olymer volume fraction, and anion volume fraction. The resulting data is then interolated and numerically differentiated to yield necessary equation of state derivatives with resect to comositions. The locus of oints satisfying the sinodal condition is then determined numerically. Further details of this aroach, including Mathematica code, may be found in aendix A.3.1. Results As shown by Figure 9, redicted hase behavior is characterized by an LCSTtye sinodal curve which at higher temeratures curves bac over into a UCST tye sinodal to form a closed immiscibility loo. With the addition of free salt, the LCST shifts either uward or downward in temerature, corresonding to salting-in or saltingout, deending on salt roerties. This shift of the sinodal curve can also be understood 61

as corresonding to an isothermal ionic strength-triggered transition, as shown by Figure 10. Figure 10 is essentially a arallelogram-shaed cutout of the overall water-peo-salt ternary hase diagram, as shown by Figure 11. 480 T(K) 460 440 420 400.00178.0373.0728 380 360 0 0.05 0.1 0.15 φ 2 Figure 9: Sinodal curves for aqueous PEO in the resence of salt at various concentrations. Salt roerties have been manually adjusted to aroximate exerimental results for the LCST of PEO with KI. Numbers on curves are salt occuied volume fraction corresonding to that curve. Note that the aarent meeting oint of the three curves is not truly a single oint uon closer insection. 62

0.08 0.07 0.06 360 φ 4 0.05 0.04 370 380 0.03 0.02 0.01 0 0 0.05 0.1 0.15 0.2 φ 2 Figure 10: Cutout of isothermal ternary hase diagram for water-peo-salt system, shown in Figure 11. Curves denote sinodals at indicated temeratures. The vertical axis corresonds to anion occuied volume fraction while the horizontal axis corresonds to olymer occuied volume fraction. Salt roerties have been manually adjusted to aroximate exerimental results for the LCST of PEO with KBr. Numbers on curves denote the system temerature for that curve. Dashed line denotes an isothermal saltinduced LCST transition at 370 K. 63

H 2 0 0 1 φ 2 φ 1 PEO 1 0 φ + φ 3 4 1 0 Salt Figure 11: The ternary hase diagram for the resent system. The white region corresonds to the region shown in Figure 10. Results have not been obtained for the grey region, and the resent model would liely be inaroriate for treatment of the region corresonding to higher salt concentration due to breadown of the Debye-Hucel model. There are three mechanisms in this model by which the addition of salt alters solution stability, corresonding to electrostatic, hysical and associating contributions. As suggested in the theory develoment, for these modest charge concentrations the electrostatic effect is quite small and lays little role in affecting stability. In contrast,, the effect of salt hysical interactions on hase stability is tyically one of ronounced destabilization. For normal Berthelot arameters (near one) it is a good rule of thumb for the lattice fluid model that the addition of a third comonent is energetically unfavorable and tends to lead to destabilize a mixture. The effect of salt associating interactions is somewhat more comlicated and can consist of either stabilization or destabilization of the solution, deending on salt roerties. As shown by Figure 12, the central associative interaction that comatibilizes the mixture is the PEO-water hydrogen bond. Cation-PEO and anion-water ion-diole 64

interactions comete with this interaction and thus tend to destabilize the mixture. In contrast, the cation-water interaction cometes only with the solution-destabilizing waterwater hydrogen bond, thereby stabilizing the mixture. The balance of these effects determines the associating contribution to system stability: when cation-peo and anionwater interactions dominate, they tend to destabilize the solution; when cation-water interactions dominate, it tends to stabilize the solution. The redicted salting-in effect can thus actually be understood as a salting-out of water from water. C + H O H H O H O A - Figure 12: Schematic of ossible associations in PEO-water-free salt system. Heavy dashed lined denote hydrogen bond tyes, which are resent in the absence of free ions. Light dashed lines denote ion-diole bond tyes. Bonds that share a locus comete with one another for use of that tye of site. The heavy curve denotes a PEO chain, with a articular oxygen site note. C+ and A- denote free cation and anion, resectively. The above hysics ermit the model to reroduce asects of the lyotroic series of salts. By tuning the above balance of ion-binding interactions towards stabilization or destabilization, it is ossible to obtain theoretical salts along a sectrum of salting in and salting out behavior. As shown in Figure 13 for aqueous PEO, the model is caable of at 65

least semi-quantitative matches with exeriment for a variety of salts. Note that since exerimental data in this figure is from cloud-oint exeriments and redictions are for the LCST, an exact quantitative match is not exected. Furthermore, redicted curves are roduced by a manual best-fit of salt arameters, and numerically otimized best fits resently underway are exected to yield imroved agreement. However, interaction arameters for each ion are allowed to differ deending on their counterion; an imortant future test of the model will be whether a fit comarable to that shown in Figure 13 is ossible while using consistent ion arameters for each system. LCST (K) 400 390 380 370 360 350 340 330 320 310 300 290 280 270 0 0.5 1 1.5 Salt Concentration (molality) 2 Figure 13: Exerimental cloud oints 80 (oints) and manual LCST fits (curves) based on the above theory, for aqueous PEO as a function of salt concentration for various salts. Blac curve and oints corresond to the LCST in the resence of KI, green to KBr, blue to NaCl, and red to KF. The urle line demonstrates a tyical salting-in case as redicted by this model for a hyothetical salt characterized by a strong favorable cationwater hydration interaction and wea anion-water and cation-olymer interactions. 66

As exected, due to the neglect of ternary interactions, the resent model does not cature the more extreme lyotroic behavior that is characterized by ronounced saltingin at low salt concentrations followed by salting out at higher. Furthermore, it is not clear that the model is consistent with the observation that salt s anion controls its behavior. Desite these exected shortcomings, it is notable that the model is able to reroduce much of the range of the lyotroic series with binary interactions only. This result bolsters the contention, for examle by Par and Hoffman 81, that binary ion-diole interactions lay a significant role in the Lyotroic series. Furthermore, a framewor for extension of the resent model to incororate ternary hydration interactions has been described in section 2.2.2, and such an extension would be exected to address these shortcomings. 2.3.2. Polyelectrolyte System definition In order to study a olyelectrolytic system, the above PEO-water-salt system is modified by relacing the hydrogen bonding site on a small fraction f I of PEO monomers with a negatively charged grou. The counterion of this charged grou is taen to be the same secies as the cation of the free salt. As described in the theory section above, this system requires more comlex variable sets and indexing conventions than the neutral olymer case. Furthermore, olyelectrolytes of the low charge density sort addressed here are in reality coolymers of charged and uncharged monomers of different tyes. However, for low olymer charge fractions the effect of differences in the van der Waals interactions of the substituted monomers should be small enough to be neglected, as they are here. Finally, in order to establish the basic behavior of this model with the minimum number of interaction arameters, interactions involving ion airs will be neglected (i.e. d lu i and a will be treated as zero for all ion airs). lu j For this model, PEO, water, cations, and anions will be considered comonents 1, 2, 3, and 4 resectively (note the reversal of PEO and water with resect to the above 67

system). Each chain of PEO will contain a number of anionic grous 1 A 1 equal to f I times A the degree of olymerization of the chain. Each such anionic grou will have a 1 hydration sites that function as association accetors. The olymer chain will furthermore contain a 1 accetor sites (oxygen atoms) equal to 1 times the degree of 4 olymerization. Each free anion molecule will corresond to one anion grou, such that 4 A A 2 = 1. Each grou will resent a 2 hydration sites that serve as association accetors. Similarly, each free cation molecule will corresond to one cation grou, such that 3 C C 1 = 1, and each such grou will resent d1 association donors. Finally, as in the rior fi model each water molecule will contain two donors and two accetors, such that d = 2 2 2 and a = 2 3 2. The ionic grous will form one tye of ion bond: 1-1, which corresonds to a olyelectrolyte-cation bond. The associating grous will form six tyes of association bonds:1-3 (cation-water); 1-4 (cation-peo oxygen); 2-1 (water-olyelectrolyte anion); 2-2 (water-free anion); 2-3 (water-water); and 2-4 (water-peo oxygen). Values of s and the Berthelot arameters shall be as in the neutral PEO model. In this case, one density equation of state and six hydrogen bonding equations of state characterize the system s state. The electroneutrality constraint from equation (2.75) becomes φ φ φ Cz + Az + Az = 0 (2.151) r r r 3 3 C 1 1 A 4 4 A 1 1 1 1 2 2 3 1 4 As in the neutral olymer case, the sinodal condition is given by the determinant of a two-by-two matrix, and the necessary relations to write this condition in terms of chemical otentials are given by equations (2.146) through (2.148). Numerical solution methodology for this system is essentially the same as that for the neutral olymer case, albeit with the necessary alterations to arameter and variable definitions. Finally, for any articular combination of associating and ion-binding arameters, the olymer dissociable grous may be characterized by a C which is the negative base 68

ten logarithm of the counterion concentration and is the same as H when the counterion is hydrogen. For simlicity, the salt used in generating the below results has lattice fluid arameters based uon NaCl, and all ion-diole interactions are taen to have energies of formation 0 E ij equal to about two thirds that for the water-water hydrogen bond while and entroies of ion-diole bond formation water-water hydrogen bond formation. Results 0 S ij are taen to be about the same as that for The addition of charges to the olymer chain stabilizes the system, increasing the LCST temerature, as shown by Figure 14 for olymers with u to 1% substitution of electrolytic grous. The electrostatic interactions are calculated to lay only a small role in such cases, and stabilization is instead driven by enhanced chain hydration. This is consistent with the scaling observation made in the above theoretical develoment to the effect that the electrostatic term in the free energy is quite small relative to other contributions at small to modest charge concentrations. 69

405 400 395 LCST (K) 390 385 380 375 370 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 f I Figure 14: Effect of fraction of electrolytic monomers on PEO LCST, for monomers with a very low K 0 corresonding to a strong electrolyte, in 0.05 molar aqueous salt solution. The roerties of the salt do not corresond exactly to any real salt but fall within the range of lyotroic salting-out behavior. The addition of free salt to the solution counters this effect, destabilizing the solution and lowering the LCST. Two mechanisms drive this trend. The first is ionic bonding of counterions to the olymer charged grous, which reduces the ionization fraction of the olymer. Figure 15 demonstrates this mechanism for a PEO chain with 1% of its monomers relaced with dissociable ionic grous. As the K 0 of the dissociable grous is increased through the solution C at fixed salt concentration, there is a stelie reduction in LCST temerature that is driven by the reduction in unbound ions on the olymer. Note that for high K 0 the solution is destabilized with resect to the neutral solution, in qualitative agreement with exerimental results. 70

