In a

polymer A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...

solution, a theta solvent (or θ solvent) is a

solvent A solvent (s) (from the Latin '' solvō'', "loosen, untie, solve") is a substance that dissolves a solute, resulting in a solution. A solvent is usually a liquid but can also be a solid, a gas, or a supercritical fluid. Water is a solvent for ...

in which polymer coils act like

ideal chain Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...

s, assuming exactly their

random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...

coil dimensions. Therefore, the

Mark–Houwink equation The Mark–Houwink equation, also known as the Mark–Houwink–Sakurada equation or the Kuhn–Mark–Houwink–Sakurada equation or the Landau–Kuhn–Mark–Houwink–Sakurada equation or the Mark-Chrystian equation gives a relation between intr ...

exponent is

1/2

in a theta solvent. Thermodynamically, the excess

chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...

of mixing between a polymer and a theta solvent is zero.

Physical interpretation

The conformation assumed by a polymer chain in dilute solution can be modeled as a random walk of

monomer In chemistry, a monomer ( ; ''mono-'', "one" + '' -mer'', "part") is a molecule that can react together with other monomer molecules to form a larger polymer chain or three-dimensional network in a process called polymerization. Classification Mo ...

subunits using a freely jointed chain model. However, this model does not account for

steric effects Steric effects arise from the spatial arrangement of atoms. When atoms come close together there is a rise in the energy of the molecule. Steric effects are nonbonding interactions that influence the shape ( conformation) and reactivity of ions ...

. Real polymer coils are more closely represented by a

self-avoiding walk In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) ...

because conformations in which different chain segments occupy the same space are not physically possible. This

excluded volume The concept of excluded volume was introduced by Werner Kuhn in 1934 and applied to polymer molecules shortly thereafter by Paul Flory. Excluded volume gives rise to depletion forces. In liquid state theory In liquid state theory, the 'exclude ...

effect causes the polymer to expand. Chain conformation is also affected by solvent quality. The

intermolecular interactions An intermolecular force (IMF) (or secondary force) is the force that mediates interaction between molecules, including the electromagnetic forces of attraction or repulsion which act between atoms and other types of neighbouring particles, e.g. a ...

between polymer chain segments and coordinated solvent molecules have an associated energy of interaction which can be positive or negative. For a ''good solvent'', interactions between polymer segments and solvent molecules are energetically favorable, and will cause polymer coils to expand. For a ''poor solvent'', polymer-polymer self-interactions are preferred, and the polymer coils will contract. The quality of the solvent depends on both the chemical compositions of the polymer and solvent molecules and the solution temperature.

Theta temperature

If a solvent is precisely poor enough to cancel the effects of excluded volume expansion, the theta (θ) condition is satisfied. For a given polymer-solvent pair, the theta condition is satisfied at a certain temperature, called the theta (θ) temperature or theta point. A solvent at this temperature is called a theta solvent. In general, measurements of the properties of polymer solutions depend on the solvent. However, when a theta solvent is used, the measured characteristics are independent of the solvent. They depend only on short-range properties of the polymer such as the bond length, bond angles, and sterically favorable rotations. The polymer chain will behave exactly as predicted by the random walk or

model. This makes experimental determination of important quantities such as the

root mean square In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...

end-to-end distance or the

radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentr ...

much simpler. Additionally, the theta condition is also satisfied in the bulk

amorphous In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid, glassy solid) is a solid that lacks the long-range order that is characteristic of a crystal. Etymology The term comes from the Greek ''a'' ("wi ...

polymer

phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...

. Thus, the conformations adopted by polymers dissolved in theta solvents are identical to those adopted in bulk polymer polymerization .

Thermodynamic definition

Thermodynamically, the excess chemical potential of mixing between a theta solvent and a polymer is zero. Equivalently, the

enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...

of mixing is zero, making the solution

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

. One cannot measure the chemical potential by any direct means, but one can correlate it to the solution's

osmotic pressure Osmotic pressure is the minimum pressure which needs to be applied to a solution to prevent the inward flow of its pure solvent across a semipermeable membrane. It is also defined as the measure of the tendency of a solution to take in a pure ...

(

\Pi

) and the solvent's

partial specific volume The partial specific volume \bar, express the variation of the extensive volume of a mixture in respect to composition of the masses. It is the partial derivative of volume with respect to the mass of the component of interest. :V=\sum _^n m_i \bar ...

(

v_s

): :

\Delta\mu_1 = -v_s\Pi

One can use a

virial expansion The classical virial expansion expresses the pressure P of a many-particle system in equilibrium as a power series in the density: Z \equiv \frac = A + B\rho + C\rho^2 + \cdots where Z is called the compressibility factor. This is the virial e ...

to express how osmotic pressure depends on concentration: :

\frac = \frac + Bc^2 + B_3c^3...

::''M'' is the

molecular weight A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...

of the polymer ::''R'' is the

gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...

::''T'' is the

absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...

::''B'' is the second

virial coefficient Virial coefficients B_i appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law. They are characteristic of the interaction potential ...

This relationship with osmotic pressure is one way to determine the theta condition or theta temperature for a solvent. The change in the chemical potential when the two are mixed has two terms: ideal and excess: :

\Delta\mu_1=\Delta\mu_1^+\Delta\mu_1^

The second virial coefficient, B, is proportional to the excess chemical potential of mixing: :

B=\frac{{v_s}{c^2

B reflects the energy of binary interactions between solvent molecules and segments of polymer chain. When B > 0, the solvent is "good," and when B < 0, the solvent is "poor". For a theta solvent, the second

is zero because the excess chemical potential is zero; otherwise it would fall outside the definition of a theta solvent. A solvent at its theta temperature is, in this way, analogous to a

real gas Real gases are nonideal gases whose molecules occupy space and have interactions; consequently, they do not adhere to the ideal gas law. To understand the behaviour of real gases, the following must be taken into account: *compressibility effects ...

at its

Boyle temperature The Boyle temperature is formally defined as the temperature for which the second virial coefficient, B_(T), becomes zero. It is at this temperature that the attractive forces and the repulsive forces acting on the gas particles balance out P = RT ...

. Similar relationships exist for other

experimental techniques The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...

, including

light scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...

intrinsic viscosity Intrinsic viscosity \left \eta \right/math> is a measure of a solute's contribution to the viscosity \eta of a solution. It should not be confused with inherent viscosity, which is the ratio of the natural logarithm of the relative viscosity to the ...

measurement,

sedimentation equilibrium Sedimentation equilibrium in a suspension of different particles, such as molecules, exists when the rate of transport of each material in any one direction due to sedimentation equals the rate of transport in the opposite direction due to diffusio ...

, and

cloud point In liquids, the cloud point is the temperature below which a transparent solution undergoes either a liquid-liquid phase separation to form an emulsion or a liquid-solid phase transition to form either a stable sol or a suspension that settles a p ...

titration Titration (also known as titrimetry and volumetric analysis) is a common laboratory method of quantitative chemical analysis to determine the concentration of an identified analyte (a substance to be analyzed). A reagent, termed the ''titrant'' ...

References

Polymer physics Thermodynamics Rubber properties

Physical interpretation

Theta temperature

Thermodynamic definition

See also

References