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Consider two mechanisms of response for dense states, Hookean elastic response, s₁₁ = E e₁₁, and Newtonian viscous response, s₁₂ = h d(e₁₂ )/dt, that reflect ideal solid and liquid behavior. (For a description of tensor notation used here see: Processing Chapter 1 and Processing Chapter 2 ) These two ideal equations reflect differences in two ideal mechanisms for response to a mechanical perturbation. For a Hookean elastic the dense material does not absorb energy in deformation, a simple molecular model is that atoms are affinely displaced from their equilibrium positions and return to these positions exactly on release of the stress. For a Newtonian fluid the dense material completely absorbs the energy of deformation, a simple molecular model is that atoms are completely displaced after traversing an energy barrier associated with two equilibrium positions in the liquid.

Polymers and plastics are generally not well described by either of these ideal equations when they are subjected to a mechanical perturbation. They do not display perfect Hookean behavior in the solid state partially because there is a high degree of disorder in their structure and the coupling associated with their chain structure so the energy minima associate with atomic and molecular displacement are complex and strongly coupled, compared to a simple metallic crystal for instance, hence it is always possible to find a locally deeper energy minimum and the sum of these local minima are close to equivalent after displacement. Energy is always lost in deformation of a solid polymer. Polymers do not display perfect Newtonian behavior for some of the same reasons, i.e. atomic motions in polymeric liquids are coupled due to the chain like nature and there is a wide dispersion of energy states in these disordered liquids. (See for example Processing Chapter 3 )

Polymers display behavior intermediate between Hookean and Newtonian behavior both in the liquid (melt) and solid states. The behavior of polymers has been called viscoelastic or elastoviscous depending on the dominance of solid like or liquid like behavior. Coupling of these two behaviors is, however, far from linear and the behavior can not be completely modeled by superposition of solid and liquid behaviors. The behavior of polymers has sometimes been refered to as anelasticity reflecting the coupling of viscous and elastic behavior in a complex way. The chief feature of anelasciticty is a time dependence to properties, that is, the behavior can be characterized as strongly solid-like at short times and strongly liquid-like at long times.

Time dependent mechanical response is usually covered in introductory physics courses in terms of the harmonic oscillator and damped harmonic oscillator. The perturbation considered in study of such systems is usually considered to be a mechanical pulse rather than a continuous stress. Striking a bell (perturbation) and observation of the tone or sound frequency (time response) and amplitude (magnitude of response) is a typical measurement. Understanding time dependent phenomena in polymers involves drawing a fundamental connection between time dependent mechanical response such as in a harmonic oscillator and simple constitutive equations such as Newtonian fluid or Hookean elastic equations.

The response of polymers to mechanical deformation is time dependent, temperature dependent and dependent on the total strain applied to the sample. At high elongations complex anelastic behavior is observed that is beyond the scope of a general understanding of viscoelastic response in polymers. For this reason, we will only consider small deformations in this class. The understanding of the dynamic (time dependent) response of polymers at low strain has impact on a eclectic group of fields ranging from mechanical creep measurements to inelastic neutron scattering measurements. In this course an attempt will be made to link these diverse observations of anelastic phenomena with the basis being the nature and molecular models of the polymer material itself and the physics of anelastic phenomena.

Two related, simple mechanical experiments can be performed on a mechanically anelastic material to demonstrate reversible and irreversible response to stress. A creep measurement involves the application of a fixed load followed by observation of strain as a function of time. For a Hookean material the response is instantaneous strain to e₁₁ = s₁₁/E. For a Newtonian material the response is an instantaneous constant rate of strain, de₁₂/dt = s₁₂/h. Neither of these materials display the time dependence characteristic of polymeric materials. An anelastic material will display a complex response that can be thought to include both of these instantaneous responses as well as a time dependent response usually characterized by and exponential function of time, e₁₁ = k (1 - exp(-t/t)), where t reduces time in a manner characteristic of the material and the temperature of observation. t is called the relaxation time of the material. The figure below, from Strobl (p. 193 Figu. 5.1), generically shows these three contributions to anelastic creep.

The relaxation time, t, for a simple harmonic oscillator is the inverse of the natural frequency of oscillation, t = 1/w. This is measured by a pulse (striking a bell) perturbation and a timed observation of the response (listening to the tone or frequency of vibration). For the creep measurement, shown above, the continuous application of stress (force) could be considered as a series of pulses of infinitely small duration, superimposed to create a continuous stress as will be discussed below.

If a measurement such as that shown in the figure above could be made form an infinitesimally small time, t₀, to an infinitesimally long time, t_°, then the observed creep compliance, D(t) = e₁₁(t)/s₁₁ would completely describe the anelastic mechanical behavior of the material. The creep experiment is natural to more solid-like materials.

A second simple experiment is to fix the strain on a sample rather than the stress. Over time the stress will relax due to Newtonian flow as well as due to the exponential decay of the time dependent component discussed above. Again, if stress relaxation measurements could be made from an infinitesimally small time, t₀, to an infinitesimally long time, t_°, then the observed time dependent tensile modulus, E(t) = s₁₁(t)/e₁₁ would completely describe the anelastic mechanical behavior of the material. Stress relaxation experiments are more natural to liquid-like materials.

The creep and stress relaxation experiments give direct insight into anelastic mechanical behavior but do not allow for controlled measurements over a wide range of time because it is difficult to measure at very short times or at very long times. A dynamic mechanical measurement allows for the measurement of extremely short time relaxations that are characteristic of polymeric materials by a repeated cycling of a low strain deformation at relatively high frequency (10 Hz). In the dynamic tensile stress measurement a sinusoidal stress is applied to a strip of elastomer, for instance, and the strain is measured as a function of time. The ratio of the time cycled stress to the time cycling strain is called the dynamic tensile modulus or complex modulus, E*(w) = s₁₁(t)/e₁₁(t), where the applied stress follows a sinusoidal function, s₁₁(t) = s⁰₁₁ exp(iwt) and the strain response follows a time lagged sinusoidal function, e₁₁(t) = e⁰₁₁ exp(-id) exp(iwt), where d is a phase factor for this lag. (Note: e^-iA = cos A - i sin A and e^+iA = cos A + i sin A.)