LCST (K) 405 400 395 390 385 380 375 370 365 360 355-6 -4-2 0 2 4 K 0 Figure 15: LCST of aqueous PEO with electrolytic subunits substituted for 0.1% of its monomers, as a function of the K 0 of the electrolyte subunits. The vertical dashed line indicates the C of the solution (the negative base ten logarithm of the counterion concentration) in solution. The free ions do not corresond to any real salt, but are hyothetical ions constructed to fall into the tyical range of salting out behavior described above. In the resent results, the dro in LCST with bonding of counterions at high K 0 results simly from the reduction in otential hydration sites on the chain. Because ternary interactions between ion airs and dioles are not considered, these sites effectively disaear uon ion binding. At the molecular level, this mechanism differs from that roosed for exerimental systems, namely, favorable interactions between bound ion airs on the chain with each other and with other sites on the chain. However, at a thermodynamic level the two mechanisms are comarable: as ion airs form, 71

favorable interactions with water become dominated by favorable self-interactions, and the system is destabilized. A second mechanism for reversal of the olyelectrolyte stabilization effect is that of lyotroic salting out, discussed in section 1.3.1 and for the neutral chain model in section 2.3.1. This effect is not strictly related to the charges on the chain; rather it reresents a swaming out of the olyelectrolyte stabilization by lyotroic destabilization. In other words, whereas the addition of charges to the chain enhances hydration by adding ion hydration sites, the lyotroic effect in this model diminishes hydration by occuying hydrogen-bonding hydration sites. This effect can be isolated from the counterion effect for charged PEO chains with a very low K 0, as such chains do not become significantly de-ionized as the free charge concentration increases. Results for such a system can be seen in Figure 16. Desite the unaltered resence of charges on the chain, there is a ronounced salting out with increased system charge that is comarable to that demonstrated for neutral chains in section 2.3.1. 405 400 395 390 LCST (K) 385 380 375 370 365 360 0 0.005 0.01 0.015 0.02 φ 4 Figure 16: LCST of PEO with strong electrolytic subunits substituted for 0.1% of its monomers, as a function of anion volume fraction. The equivalent range of salt molarity is zero to about 0.7. 72

2.4. Conclusions The LCST transition of uncharged aqueous olymers in the resence of salt is shown to be amenable to a two art aroach that combines a mean field lattice fluid model with a combinatorial method for strong air interactions. This aroach is an extension of the earlier hydrogen bonding lattice fluid model 128, and its ey addition is the treatment of ion-diole interactions in the same manner as hydrogen bonding. A cometition results between ion hydration and hydrogen bonding hydration, through which salting-in and salting-out effects emerge. Solution destabilization results from dominance of ion diole-interactions that are in cometition with water-olymer hydrogen bonding. Conversely, stabilization emerges from dominance of ion-diole interactions that comete with water-water hydrogen bonding. Via manual tuning of ion-diole interaction arameters, this model is shown to roduce results in semi-quantitative agreement with exerimental cloud oint data for PEO in various salts. Results reliminarily suggest that it is ossible to reroduce imortant exerimental asects of the lyotroic series by considering only binary interactions. However, further wor is necessary in order to demonstrate that this result is ossible while retaining identical arameters for each ion in differing systems. Furthermore, the aarent inability of this binary-only aroach to reroduce more extreme lyotroic behavior, characterized by both salting in and salting out at different salt concentrations, bolsters that argument that ternary or higher interactions are central in roducing the full lyotroic series. By exlicitly incororating such ternary interactions into this model, it seems robable a more comlete theory for lyotroic behavior could be roduced. A method of doing so is offered as art of the develoment of a model for olyelectrolytes. Alication of this extended aroach to the lyotroic series would liely be a fruitful avenue of future research. Beyond otentially offering a quantitative model for the lyotroic series, it could hel resolve the question of whether salt effects on hydrohobic hydration or hydrogen-bonding hydration are dominant in modulating solution stability. 73

An extension of this model is emloyed in order to model lightly charged olyelectrolytes. In addition to the combinatorial aroach to hydrogen bonding and iondiole interactions, a second, arallel combinatorial networ is established to treat ionic bonding. The model redicts that addition of charges to a chain stabilizes the solution (raises the LCST) as a result of enhanced chain hydration, consistent with qualitative exerimental observation. It further redicts that there is a ste-lie reduction in the LCST as the olyelectrolye K 0 becomes greater than the H of the solution, in qualitative agreement with exeriment. In both of the above models, future wor should include otimization of charge interaction arameters with resect to various exerimental results. Ideally, exerimental roerties such as heats of mixing and ion dissociation constants could be used to avoid fitting directly to hase behavior. Such an aroach would broaden the number of systems for which arameters could be raidly obtained and would establish a greater redictive (rather than merely interolative) caacity of the model. 74

Chater 3. Single Chain Conformational Behavior In highly dilute solution, the analogue of the thermally induced hase searation addressed above is exected to be a thermally induced chain collase. This mechanism is believed to drive the functionality of many bioolymers 66 and is relevant to the design of synthetic olymer systems in a variety of alications. However, rior to this wor no model was available in the literature with a broad ability to semi-quantitatively redict this transition. In addition to the hysics resent in models for the lower critical solution temerature (LCST) transition, the treatment of the single chain conformation mainly requires the addition of intrachain excluded volume interactions. From a statistical thermodynamic standoint, this amount to counting the number of non-self intersecting chain conformations rather than simly all conformation. As demonstrated by a successful model 27 for the coil-globule transition (CGT) that is associated with the uer critical solution temerature (UCST), the Widom insertion arameter aroach to the chemical otential rovides a far simler calculation of this quantity than that ossible by calculation of the system artition function. The resent wor follows this aroach within both lattice fluid and scaled article theory framewors. The lattice fluid aroach offers the advantage of greater simlicity. However, unublished results by Sanchez 132 suggest that ure comonent arameters for the SPT model have a much closer lin to comonent molecular characteristics, offering a stronger lin to underlying hysics and otentially greater redictive ower. 75

3.1. Theory 3.1.1. General model Consider a olymer-solvent system at infinite dilution. In such a system, the olymer chains do not interact, and each will indeendently ervade a volume characterized by a gyration radius R. Within this domain the olymer chain will occuy a volume fraction η, which scales as 3 r σ η ~, (3.1) 3 R 3 where r is a measure of chain length and σ is a measure of chain occuied volume er length. Furthermore, within the ervaded volume N solvent molecules will occuy volume fractionη s. Accounting for vacant sace, the total occuied volume fraction within this domain, given by ρ = η + ηs, will in general be less than one. At any given thermodynamic state, two equilibrium conditions will describe the chain and its immediate surroundings. The first is an equilibrium condition on the gyration radius or occuied volume fraction of the chain. The second is an equilibrium condition on the occuied volume fraction of the solvent within the ervaded volume The equilibrium gyration radius corresonds to the mean value of the artition function with resect to R. In the thermodynamic limit such a condition is tyically determined by simly finding the minimum of the free energy (or maximum of the artition function) with resect to the variable of interest. However, the artition function Q of an ideal chain is nown to be sewed with resect to gyration radius. On this basis the equilibrium chain gyration radius for the above system is given in terms of free energy G by the Hermans-Overbee aroximation 27 : ( ln R βg N ) s + = 0, (3.2) R TP,, ηs,{ vij} 76

where N is the number of such isolated olymer chains in the system. The second equilibrium condition is that the solvent occuied volume fraction must satisfy equality of chemical otential with resect to the bul solvent: where μ = μ, (3.3) μ s is the chemical otential of the solvent in the ervaded volume and s B s B μ s is the chemical otential of the solvent in the bul. This constraint may be used to simlify the minimization in equation (3.2), as follows. where The system free energy may be written in terms of chemical otentials as μ is the olymer chemical otential, the system, and B ( ) G = N μ + N N μ μ + N μ B, (3.4) T s s s s s N is the total number of olymer chains in T N s is the total number of solvent molecules in the system. Since the B system is at infinite dilution, μ s must be constant with resect to all roerties of the chain and thus equation (3.2) reduces to 1 + β μ = 0. (3.5) R R η The olymer chemical otential may be calculated via the calculated via the Widom insertion arameter: βμ ln TP,, = P 0 ( R B ) s 3 ρλ, (3.6) where ρ is the number density of olymer in the entire system volume and λ is the thermal wavelength, both of which are constant with resect to chain gyration radius and may be neglected for the uroses of this model. P0 ( R ) is the ideal chain gyration radius distribution function and B is the insertion arameter for the olymer chain. The ideal chain gyration radius is well aroximated by the modified Flory-Fis distribution: 77

where r is a measure of chain length. 6 7 P ( R) dr~ R ex 0 13 23 2 r η a, (3.7) It follows from equations (3.5) and (3.6) that the condition for the equilibrium chain gyration radius is ( B P0 ( R) dr) 1 ln β R R TP,, ηs = 0. (3.8) Since the CGT is defined as the oint at which the chain is in its ideal state, the contributions to equation (3.8) that aly to the ideal state are zero at the CGT by definition. This leaves simly ln B α = 1 = 0 R T, P, ηs (3.9) 2 as the condition for the CGT, where α is nown as the exansion factor and is a dimensionless square gyration radius defined with resect to the ideal chain state: 2 2 2 α R R 0, (3.10) where 2 R 0 is the mean square gyration radius of an ideal chain. For such a chain, 2 ~ 2 R 0 rσ, and combining equations (3.10) and (3.1) yields a relationshi between the exansion factor and olymer occuied volume fraction within the ervaded volume: 2 a α = r η, (3.11) 13 23 where a is some dimensionless constant. Most imortantly, by definition of the CGT, Thus at the CGT itself, << 1 collased globule 2 α = 1 CGT (ideal coil). (3.12) >> 1 exanded coil 78

32 a η =. (3.13) r It is imortant to note that the chain occuied volume fraction is thus zero in the limit of an infinite chain at the CGT. where The insertion arameter, required by equations (3.8) and (3.9), is given by P 12 B = P ex[ βψ ], (3.14) is the hard-core insertion robability of a molecule of secies and average interaction energy of the molecule uon insertion. In general, the hard core ψ is the insertion robability of the chain may be related to the insertion robability P, j of a chain subunit by taing the roduct of P, j over all subunits, accounting for the increasing occuied volume fraction as the chain is inserted: P r 1 P, j j = 0 =, (3.15) where r is the number of subunits and where short range chain correlations have been neglected. U until this oint, no articular model for the chain or fluid have been required. Furthermore, no exlicit definition has been rovided for the number of chain subunits. Thus, two alternate models for the mixture will be alied in order to obtain P, j and ψ, as well as the bul solvent roerties. These models will also rovide exlicit definitions for the number of chain subunits. The first model is the lattice fluid model, and the second is a scaled article model. 3.1.2. Lattice model Within a lattice-fluid framewor, each molecule of secies occuies r lattice sites, each of volume v *. In these variables the olymer occuied volume fraction is related to the gyration radius by 79

Furthermore, P, j is simly the fraction of unoccuied sites: * rv η =. (3.16) 3 R j P = η η, (3.17), j 1 s r where j is the number of reviously inserted chain subunits. Adjacent sites occuied by molecules of secies i and j interact with a characteristic energy of ε ij. Simly by summing over the average number of interactions er site, the chain interaction energy is then given by where r 1 j ψ = z rεsηs + ε η j= 1 r, (3.18) * * ( 2ε η ε η ) = r + s s zε * ij ε ij = (3.19) 2 is a characteristic interaction energy between secies i and secies j and where z is the coordination number of the lattice. Substitution of the insertion robability and energy above into equation (3.14) yields ( 1 η ) ( 1 ηs η) 1 η s r η 1 ηs η r η ( * * s s r s B = e ex βr ε η + 2ε η. (3.20) Equation (3.6) then yields the olymer chemical otential: 1 η η s 1+ ln 1 ln( 1 ηs η) η 1 ηs βμ = r *. (3.21) * * 2 rv 7 a β( ε η + 2εsηs ) ln + 43 23 r η 2 r η The equilibrium gyration radius may now be obtained from equation (3.8): 7 3 2 * ( 1) { ln( 1 ) α = r βε η + ϕ + ϕ ϕ}, (3.22) 80 )

where η ϕ < 1. (3.23) 1 η s As noted above, ηs is determined by equating solvent chemical otential in the ervaded volume and the bul. The solvent chemical otentials are obtained in a similar fashion to that of the olymer, through the insertion arameter, albeit without an ideal chain gyration radius contribution. Within the ervaded volume, the insertion arameter for the solvent is given by In the bul, it is given by r ( ) s * * 1 η η ex βr ( ε η 2ε η ) B s = s s ss s + s. (3.24) B * ( 1 ) r s ex r B η β ε η s. (3.25) B B S = s s ss B where η s is the bul solvent density. Equations (3.24) and (3.25) yield the chemical otential equality constraint on the solvent density in the ervaded volume: 1 B η ηs ( * s r B s ss s 1 η s r s B ( ) * η = ρ ex β ε η ηs + 2εsη. (3.26) At the CGT itself, equation (3.22) is simlified in accordance with equation (3.9) to yield the condition for the CGT: * ε η,0 + ln ( 1 ϕ0 ) + ϕ0 ϕ0 = 0 Θ, (3.27) 2 where 0 subscrits denote values at the CGT, where α = 1. Also, it has been reviously shown 27 that for this model ( ) 13 19 27 a =, which is needed to determine η,0 via equation (3.13) at the CGT. Exanding the logarithmic term in equation (3.27) gives * 2 * T ϕ0 ϕ T 0 1 η,0 η,0 [ + +...] = η... 0 2 =, (3.28) Θ 2 3 Θ 2(1 ηs) 3(1 ηs) ) 81

where T = ε is a characteristic temerature of the olymer and Θ is the CGT * * temerature. In the limit of infinite molecular weight, r, φ 0 0, ρ ρ B, and from eq. (3.28), the coil-globule transition temerature is given by the simle equation: ( ) Θ = Θ * B 2T 1 ηs. (3.29) The first order correction δθ to this transition temerature for a chain of finite length is obtained from eq. (3.28) and yields where δθ=θ( r) ( ) δθ 4 = φ0 ~ 12 r, (3.30) 3 * T Θ. Furthermore, in the infinite chain length limit the olymer volume fraction goes to zero within the ervaded volume. This follows from the fact that at the coil-globule transition the chain is in its ideal state and the gyration radius scales as 12 R r, and thus η 12 r, which goes to zero as chain length goes to infinity. Thus the solvent in the ervaded volume is effectively in its ure state and equation (3.26) for solvent chemical otential equality reduces to equality of solvent density in the ervaded volume and the bul. For the bul solvent, the most self-consistent aroach would be to use an equation of state derived based on the solvent insertion arameter above. However, this is not strictly necessary all that is required is a relation for solvent density as a function of temerature and ressure. The Sanchez-Lacombe (S-L) equation of state is based u on the same general hysical aroach as the resent one, albeit via the overall artition function as oosed to the Widom Insertion Parameter as described in the introduction. Furthermore, arameters for the S-L model are tabulated for a wide variety of common olymers and solvents. Therefore the S-L EOS will be used to obtain the density of the bul solvent. This is given by B 2 B ( ρ ) P s T s ( ρ ) ( rs ) B + + ln 1 + 1 1 ρ = 0, (3.31) 82

where T and P * are the solvent reduced temerature and ressure TT and aroriate equation of state arameters (T s *, P s *, r s ) for many solvents have been tabulated 15. The reduced solvent density * * * arameters by ρ B = ρ / ρ with ρ = ( M / r ) P / T *. B ρ is also related to equation of state s s s s * PP. s The By combining equation (3.29) for the CGT and equation (3.31) for bul solvent density, it is ossible to obtain an equation directly relating the temerature and ressure of the CGT: ( 1 2 ) 2 ln ( 1 1 )( 1 s ) P s = Θ + ζθ Θ+ r Θ, (3.32) where Θ Θ T and the arameter * * ζ T T is the ratio of characteristic * 2 s temeratures of the olymer and solvent and characterizes the relative strength of their self-interactions. Alternatively, for the case of a finite chain, a numerical result for the P- T behavior of the CGT may be obtained via simultaneous solution of equations (3.26), (3.27), and (3.31). An additional value of interest is the sloe of the P-T curve at zero ressure. From equation (3.32), the sloe Θ P of the CGT for an infinite chain is P = 0 given by * Θ T 1 1 = 1 * +Θ 0 2ζ 1 1 ζ 2 + lnθ 0, (3.33) P P= 0 2Ps rs rs where Θ 0 =Θ. Obtaining a numerical result for equation (3.33) requires solving P=0 equation (3.32) for temerature at zero ressure and substituting the result into equation (3.33). For a finite chain Θ P may be obtained numerical via simultaneous P = 0 solution of equations (3.26), (3.27), and (3.31) followed by numerical determination of the sloe at P = 0. An equation for the temerature of P max of an infinite chain can be established through a similar aroach. From equation (3.32), this maximum must satisfy the condition 1 83

P s 1 1 = 0= 2 1+Θ 2ζ 1 1 ζ 2 + lnθ. (3.34) Θ rs rs 3.1.3. Scaled article theory Consider the more general case in which the system may, in addition to the olymer chain of interest, contain multile solvent secies. Let molecules of each secies in such a solution be reresented by a hard shere or chain of r hard sheres, each of diameter σ. The volume fraction η occuied by a secies is then given by 3 π rn σ η =. (3.35) 6V It follows that the total occuied volume fraction is 3 π rn σ ρ = η =. (3.36) 6V In the general case of multile solvent secies, this can be searated into two contributions, one accounting only for the olymer chain and the other accounting for all solvent secies: where the solvent occuied volume fraction is ρ = η + η, (3.37) and the olymer occuied volume fraction is s 3 π rn σ ηs = = η, (3.38) 6V 3 π rn σ η =. (3.39) 6V Once the entire chain is inserted, it will occuy a volume fraction within the ervaded volume of 84

3 rσ η =. (3.40) 3 8R Uon insertion of the jth monomer the olymer occuied volume fraction will be η, j j = η, (3.41) r which is of the same form as that for the lattice model. It follows that the total occuied volume fraction uon insertion of the jth monomer will be ρ j = η + η s r j. (3.42) For the urose of calculating the insertion robability of any one hard shere into such a solution, the connectivity of the hard sheres constituting the solution may be neglected. The insertion robability of a single such shere has then been shown to be 132, 133 : where y = ρ ( 1 ρ ) ( y) 2 3 ( 3σ 1σ + 3σ 2σ + σ 3σ ) ( ) ln P ln 1, (3.43),1 = + + y 2 2 3 3 3 2 + 9σ 1σ 2+ 3σ 1σ 2σ y+ 3σ 1σ y, and where σ η i l l σ i. (3.44) i ρ Equation (3.43) may then be cast in a form that is exlicitly deendent uon j and combined with equation (3.15) to yield the insertion robability for the entire chain (see aendix A.2.2 for more details). The interaction energy of a shere of tye i with surrounding sheres of tye j is given by 132 where ψ 6 2 σ ij ij = 4πε ijρ j x σij = 2εη * ij j dx x, (3.45) 85

3 * 2π σ ij εij εij. (3.46) 3 σ jj Equation (3.45) is of the same form as that for the lattice fluid model. Thus the average insertion energy of the chain is of the same form as that for the lattice fluid model: * * β ψ = βr ε η + εiη i, (3.47) i noting that equation (3.46) rather than (3.19) now alies and the Berthelot rule thus gives where ε ε. * * i ii σ * ij εij = ξij εi ε j σ j 3 * * ( ) 12, (3.48) Alying equations (3.43) and (3.47) for a sufficiently long chain, equation (3.8) for the equilibrium gyration radius now yields the condition where ( ) ϕ+ ln 1 ϕ * 1 ς 3 β( 1 ηs) ϕε + ( ϕ1,1 + ln( 1 ϕ) ) 1+ ϕ ϕ 1 ηs 7 22 24 9 + ϕ1,1 3 + ϕ2,2 3 ϕ3,3 ϕ 4,4 r r r r 5 1 2 3 2 ς + ς 15ς1+ 6ς2 + ς3 2 3ς1 3ς1 + ς1ς 2 + ϕ1,2 + 3 + + 3 3 2 r( 1 ηs) 1 ηs 21 ( ηs) ( 1 ηs) 3ς + 2ς ς + 3ϕ 1,3 r 1 η 2 3 2 1 1 2 1 1 + 3 2 3 2 ( ) ( 1 η ) r ( 1 η ) s s s 3ϕ 14ς + 2ς 1 3 ς 3 ς 2 1 2 1 1 2,3 + + + 2 r( 1 ηs) ( 1 ηs) ( 1 ηs ) ς ς ς 7 9ϕ 27ϕ 27ϕ 1 α 3 1 1 1,4 3 2,4 3,4 r ( 1 ) ( 1 s) 3 η r η r s ς 2 ( ) =0, (3.49) 86

η ϕ, (3.50) 1 η s and ϕ i, j = i ϕ ( 1 ϕ ) j, (3.51) Similarly, equation (3.9) for the CGT yields σ ς ηi. (3.52) i σ i ( ) ϕ+ ln 1 ϕ * 1 ς 3 β( 1 ηs) ϕε + ( ϕ1,1 + ln( 1 ϕ) ) 1+ ϕ ϕ 1 ηs 7 22 24 9 + ϕ1,1 3 + ϕ2,2 3 ϕ3,3 ϕ 4,4 r r r r 5 1 2 3 2 ς + ς 15ς1+ 6ς2 + ς3 2 3ς1 3ς1 + ς1ς 2 + ϕ1,2 + 3 + + 3 3 2 r( 1 ηs) 1 ηs 21 ( ηs) ( 1 ηs) 3ς + 2ς ς + 3ϕ 1,3 r 1 η 2 3 2 1 1 2 1 1 + 3 2 3 2 ( ) ( 1 η ) r ( 1 η ) s s s 3ϕ 14ς + 2ς 1 3 ς 3 ς 2 1 2 1 1 2,3 + + + 2 r( 1 ηs) ( 1 ηs) ( 1 ηs ) 3 ( 1 η ) r( 1 ηs) 3 ς1 ς1 9ϕ1,4 27ϕ2,4 27ϕ3,4 = 0 r. (3.53) s ς However, in the infinite chain length limit, this reduces to the much simler exression: ( 1 η ) Θ * = T 3 15ς + 6ς + ς 2 2 9ς 41 ( ηs) + ( 1 η ) + ( 9ς + 3ς ς )( 1 η ) + 2 2 4 ς s 3 1 2 3 1 s 1 1 2 s If all molecular diameters are the same, it further reduces to ( 1 η ) 4 s * 2 T 4 η s 2η s η s. (3.54) Θ = + 3 2. (3.55) 87

As in the lattice fluid case, an equation of state for the solvent is necessary. In this case, the insertion arameter aroach yields an equation of state consistent with the above model, for which arameters are available in a limited range of olymer and solvents. An equation of state can be directly derived from the insertion arameter via ρ P 1 z = β 1 ln ln dρ ρ = B + ρ B, (3.56) where for a multicomonent mixture in which self-interactions are not exlicitly considered as they are for the CGT: This has been shown 132 to give for the equation of state: 0 ln B= x ln B r. (3.57) ( 3 3 1 2 3 1 ) 3 ( 1 1 ) Pv * T y y 3 y 2 s rs s P v = σ + σ σ + σ σ η η 2 β * *. (3.58) Furthermore, for a finite length chain, it is necessary to obtain equations for the chemical otential of the solvent in the ervaded volume and in the bul in order to imlement chemical otential equality. The chemical otential of any comonent for which self interaction need not be exlicitly considered may be calculated simly by scaling equation (3.43) aroriately with the number of beads comrising a molecule of the comonent.. The chemical otential has thereby been shown 132 to be given by where and * * l l l 2 3 ( + y) + ( σ 1σ + σ 2σ + σ 3σ ) ln 1 3 3 y η βμ = + r * rv 9 2 2 3 2 3 3 3 + σ 1σ + 3σ 1σ 2σ y + 3σ 1σ 2 y, (3.59) 2βrv η P P v * 3 πσ i 6 * * ij ij =, (3.60) 3 σ ij = 4ε. (3.61) σσ i j 88

For a ure bul solvent, equation (3.59) reduces to βμ η B B s B B B2 B3 B * s = + r ln * s ( 1 y ) 7 y y 3y 2βrs s s rv + + + + s s 2 η ε ) 15, (3.62) where y B = ρ B (1 ρ B. Within the ervaded volume, equation (3.59) is not subject to simlification. 3.2. Results 3.2.1. Overall ressure-temerature behavior Tyical conformational behavior as redicted by equation (3.32) for the lattice fluid aroach is consistent with the high temerature behavior redicted in the literature as shown in Figure 6. As shown in Figure 17, it is characterized by an HCGT that smoothly asses into negative ressure at low temeratures and curves over into a maximum and a CCGT at high temeratures. Furthermore, for an infinite chain the ressure of the CCGT aroaches zero as the system aroaches the theoretical vaor/liquid critical temerature of the chain ( * 2 ). As shown in Figure 18 for the CGT behavior of olyisobutylene in n-entane, the SPT model liewise qualitatively conforms to the behavior redicted by Figure 6. One additional interesting, if exerimentally unractical, observation is that as the solvent size goes to infinity, the CGT ressure actually reaches a minimum at Θ= 1 so that this oint reresents a theoretical hyercritical oint rather than a CCGT. The SPT and LF models are in reasonable agreement with each other and with exeriment at temeratures well below the high ressure hyercritical oint. As shown in Figure 19, the P-T behavior of the CGT in this region as redicted by the LF model is in good quantitative agreement with exerimental results for several systems for which such data are available. The SPT model is in somewhat weaer but still reasonable quantitative agreement with the same exerimental results. However, the LF and SPT models quantitatively diverge as the high ressure hyercritical oint is aroached. In 89 T

articular, the SPT model tyically redicts the hyercritical oint to be at much higher ressure and temerature than the LF model. Given the fact that equation of state arameters for both models are by necessity fit to the extant data in the lower temerature region, it is unsurrising that they differ in the higher temerature region where behavior is urely extraolated. Due to the lac of exerimental data for olymers in this higher temerature domain, it is resently difficult to establish whether the difference emerges simly from a oor fit of the arameters to this region or from some more fundamental weaness in one or both of the models. As additional data for the CGT or LCST in very high ressure high temerature systems become available in the literature, it may become ossible to better resolve this discreancy. 0.4 0.2 Coil P max 0-0.2-0.4 0 0.5 Globule 1-0.6-0.8-1 Figure 17: Plot of dimensionless CGT ressure versus temerature as redicted by equation (3.32) for an infinite chain in solution with r s = 10 and * Because Θ Θ 2T = ( T 2) α = 1 ζ = T T = 2.0. * * P S, the scaling of the temerature axis is such that the temerature value given for any oint on the curve corresonds to the value of Θ ~ at that ressure. The solid square denotes the critical oint of the solvent. Values to the left of 90

the hyercritical oint Pmax corresond to an HCGT while values to the right corresond to a CCGT. 1000 500 P (Bar) 0-500 -1000 0 1000 2000 3000 4000 5000 6000-1500 -2000 T (K) Figure 18: Plot of CGT of olyisobutylene in n-entane. The solid line is the lattice fluid rediction, while the blue line is the scaled article theory rediction. Pressure (Bar) 400 350 300 250 200 150 100 50 0 Butane Pentane Hexane 0 50 100 150 200 Temerature ( o C) 91

Figure 19: P-T lot of the HCGT of olyisobutylene (M w = 1.66 10 6 g/mol) in various solvents. Solid lines corresond to redictions based uon equation (3.32) for the lattice fluid model for an infinite chain. Dashed lines corresond to redictions based uon equation (3.54) for the SPT model for an infinite chain. Points corresond to exerimental data 11. 3.2.2. Physics of the transition The redicted transitions can best be understood hysically by examining equation (3.22), which has a straightforward interretation. The LHS is the chain elastic force that is balanced at equilibrium by the thermodynamic driving forces on the RHS. The RHS is the sum of 2 terms: a negative term corresonding to the attractive energy of the chain with itself and a ositive term corresonding to the excluded volume interaction of the chain with itself and with solvent. The self interaction energy term always favors chain collase, while the excluded volume term always favors chain exansion. Equivalently, a ositive net driving force corresonds to an exanded coil, a negative net driving force to a collased globule, and a zero net driving force to a coil-globule transition. The qualitative contributions of these two thermodynamic forces are shown in Figure 20. As can be seen there, the derivative of the net driving force with temerature will be ositive at a CCGT and negative at a HCGT; this rovides an additional mathematical means of distinguishing between the two. By rearranging equation (3.54), it can be shown that these same hysical arguments aly to the SPT model. The collased states associated with both transitions are characterized by dominance of self interaction energy over excluded volume. Since the olymer selfinteraction effect is the only thermodynamic driving force that favors the globule state, this model redicts that a chain with no attractive self interactions will have no equilibrium globule state. This reresents a limitation of this model that is also characteristic of the S-L model, which does not redict an LCST in the absence of attractive interactions. In contrast, simulation studies have redicted a CGT in the absence of attractive interactions 60, 61. 92

Thermodynamic Driving Force Excluded volume contribution 0 Sum Θ l Θ sc T Interaction energy contribution Figure 20: Qualitative contributions by excluded volume and olymer self-interaction energy to the RHS of equation (3.22) as a function of temerature for a system in which the olymer S-L characteristic temerature is greater than the solvent S-L characteristic temerature. Coil-globule transitions occur when the sum of these contributions is zero. Both the LF and SPT models mae the qualitative rediction that for similar interaction energies (and thus similar characteristic temeratures), higher bul solvent occuied volume fractions will yield warmer HCGTs and cooler CCGTs. For the lattice fluid model, this can be seen by noting the series exansion of equation (3.22) shown in equation (3.28). Since all contributions to the excluded volume scale as ( 1 ) η s x,where x is an integer, the excluded volume always increases with increased solvent occuied volume fraction. By rearranging equation (3.54), it can be straightforwardly shown that the same holds true for the SPT model. Since, as argued above, the excluded volume interaction always contributes to the exanded coil state, such an increase in the excluded volume will tend to exand the temerature range of the globule state, increasing the HCGT temerature and lowering the CCGT temerature. 93

Since the characteristic density of each solvent is different, this trend cannot be exected to be exactly true in terms of dimensional density. Nevertheless, as most solvent characteristic densities fall within a reasonably narrow range, it is exected to be a good rule of thumb that increased solvent density will tyically correlate with an elevated HCGT temerature. As shown in Figure 21 for olyisobutlyene in various solvent, this rediction is reasonably born out both in numerical results of these two theories as well as in exerimental LCST results, with some aberrations where interaction energies greatly differ. As will be shown later in Table 1, this trend is even more aarent when solvents are groued by carbon number. 300 250 CGT Temerature ( o C) 200 150 100 50 LF CGT Ex. LCST SPT CGT LF LCST 0 500 600 700 800 900 Solvent Density (g/m 3 ) Figure 21: LCST and CGT data for olyisobutylene in various solvents as a function of solvent density. 94

3.2.3. The high ressure hyercritical oint Via numerical solution, equation (3.34) yields an estimate for the dimensionless temerature Θ of the high ressure hyercritical oint for the LF model (the SPT Pmax model cannot be reduced to two arameters and thus it is not ossible to mae a comarably universal two dimensional lot for this model). This is of articular interest because systems with relatively low values of are exected to be most liely to exhibit and exerimentally accessible high temerature UCST. As shown in Figure 22 for long chains, Θ Pmax Θ Pmax generally decreases with increasing ζ and r s. Solvents associated with the lowest values of Θ Pmax tend to be small, symmetric molecules with relatively wea self interactions; for examle, as shown in Figure 22, carbon dioxide, nitrogen, and methane (as well as oxygen and ethylene, not shown) yield values of low value of Θ Θ Pmax considerably below the more tyical range offered by larger organic solvents such as roane, benzene, and hexanes. However, as shown in Figure 17, P max is tyically well into the solvent suercritical region, which can resent additional exerimental challenges. Since solutions of olymer in suercritical carbon dioxide in articular are resently an area of active study, they might rovide a convenient system in which to begin a search for a high temerature UCST. The CO 2 olystyrene system in articular has a articularly Pmax for this solvent and could be of articular interest for this urose. 95

ζ 6 5 4 3 2 1 0.400.352.528.616 CH 4 N 2.704.792.880 CO 2 1 10 Figure 22: Contour lot of temerature a function of interaction ratio Θ Pmax Proane Benzene Cyclohexane n-hexane at the high ressure hyercritical oint as ζ and solvent size r s, for the LF model in the limit of infinite chain legnth. The numbered lines indicate the value of Θ r s along that contour. White oints indicate the osition of various olymer / solvent systems. Each vertical trilet of oints corresonds, from to to bottom, to olystyrene, olyisobutylene, and PDMS in the labeled solvent. 3.2.4. The CGT near the solvent vaor ressure Pmax As suggested above, a oint of articular interest that is the focus of Table 1 is the CGT at or near the vaor ressure of the solvent, which is nearly at zero ressure. In general, such a CGT may theoretically occur either in the gas hase or the liquid hase. 96

This behavior may be elucidated in the following way. By combining equation (3.29) with the S-L equation of state, it is ossible to arrive for the LF model at a form of the dimensionless CGT temerature * Θ T s that for an infinite chain is deendent only uon r s and the ratio of the olymer and solvent characteristic temeratures defined as ζ T T * * s. As noted above, it is not ossible to obtain a similar two arameter reduction of the SPT model, and this investigation will therefore focus on the LF model alone. The behavior of the transition temerature as a function of these LF arameters is shown in Figure 23. These lots can be understood in the following way. Within a P-T lot such as Figure 17, follow the vaor ressure curve of the solvent to its critical oint, and then follow an isobaric line to higher temerature. The curves in Figure 23 denote the CGT transitions that are encountered while travelling this ath, as a function of ζ as a given r s. As shown in these lots, for any given r s there will be a critical value ζ c at which the transition will occur at the liquid-vaor critical oint of the solvent. Based on the S-L equations for critical temerature and density of the solvent this value can be shown to be c rs ( 1 s ) ζ = + r for an infinite chain, so that it goes to unity as r s goes to infinity and one half as r s aroaches unity. Furthermore, the temerature of this unique oint will be also given by Θ ccgt, = rs ( 1+ s) r. For ζ > ζ c, there will be an HCGT in the solvent liquid state and a CCGT in the solvent suercritical state at higher temerature. For ζ significantly less than ζ c, there will be only a CCGT in the solvent gas hase. However, from a ractical standoint, there is no olymer in the solvent gas hase, and as such ζ c reresents an effective minimum condition at which a CGT will occur. 97

1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 1 2 3 4 0 0 1 2 3 4 Figure 23a Figure 23b Figure 23: Quantitative lot of dimensionless transition temerature vs. the ratio of the S- L characteristic temeratures of the olymer and solvent for infinite chain length. The vertical axis is the dimensionless transition temerature Θ. Figure 23a is for infinite r s while Figure 23a is for r s equal to 10. The oint mared in red on each lot is the liquidvaor critical oint of the solvent. The green curve denotes the critical oint of the solvent as a function of ζ. Each branch from the critical oint corresonds to a coil globule transition for a chain in a different solvent hase, as labeled on the lots. Note that in the limit of infinite r s (Figure 23b) both the gas and suercritical hases are at zero solvent density. Sub-critical data is at the solvent saturated vaor ressure while suercritical data is at the solvent critical ressure. Numerical solutions of equation (3.29) for the lattice fluid model and equation (3.54) for the SPT model yield good corresondence between redicted HCGT temeratures and exerimental and S-L LCST temeratures, as shown in Table 1. Furthermore, theoretical redictions of T P P =, as shown in Table 2 for a variety of 0 systems, exhibit a generally good match with exerimental results, albeit with a moderate bias towards under-rediction, as shown in Figure 24. 98

Table 1: Comarison of theoretical HCGT temeratures at solvent vaor ressure with exerimental and theoretical LCSTs for olyisobutylene in various solvents. Solvents are groued by carbon number. SPT results are unavailable for some solutions due to resent limitations on arameter availability for this model. Polyisobutylene/ Pentanes Density ρ, 25 o C (g/m 3 ) Reduced Density ρ/ρ, LCST, o C HCGT, o C 25 o C Ex. Theory LF SPT Neoentane 585 0.786 Immiscible at 25 o C -40 42 - Isoentane 614 0.802 54 53 60 87 n-pentane 619 0.82 75 72 82 90 Cycloentane 746 0.86 188 157 147 Hexanes 2,2-Dimethylbutane 644 0.833 103 7 101-2,3-Dimethylbutane 657 0.841 131 64 114 - n-hexane 660 0.852 128 99 134 120 Cyclohexane 783 0.868 243 189 168 177 Other n-hetane 691 0.864 168 136 163 151 n-octane 713 0.875 180 162 194 167 Benzene 877 0.882 260 224 198 240 99

Table 2: Comarison of theoretical and exerimental results for Θ for various P P = 0 systems. For the LF model, theoretical calculations are based on olymer molecular weights chosen to match those associated with each exerimental result. For the SPT model, calculations are at infinite molecular weight, which is exected to cause little error due to the large size of the chain. SPT results are unavailable for some solutions due to resent limitations on arameter availability for this model. Polymer Solvent Θ ( C / bar) P P = 0 Exeriment LF Theory SPT Theory Polystyrene Tert-butyl acetate 134 0.68 0.50 - Methyl cyclohexane 135 0.80 0.65 - Methyl acetate 136 0.47 0.22 - Polyisobutylene 11 Proane 0.33 0.23.27 n-butane 0.37 0.27.32 n-pentane 0.45 0.36.37 n-hexane 0.61 0.47.42 100

0.8 o C / bar) Theoretical ( 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 Exerimental ( o C / bar) Figure 24: Plot of theoretical vs exerimental values of Θ P = P 0 for systems shown in Table 2. Blac diamonds denote LF-CGT results while blue squares denote SPT-CGT results. The 45 degree line indicates the locus of oints along which theoretical and exerimental values would agree. 3.2.5. Chain conformation through the CGT Chain collase or exansion will occur any time the system crosses the CGT curve in Figure 6 (or equivalently Figure 17) through any combination of ressure and temerature changes. The two CGT transitions of articular interest are the cases of an isobaric thermally triggered CGT and an isothermal ressure triggered CGT. The 101

isobaric CGT will tyically be nearly the same as a third case of interest, which is a CGT triggered along the vaor ressure curve of the solvent. Equation (3.22) can be used to solve for the Lattice-Fluid gyration radius and chain mer density around an isobaric HCGT transition, yielding results such as those shown in Figure 25 and Figure 26 for a olyisobutylene / n-entane system. As exected, the coil state is redicted at temeratures below the transition while the globular state is redicted at temeratures above the transition. Numerical fitting of results shows that 13 R ~ r in the globule state and 35 R ~ r in the coil state, consistent with Flory s redicted scaling. In addition, the chain occuied volume fraction η is an aroriate order arameter that is bounded between zero and unity. As shown in Figure 26, as the chain aroaches infinite density there is a discontinuity at the transition in the sloe of η but not in η itself. This is consistent with a second order thermodynamic transition. Figure 27 shows tyical results from the LF model for the exansion factor of the chain through isothermal CGTs at temeratures near solvent vaor-liquid equilibrium. The transition is qualitatively similar to exerimental results for ressure induced swelling of aqueous olymer networs 137. As exected, an increase in ressure triggers exansion of the chain. This can be equivalently understood as a ressure induced shift of the CGT temerature uward through the system temerature. In addition, it is aarent that for the small differences in temerature shown in this figure, the P-T relationshi is aroximately linear, which is consistent with exerimental results over small temerature ranges. 102

4 3.5 3 2.5 α 2 1.5 1 0.5 0 300 320 340 360 380 400 Temerature (K) Figure 25: Exansion factor as a function of temerature for various molecular weights of olyisobutylene in n-entane near the HCGT. The red line corresonds to a molecular weight of 10 6, the blue to a molecular weight of 10 7, and the blac to a molecular weight of 10 8. η 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 350 352 354 356 358 360 Temerature (K) Figure 26: Polymer volume fraction as a function of temerature for various molecular weights of olyisobutylene in n-entane near the HCGT. The red line corresonds to a molecular weight of 10 6, the blue to a molecular weight of 10 7, and the blac to a molecular weight of 10 8. 103

2.5 2 1.5 α 1 0.5 0-6 -4-2 0 2 4 6 8 10 Pressure (bar) Figure 27: Tyical lot of chain exansion factor through the ressure induced globuleto-coil transition as redicted by this model, shown as calculated for the system olyisobutylene / n-entane. Curves corresond to different temeratures. From leftmost to rightmost curve, corresonding temeratures are 353 K, 354 K, 355 K, 356 K, and 357 K. 3.3. Conclusions Proosed lattice fluid (LF) and scaled article theory (SPT) models for single chain conformational behavior with ressure and temerature offer semi-quantitative agreement with exeriment without the use of adjustable mixture arameters. Coilglobule transition (CGT) ressures and temeratures are well redicted for a variety of olymer / solvent systems, and imortant asects of a roosed 3 master hase boundary for wealy interacting olymers are reroduced. Predicted sloes of the CGT with ressure exhibit reasonable agreement with exerimental results, albeit with a tendency towards under rediction. Qualitatively, redicted behavior is characterized by a heating induced coil-toglobule transition (HCGT) that smoothly asses into negative ressure at low 104

temeratures and curves over into a maximum and a cooling induced coil-to-globule transition (CCGT) at high temeratures. The behavior of the redicted transitions is consistent with a second order thermodynamic transition. The maximum at which the CCGT and HCGT meet corresonds to a high ressure hyercritical oint P max in the olymer / solvent hase behavior 10. This behavior is often exerimentally inaccessible due to olymer degradation; however, results suggest that solutions of olymer in very small molecule suercritical solvents such as O 2, CO 2, N 2, and CH 4 may exhibit a P max at sufficiently low temeratures to allow observation. The single chain lattice fluid aroach redicts a critical olymer/solvent interaction energy ratio c rs ( 1 s ) ζ = + r, significantly below which no exerimental CGT will be found for most systems. For systems with ζ > ζ c it redicts two CGTs: an HCGT in the solvent liquid hase and a CCGT in the solvent suercritical hase. Within both models, the collased states associated with both the CCGT and HCGT are characterized by dominance of olymer self interaction energy over excluded volume effects. All else being equal, solvents with higher reduced density yield warmer HCGTs and cooler CCGTs due to greater excluded volume effects. Conversely, increasing chain length correlates with cooler HCGTs and warmer CCGTs due to reduction of the excluded volume effect. Results from the LF model aear to comare to exeriment similarly or even slightly better than those for the SPT model. Furthermore, the governing equations for the LF model are far simler than those for the SPT model. This outcome emhasizes the surrising success of the lattice fluid model in obtaining good quantitative correlation with exeriment. However, one outstanding oint favors the SPT model. In the LF model, it is very difficult to relate comonent arameters in any direct way to underlying molecular roerties of the substance. In contrast, reliminary results indicate that there is often quantitative or semi-quantitative agreement 132 between SPT arameters and molecular roerties. As a consequence, the SPT model may offer a more direct connection to the underlying hysics of the system than the LF model, and 105

corresondingly may rovide rediction of the CGT based uon a narrower range of exerimental data. 106

Chater 4. Towards a Model for the Aqueous CGT As suggested in the introduction, the CGT is of articular relevance in an aqueous setting. Many biomolecules are believed to ossess a CGT which is relevant to their function, and CGT-driven stimuli-resonsive olymers of interest in biological settings must by necessity exhibit their swelling behavior in an aqueous environment. A seemingly ideal theoretical aroach to this henomenon would straight-forwardly combine the combinatoric hydrogen bonding aroach of Chater 2 with the CGT model of Chater 3. However, such an aroach is comlicated by the fact that the former results from a system artition function aroach while the latter results from an insertion arameter aroach. As noted in section 1.4, such combinations may be roblematic at best due to sometimes contradictory outcomes of these two aroaches. Nevertheless, such an attemt is described below in an attemt to establish groundwor towards the develoment of a successful model for this henomenon. 4.1. CGT with Veytsman Statistics Consider a olymer-solvent system at infinite dilution. In such a system, the olymer chains do not interact, and each will indeendently ervade a volume characterized by a gyration radius R. Within this domain the olymer chain will occuy a volume fraction η and Ns solvent molecules will occuy a cumulative volume fractionη s. Accounting for vacancies, the total occuied volume fraction ρ = η + ηs within this domain will in general be less than one. Furthermore, hydrogen bonding within the system will be characterized by the set of variables { ij } v, where is an intrinsic measure of the number of hydrogen bonds between roton donors of tye i and roton accetors of tye j. The artition function Q of an ideal chain is nown to be sewed with resect to gyration radius, and on this basis the equilibrium chain gyration radius for the above v ij 107

system is given in terms of free energy G by the Hermans-Overbee aroximation 27, as in the above models: ( ln R βg) + R T, P, ˆ φs, = 0 { vij} (3.63) The system free energy may be written in terms of chemical otentials as B ( ) G = N μ + N N μ μ + N μ B (3.64) T s s s s s where μ and μ s are olymer and solvent chemical otential within the ervaded volume, resectively, B μs is the solvent chemical otential within the bul, N is the total number of olymer chains in the system, and T N s is the total number of solvent molecules in the system. Throughout this develoment, B suerscrits on any variable will denote its bul value. Furthermore, the solvent within the ervaded volume must satisfy equality of chemical otential with the bul solvent, such that μ = μ (3.65) Maing use of the fact that the bul solvent chemical otential at infinite olymer dilution is not a function of chain gyration radius, equation (3.2) now reduces to s B s 1 μ + β N = 0 R R T, P, ˆ φs, { vij} As in the hydrogen bonding lattice fluid model (HBLF) described in section 2.1.2, the artition function will taen as factorable into searate contributions from hydrogen bonding (Q HB ) and hysical interactions, such as excluded volume and van Der Waals forces (Q P ): P HB (3.66) Q= Q Q. (3.67) The free energy will, as in the HBLF model, liewise be searable into corresonding contributions, such that G= G + G, (3.68) P HB 108

as will the chemical otential of any comonent ( being s for solvent or for olymer): μ = μ + μ. (3.69), P, HB 4.1.1. Physical contribution The hysical contribution to the chemical otential is given as in the CGT model of Chater 3. It follows from that develoment that the hysical contribution to olymer chemical otential is given by βμ 1 η η s 1+ ln 1 ln( 1 ηs η) η 1 ηs = r. (3.70) P, * * * 2 rv 7 a β( ε η + 2εsηs ) ln + 43 23 r η 2 r η The hysical contribution to the solvent chemical otential is similarly calculated via the Widom insertion arameter: 3 ρsλ μsp, = T ln. (3.71) Bs Within the ervaded volume, this is given by Within the bul, it is r (( ) s r ( ) ) * * βμsp, ln ρs ln 1 η ηs ex β s εssηs+ 2ε sη. (3.72) r * ( ) s r B B B B βμsp, = ln ρs ln 1 ρ ex β sεssρ. (3.73) 4.1.2. Hydrogen bonding contribution Based on the HBLF model, the contribution to free energy from hydrogen bonding, within the ervaded volume is βg rn v G v v v v md ma md ma PV 0 ij i i0 j 0 j HB = ij 1+ β ij + ln + d ln + a ln i j i= 1 j= 1 ρvi0v 0 j i= 1 vd j= 1 va v, (3.74) 109

where G = E TS + PV 0 0 0 0 ij ij ij ij, and v i d = i Nd rn N rn N i0 ij, vi 0 =, vij =, and so on. Similarly for rn the bul solvent: βg rn v G v v v md m B a md B m B a B B B 0 ij Bi vi 0 j 0 j HB = s s ij 1+ β ij + ln + d ln + a ln B B B Bi Bj i= 1 j= 1 ρ vi0v 0 j i= 1 vd j= 1 va v.(3.75) The contribution to the overall system free energy from hydrogen bonding is then given by βg βg N βg B PV HB = HB +. (3.76) HB It follows that the hydrogen bonding contribution to the olymer chemical otential is βμ rη v v md m B a md B m B a s B 0 ij Bi i0 j 0 j HB, = vij 1+ βgij + ln + vd ln + valn B B B Bi Bj η i= 1 j= 1 ρ vv i0 0j i= 1 vd j= 1 va v v + rn v + G + + v + v md ma md ma 0 ij i i0 j 0 j ij 1 β ln ln ln ij d i a j i= 1 j= 1 ρvv i0 0 j i= 1 vd j= 1 va (3.77) The hydrogen bonding contribution to solvent chemical otential in the ervaded volume is and in the bul is βμ βμ v n m n m s i0 s 0 j shb, = rs vij+ di ln + jln i a (3.78) j i= 1 j= 1 i= 1 vd j= 1 va n m n B m B B s i0 s shb, = rs vij + di ln + a Bi, j i= 1 j= 1 i= 1 vd j= 1 v v v ln v B 0 j B j a v v.,. (3.79) 4.1.3. Chain gyration radius and coil-globule transition Substitution of equations (3.70)and (3.77) into equation (3.66) yields as a condition for the equilibrium gyration radius: 110

md ma η B vij 1+ v ij i 1 j 1 η η = = s s m B η d B ma i η vi0 Bi v i0 j η v0 j v j 0 j vd 1 ln vd ln va 1 ln va ln + + + + i Bi j Bj i= 1 ηs vd v. (3.80) d j= 1 ηs va v a m a d v a j v ln ln r v r v 1 η η 71 md i j i d a s * 2 + ln 1 + 1+ βε η = 1 i= 1 i0 j= 1 0 j η 1 ηs 3 r ( α ) The chain coil-globule transition is defined as the condition at which the chain gyration radius is equal to that of an ideal chain, such that Thus at the CGT equation (3.80) reduces to the condition α = 1. (3.81) md ma ˆ φ B vij 1+ v ij ˆ ˆ i 1 j 1 φ = = φ s s ˆ φ ˆ φ ˆ φ v + + + + md i m * a j di v a ˆ ˆ d j va 1 φ φ T s ln ln ln 1 1 ˆ + + + φ = 0 i= 1 r v ˆ ˆ i0 j = 1 r v0 j φ 1 φ Θ s md B m B a i vi0 Bi vi0 j 0 j j 0 j vd 1 ln vd ln va 1 ln va ln ˆ i Bi ˆ j i= 1 φ v s d v d j= 1 φ Bj v s a va CGT v, (3.82) where Θ is the CGT temerature. Note that in the absence of hydrogen bonding, equation and for an infinite chain, equation (3.82) reduces to Θ= 2T 1 (3.83) * ( η ) which is the same result as that given by the lattice fluid CGT model. However, a crucial roblem arises in the resence of hydrogen bonding interactions within this model. As the chain length goes to infinity, the hysical terms in equations (3.80) and (3.82) go to zero faster than the hydrogen bonding terms. Thus, this model indicates that in the infinite chain length limit hydrogen bonding alone determines the chain conformation. This is a clearly ahysical result that liely results from discreancies between the system artition function and insertion arameter aroaches. In this case, the contribution of hysical interactions (including excluded volume) has s 111

been calculated via the insertion arameter, whereas the contribution of hydrogen bonding interactions has been calculated from the system artition function. A model for this henomenon thus aears to require either an insertion arameter model for hydrogen bonding or a comutation of the system artition function that accounted exlicitly for olymer intramolecular interactions. 4.2. Basis for a Consistent Aroach A self-consistent insertion arameter aroach that would satisfy this requirement could be develoed in the following way. Begin with equation (1.19) for the insertion arameter, in which insertions resulting in a reulsive interaction (corresonding to overlaing bodies in the hard shere case) have already been searated out in the form of an insertion robability: B = P ex( βψ ). (3.84) i i i Now note that the interaction energy of insertion can be searated into hysical and hydrogen bonding contributions: ( βψ (, ψ, )) B = P ex +, (3.85) i i i P i HB or equivalently ( βψ, ) ex( βψ ) B = P ex,. (3.86) i i i P i HB Recall that in the hydrogen bonding lattice fluid model, the artition function was taen to be searable into exlicitly indeendent factors: one accounting for hysical interactions and ignoring hydrogen bonding, and one accounting for hydrogen bonding and ignoring hysical interactions. An equivalent searation may be imlemented in the insertion arameter by searating the average over a roduct of Boltzmann factors in equations (3.86) into the roduct of two averages, one including only hydrogen bonding interactions and the other including only hysical interactions. Without aroximation, this is equivalent to ( βψ, ) ex( βψ ) B = P ex,. (3.87) i i i P i HB 112

( βψ, ) ex( βψ, ) ( βψ ( ip, ψip, )) ex( βψ ( ihb, ψihb, )) Piex i P i HB B i =. (3.88) ex Under a mean field aroximation, the latter two factors are neglected, yielding ip, ihb, ( βψ, ) ex( βψ ) B = P ex i i i P i HB = B The hysical contribution to the insertion arameter given in exactly the same way as in section 4.1.1. B B,. (3.89) ( βψ ) = P ex ip, i ip, is then The hydrogen bonding contribution to the insertion arameter B ihb, = ex( βψ ihb, ) requires a new treatment recasting Veytsman statistics in a single molecule basis. A framewor for such in aroach is as follows. As in the hydrogen bonding lattice fluid model, consider a system containing m d tyes of roton donors and m a tyes of roton accetors. Each molecule of secies contains d i such donor sites of tye i and a such accetor sites of tye j. Such a molecule may in general articiate in j both intramolecular bonds and in inter-molecular bonds as both donor and receiver. The number of bonds between a donor i on the molecule and an accetor j on a different molecule will be denoted d ij. Liewise, the number of bonds between an accetor j on the molecule and a donor i on a different molecule will be denoted a ij. The number of intramolecular bonds articiated in by the molecule will be denotes as b ij. If the energy of formation of a hydrogen bond between a donor of tye i and an accetor of tye j is given by E, the hydrogen bonding interaction energy of the molecule in any articular 0 ij state is given by ψ 1 ( + ). (3.90) n m 0 ihb, = Eij dij+ aij 2bij 2 i= 1 j= 1 The hydrogen bonding contribution to the insertion arameter is then, trivially 113

n m 1 0 BiHB, = ex β Eij( dij + aij+ 2b ij). (3.91) 2 i= 1 j= 1 Physically, the average over insertion energies is tyically taen to be over all ossible article ositions for which the interaction energy is negative. However, for the hydrogen bonding insertion energy, it is aroriate to treat this average as being over all ossible hydrogen bond states of the system. 114

Chater 5. Conclusions The novel theoretical aroaches resented herein have the otential to extend understanding and rediction of behavior of LCST-driven stimuli resonsive olymers. The aroaches address the LCST henomenon on two scales: the macroscoic scale, at which overlaing chains in semi-dilute solution exerience a hase transition at the LCST; and the nano-scale, at which isolated chains in dilute solution exhibit a coilglobule transition (CGT) near the LCST. With modest further develoment, the models have great otential to guide develoment of stimuli resonsive olymers for a variety of alications. For examle, the lattice fluid model for olyelectrolytes could be quantitatively fit to Poly(N-isoroylacrylamide) in order to allow targeted design of PNIPAAM coolymers via inclusion of charged grous in order to control the LCST. Such a model would facilitate design of PNIPAAM based drug delivery systems, among other alications. Of additional interest would be the develoment of a model for the single chain LCST-driven CGT of aqueous olymers and low charge olyelectrolytes. However, a straightforward combination of the single chain lattice fluid model with the hydrogen bonding model described above has been shown to fail due to inconsistencies between insertion arameter and system artition function aroaches. A framewor has thus been described for the develoment of a self-consistent model via calculation of the hydrogen bonding contribution to the Widom insertion arameter. Such a model could be further extended to address the behavior of aqueous olyelectrolytes, including swelling transitions of smart synthetic olymers, elements of rotein cold denaturation, and asects of DNA conformational behavior. Finally, by incororating crosslins, the models could be generalized to quantitatively treat networ olymers. 115

Aendix 1. Nomenclature A u a A a j a j lu a j B C l C d i d i lu d i 0 E ij I E lu E L, i 0 F ij I F lu f I number of anionic sites of tye u on a molecule of secies intensive Helmholtz free energy in the olyelectrolyte model, the number of association accetors of tye j on an anion of tye j number of association accetors of tye j on a molecule of secies ; in the model for olyelectrolytes this excludes ion hydration sites number of association accetors of tye j on an ion air consisting of a cationic grou of tye l and an anionic grou of tye u insertion arameter of a molecule of secies number of cationic sites of tye l on a molecule of secies in the olyelectrolyte model, the number of association donors of tye i on a cation of tye i number of association donors of tye i on a molecule of secies ; in the model for olyelectrolytes this excludes ion hydration sites number of association donor sites of tye i on an ion air consisting of a cationic grou of tye l and an anionic grou of tye u energy of formation of association bond between donor of tye i and accetor of tye j energy of formation of ionic bond between cation of tye l and anion of tye u ionic lattice energy of salt i Helmholtz free energy of formation of association bond between donor of tye i and accetor of tye j Helmholtz free energy of formation of ionic bond between cation of tye l and anion of tye u fraction of dissociable subunits in a olyelectrolyte chain 116

f Au f Cl s f Au s f Cl G g G A G E fraction of anions of tye u that are on olymeric molecules fraction of cations of tye l that are on olymeric molecules fraction of anions of tye u that are on small molecules fraction of cations of tye l that are on small molecules extensive free energy intensive free energy associating contribution to extensive free energy electrostatic contribution to extensive free energy G HB hydrogen bonding contribution to extensive free energy G I G P 0 G ij I G lu H[ x ] I I I f ion-binding contribution to extensive free energy hysical contribution to extensive free energy Gibbs free energy of formation of association bond between donor of tye i and accetor of tye j Gibbs free energy of formation of ionic bond between cation of tye l and anion of tye u Heaviside ste function ionic strength dimensionless ionic strength dimensionless contribution to ionic strength from free ions I κ dimensionless contribution from ions fixed on olymer to screened ionic strength I κ dimensionless screened ionic strength B l B m A m a Boltzmann s constant Bjerrum length number of tyes of anionic sites in system number of tyes of association accetor sites in system 117

m C m d m I N N 0 N l N C u N A N ij N i0 N 0 j I N ij I N l 0 N P P P P * P Q I 0u Q A Q E number of tyes of cationic sites in system number of tyes of association donor sites in system number of tyes of ions in system total number of molecules of all secies unoccuied lattice sites molecules of comonent total number of cationic sites of tye l total number of anionic sites of tye u number of association bonds between donor of tye i and accetor of tye j number of unbonded association donors of tye i number of unbonded association accetors of tye j number of ionic bonds between a cation of tye l and an anion of tye u number of unbonded cationic sites of tye l number of unbonded anionic sites of tye u ressure reduced ressure reduced ressure based on characteristic ressure of secies insertion robability of a molecule of secies characteristic temerature of secies system artition function associating contribution to system artition function electrostatic contribution to system artition function Q HB hydrogen bonding contribution to system artition function Q I Q P q ion-binding contribution to system artition function hysical contribution to system artition function elementary charge 118

R r r 0 S ij I S lu s s T T T * T t t t s V 0 V ij I V lu v v ij I v ij v 0 j v i0 gyration radius number average molecular size arameter molecular size arameter: lattice sites er molecule in lattice theory; hard sheres er molecule in scaled article theory entroy of formation of association bond between donor of tye i and accetor of tye j entroy of formation of ionic bond between cation of tye l and anion of tye u lattice fluid hard core volume fraction mixture average surface to volume ratio arameter; also intensive system entroy in lattice fluid model, surface to volume ratio arameter for secies absolute temerature reduced temerature reduced temerature based on characteristic temerature of secies characteristic temerature of secies number of mixture comonents number of olymeric mixture comonents number of small-molecule mixture comonents extensive volume volume change of formation of association bond between donor of tye i and accetor of tye j volume change of formation of ionic bond between cation of tye l and anion of tye u intensive volume number of association bonds between a donor of tye i and an accetor of tye j er lattice site number of ionic bonds between a cation of tye l and an anion of tye u er lattice site number of unbonded association accetors of tye j er lattice site number of unbonded association donors of tye i er lattice site 119

v I 0u I v l 0 u v A l v C * v * v v x X ij x z 2 α α l A α u C α l β ε 0 ε r ε ij ζ ζ c η Θ number of unbonded anionic grous of tye u er lattice site number of unbonded cationic grous of tye l er lattice site number of anionic sites of tye u er occuied lattice site number of cationic sites of tye l er occuied lattice site mixture average volume er r ure state volume of comonent er r. intensive reduced volume equivalent to chi-interaction arameter, modified to include surface to volume ratio arameters degree of olymerization mole fraction of comonent in the hydrogen bonding with free salt lattice fluid model, charge valency of comonent ; in the olyelectrolyte mode, charge valency of ionic grou of tye olymer chain exansion factor ionization fraction of ionic sites of tye l ionization fraction of anionic sites of tye u ionization fraction of cationic sites of tye l inverse of thermal energy ermittivity of free sace dielectric constant interaction energy between a site of comonent i and comonent j ratio of olymer to solvent characteristic temeratures critical ratio of olymer to solvent characteristic temeratures occuied volume fraction of comonent coil-globule transition temerature 120

Θ dimensionless coil-globule transition temerature Θ P max dimensionless coil-globule transition temerature at the high ressure hyercritical oint θ surface fraction of comonent Ι l κ κ κ T λ i μ number of ionic grous of tye l on a molecule of tye Debye-Hucel inverse screening length dimensionless Debye-Hucel inverse screening length isothermal comressibility thermal wavelength chemical otential of comonent μ, A associating contribution to chemical otential of comonent μ E, electrostatic contribution to chemical otential of comonent μ HB, hydrogen bonding contribution to chemical otential of comonent μ I, ion bonding contribution to chemical otential of comonent μ P, hysical contribution to chemical otential of comonent ρ reduced density; equivalently, total occuied volume fraction ρ * ρ ρ I σ σ n A σ l C σ u I σ i number density of secies characteristic density of secies total dimensionless charge density of olymeric charges in the scaled article theory, the diameter of a hard shere of secies hard core volume fraction mixture average of n th order inverse hard shere diameter ionic diameter of a cationic grou of tye l ionic diameter of an anionic grou of tye u in the hydrogen bonding with salt lattice fluid model, ionic diameter of an ion of secies i; in the olyelectrolyte model, ionic diameter of an ionic grou of tye i 121

ς n I υ lu Φ ϕ φ Ψ volume fraction mixture average of n th ower of olymer shere diameter to solvent shere diameter ratio intrinsic measure of extent of ionic bonding between cationic grous of tye l and anionic grous of tye u ratio of olymer occuied volume fraction to total unoccuied volume fraction ratio of olymer occuied volume fraction to volume fraction that is not occuied by solvent hard-core occuied volume fraction of comonent Gibbs artition function ψ osition-average insertion energy of a molecule of comonent, for successful insertions 122

A.2.1. Solution Stability Aendix 2. Extended Derivations In section 2.2.3, equation (2.128) is stated to follow from equation (2.127) via the chain rule. An extended develoment of this statement follows. In general, ({ }) G = G N. (A2.1) Equivalently, G = G( { φ t( { N} )}, φt( { φ t( { N} )})), (A2.2) where the t th comosition fraction has been searated out. Comosition fractions are in general constrained by the condition that t φ = 1, (A2.3) = 1 so that the t th comosition fraction can be written as a function of the other t-1 fractions. The free energy may then be written with this substitution having been made: ({ φ ({ })}) ˆ t g = g N, (A2.4) where the hat denotes the form of g in which the constraint (A2.3) has been alied to exlicitly eliminate the t th comosition fraction. Hence gˆ g =, (A2.5) φ φ { φl } { φl, t} which yields ({ ({ })}) ˆ t 1 g g φ t N g φ u = = N u 1 { N j } N. (A2.6) = φu { φ j u, t} N { N j } { N j } Then ulling the th term out of the summation in equation (A2.6) gives t 1 g g φ g φ u = +. (A2.7) N u { N j } φ { j u, t} N φ { N j } φu { φj u, t} N { N j } 123

Recalling the form of the comosition variable given by equation (2.125), and φ r = (1 φ N { N j } rn ), (A2.8) φ N u r = u { N j } rn φ. (A2.9) Combining equations (A2.7) through (A2.9) then yields t 1 g r g r g = ( 1 φ) φu, (A2.10) N u { N j } rn φ { j u, t} rn φ φ u { φj u, t} and substitution of equation (A2.10) into equation (2.127) gives t 1 g 1 g = μ rg + r φu φ r { } ( 1 φ). (A2.11) φ u φu j, t { φj u, t} Differentiating equation (A2.11) with resect to a second comosition variable φ l then yields equation (2.128). A.2.2. Scaled Particle Theory Coil-Globule Transition As given by equation (3.43) the general SPT insertion robability for a single shere is given by 2 3 ( 3σ 1σ + 3σ 2σ + σ 3σ ) ( ) ln P = ln( 1+ y) + y. (A2.12) 2 2 3 3 3 2 + 9σ 1σ 2+ 3σ 1σ 2σ y+ 3σ 1σ y As given by equation (3.15), the overall insertion robability of the chain is given by r 1 = j j= 0 P P, (A2.13) 124

where P j is the insertion robability of segment j of the chain and is a function of the number of reviously inserted segments. Combining equations (A2.12) and (A2.13) yields r 1 r 1 ln P = ln P = ln P ( ), j, j j = 0 j = 0 r 1 = ln ( 1+ yj) + y j = 0 j 2 3 ( 3σ 1σ + 3σ 2σ + σ 3σ ) ( ) + 9σ σ 2+ 3σ σ σ y + 3σ σ 2 2 3 3 3 2 1 1 2 j 1 j (A2.14) In order to imlement the summation in equation (A2.14), it is useful to slit all variables into contributions that are and are not functions of j. r 1 ln P = ln 1 η j η s j = 0 r 2 3 ( 3( ρσ s ) ( ) ( )), σ 1 + η, j + 3 ρσ s, σ 2 + η, j + ρσ s, σ 3 + η, j, r 1 1 2 2 1 + ( 9( ρσ s σ ) 2 3( )( ) 1 ), 1 η, j ρσ s σ, 1 η, j ρσ s σ, 2 η, j ρ + + + + + j 1 ρ j = 0 j 2 3 1 + 3( ρσ s, 1 σ + η, j) 1 ρ j (A2.15) where η, j y. j = η, (A2.16) r ρ = η + η, (A2.17) j, j s y j ρ j =, (A2.18) 1 ρ j and σ η = σ, (A2.19) i s, i i ηs so that 125

1, j= s s, +, j ρ j σ ησ η σ. (A2.20) The only factors in equation (A2.15) that are a function of j are now 1 ρ, η, j, and j itself. Equation (A2.15) can thus be rewritten as 1 j where r r 1 η r r ln P = ln ( 1 ηs) ( 1 ηs) j r η j= 1 η r r r 1 Φ j + ( 3ς1+ 3ς2 + ς3) + 7 Φ η j= 0 r ϕ j r j= 0 r ϕ j 2 r 2 9 r 2 1 Φ + ς1 3ς1ς + 2 2 2 2 η j= 0 ( r ϕ j) r ( ) + 12ς + 3ς 1 2 r 2 r r j 15 j Φ 2 2 + 2 Φ η = 0 r 2 j= 0 r ϕ j j ( r ϕ j) 3 2 r 3 r 3 r 3 1 r Φ 2 j Φ + 3ς1 + 9ς 3 3 1 3 2 η j= 0( r ) r j 0 ϕ j η = ( r ϕ j) r + 9ς 3 η + j= 0 ( r ϕ j) r 2 3 r r j Φ j 1 3 r j= 0 r 3 ϕ j 3 Φ,(A2.21) η ϕ, (A2.22) 1 η s σ ς ηi, (A2.23) i σ i and η ϕ Φ =. (A2.24) 1 ρ 1 ϕ For a sufficiently long chain, the summations in equation (A2.21) can be aroximated as integrals: 126

r r 1 η r r ln P = ln ( 1 ηs) ( 1 ηs) j r η j= 1 η r r r 1 Φ j + ( 3ς1+ 3ς2 + ς3) dj + 7 dj Φ η r 0 ϕ j r r 0 ϕ j 2 r 2 9 r 2 1 Φ + ς1 3ς1ς 2 dj + 2 2 2 η 0 ( r ϕ j) r ( ) + 12ς + 3ς 1 r 2 2 r r j 15 j Φ 2 2 dj + dj 2 Φ η r 2 0 r 0 ϕ j ( r ϕ j) 3 r 2 3 r 3 r 3 1 r Φ 2 j Φ + 3ς 1 dj + 9s 3 3 1 dj 3 2 η 0 ( r ϕ j) r η 0 ( r ϕ j) r + 9ς 3 η + 0 ( r ϕ j) r 2 3 r r j Φ j 1 dj 3 r r 0 Equation (A2.25) then becomes 1 ln P = r 1 1 ln 1 + ln 1 ϕ ( ϕ) ( η ) s 3 3 dj Φ ϕ j 2 r ς 3 ς1 ϕ ϕ 1 ln ( 1 ϕ ) 1 + 9 3 r 3 + ϕ ( 1 ηs) ( 1 ηs) ( 1 ϕ ) 1 ϕ 2 9 2 3 1 3 1 ς + ς1ς 3ς 2 1+ 3ς2 + ς 3 3ς1 2 3ς1+ 3ς 3 2 + + r + + 3 2 1 ϕ ( 1 ηs) 21 ( η ) ( 1 ) 1 η s η s s 2 9 2 2 3 ϕ 1 3 1 ς + ς1ς2 + 7+ 3r + 2 3 ς 1 1 ϕ + r 2 3 1 ϕ ( 1 η ) 2 s ( 1 ηs) 2 ϕ 1 1 ς1 ς 1 + 3 2 ( 4s1+ 1s2) + r 3 3 1 ( ) ( ) + + 2 1 ϕ 1 ηs 2 ( 1 η ) ( 1 ηs) s 2 3 3 2 ς1 ς1 ϕ 3 9 3 2 3 15 ϕ 1 + 2 1 ϕ 1 ( 1 η ) ϕ ( 1 η ) ( 1 ϕ) s s 3, (A2.25), (A2.26) 127

The chain interaction energy is given by (3.47) as exlained in that section. Alying this along with (A2.26) to yield the insertion arameter and then alying equation (3.8) for the gyration radius and equation (3.9) for the CGT yields equations (3.49) and (3.53), resectively. 128

Aendix 3. Mathematica code A.3.1. LCST in Aqueous Solution with Salt The numerical method for calculation of the LCST of uncharged olymers in salt solutions is as follows. The solution density and hydrogen bond counts are first calculated from the equations of state as a function of olymer and anion volume fractions over a range of temerature. The equations of state are often very sensitive to guess values, and several tactics are thus emloyed to obtain correct results. Firstly, as EOS calculation roceeds over the grid of comositions and temeratures, revious EOS results are used as guesses via finite-difference-lie extraolation methods. During this rocess, any gridoint for which a numerical roblem is encountered is flagged. Once this initial solving rocess is comlete, a series of smart algorithms then attemt to find a correct solution to the equations of state at these gridoints by attemting better guesses based uon interolation or extraolation of nearby successful gridoints. Three dimensional lots are then roduced of EOS results in order to enable the user to visually determine whether any oints are still flagged as incorrect and reeat the above correction rocess as necessary. Once the equations of state are solved over comosition and temerature grids, the results are interolated in isothermal lanes using Mathematica s built-in interolation function. Numerical derivatives of these data sets are then taen with resect to olymer and anion volume fractions. These interolation objects are fed into the governing sinodal equation for the solution, yielding an interolated sinodal curve at each temerature. Sinodal data is then re-interolated in the temerature-olymer volume fraction lane in order to be consistent with the usual T-x hase diagram lane. The LCST itself is then determined as a function of salt concentration by finding the minimum of the sinodal with temerature over a range of salt concentration. Reresentative Mathematica code emloying this aroach follows: 129

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A.3.2. Lattice Fluid CGT Pressure-temerature behavior of the lattice fluid coil-globule transition of a finite chain is numerically determined as described in section 3.1.2 via straightforward numerical solution of equations (3.26), (3.27), and (3.31) over a range of temeratures. Reresentative code follows: 160

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Vita David Samuel Simmons was born in Miami and attended Miami Killian Senior High School. In 2000, he matriculated at the University of Florida and in 2005 graduated magna cum laude with a Bachelor of Science in Chemical Engineering. During this time, he conducted undergraduate research on the subject of thin film stability, and he also sent several summers emloyed as a research intern at the biomedical engineering comanies Syntheon LLC and Innovia LLC in Miami, Florida. In August, 2009 he began wor towards a Ph.D. in the Chemical Engineering Deartment of the Cocrell School of Engineering at The University of Texas at Austin. Uon comletion of his doctoral degree, David lans to join the National Institute of Standards and Technology in Gaithersburg, Maryland as a ostdoctoral research associate. Permanent Address: 9720 SW 121 St., Miami, Florida, 33176 This manuscrit was tyed by the author. 179