The application of a sinusoidal (oscillating) stress in a dynamic mechanical measurment mimics the response of a harmonic oscillator. If a harmonic oscillator were examined as a function of frequency the response would be entirely out of phase (d = 90°) for all frequencies other than 1/t. At w = 1/t the response (strain) would be entirely in phase.

For a Hookean elastic stress and strain are proportional. The dynamic mechanical response (strain) is always in-phase with the perturbation (stress) and the measurement would result in a constant value and phase for complex modulus.

For a Newtonian fluid the stress is proportional to the derivative of the strain. The derivative of a sine function is a cosine function so the stress and strain are always out of phase by 90° for a Newtonian fluid. The complex modulus is of constant value and phase for a Newtonian fluid.

Follows: M. Doi, Introduction to Polymer Physics, Oxford Science Publications, 1997

The dynamics of polymers originated in the study of dilute solutions although the main applications are in the study of polymer melts and polymers in the solid state. Dilute solutions offer a more basic situation that is amenable to modeling using relatively simple concepts. For this reason the approach to understanding particle dynamics in dilute solution is introduced at this stage.

Chapters 4 and 5 of Doi deal with Dynamics of dilute and semi-dilute to concentrated polymer systems. Doi terms the latter two "Entangled" to emphasize the importance of topological states on polymer dynamics at high concentrations. Dynamics of chains are involved in any experiment where time is a factor. These include dynamic light scattering, elastic neutron scattering, rheological measurements, and dynamic mechanical measurements. The properties of interest are of a visco-elastic nature, with always involves some kind of time constant associated with a relaxation.

The simplest approach to the dynamics of polymers is the Rouse, Bead and Spring model which you may have already been introduced to in discussions of rubber elasticity. Doi approaches this by first giving an overview of Brownian motion which serves as a basis to address the Rouse Model (bead and spring model).

A small spherical particle suspended in a solvent will display random thermally driven motion such that the average velocity for a number of particles (or for a single particle over time) as a function of time is 0, <V(t)> = 0, since the velocity is random. The velocity in the x-direction can be considered for simplicity. At two times, t₁ and t₂, the product of two velocities of the same particle will have the value V(t₁)V(t₂) and the average of this function is not necessarily zero, <V(t₁)V(t₂)> again, the average being over a single particle in time where only the difference

(t₁ - t₂) is of importance, or for a number of particles. We can write,

<V(t₁)V(t₂)>= C_v(t₁ - t₂)

where C_v(t) is the velocity correlation function. The decay of C_v(t) is described by a velocity correlation time, t_V.

The spherical particle experiences a viscous force which dampens the Brownian fluctuations which lead to its velocity. This viscous force is the product of a friction coefficient, z, and the velocity. Stokes law defines the friction coefficient for a spherical particle as

z = 6ph_sa

for a sphere of radius "a" and a solvent of viscosity h_s. This force balances with the mass * acceleration of the particle,

m (dV/dt) = -6ph_sa V

V = V₀exp(-6ph_sa (t₁ - t₂)/m) = V₀exp(-(t₁ - t₂)/ t_V)

so t_V = m/(6ph_sa)

For a sphere of density r, m = (4p/3)a³, and t_V = (2p/9)r a²/h_s.

The relaxation time, t_V, is very small for colloidal scale objects such as polymer coils, O(10^-10s) for normal solvent viscosity's and polymer sizes. This is many orders of magnitude smaller than observable time scales in the experiments mentioned above, dynamic light scattering, rheology, and dynamic mechanical experiments. This means that, for all accessible time scales, for a polymer in dilute solution, it is a safe assumption to make the approximation t_V Å 0.

We can consider the value of C_v(t) for accessible time scales for a colloidal scale particle undergoing Brownian motion in solution under the assumption that t_V Å 0. The displacement of the particle, x(t), is given by the integral of the velocity over time,

The mean-square displacement, <x(t)²>, for random diffusion is 2Dt (from the Central Limit Theorem and a Gaussian distribution in velocities as shown by Doi, pp. 67),

<V(t₁)V(t₂)>= <V(t)V(0)> = 2Dd(t)

where D is the diffusion coefficient, and d(t) is the delta function.

<V> = -(1/z)dU/dx = <dx/dt>,

where z is the friction constant for the particle discussed above, after a time t_V (Å 0) has passed.

dx/dt = -(1/z)dU/dx + g(t)

where g(t) is a Gaussian function identical to V(t) given above, <g(t)> = 0 and <g(t)g(t')> = 2Dd(t-t'). The linear sum of the velocity due to a potential field and that due to Brownian motion is referred to as the Langevin Equation.

dx/dt = -kx/z + g(t)

Using <g(t)g(t')> = 2Dd(t-t') and D = kT/z,

<x(t)x(0)> = kTexp(-t/t)/k_spr

t = z/k_spr

For t=>0, <x²> = kT/k_spr, as predicted by a Boltzman distribution, Y_equil a exp(-k_sprx²/2kT)

For t=>0 this yields (2kT/z)t = 2Dt which is the expected result from the discussion of Brownian motion above.

A photomultiplier tube of quantum efficiency Q_e records the scattered intensity associated with a separation distance R at a fixed angle q as a function of time t,

where t' is the time of irradiation and emission and t is the time of observation by the PMT (there is an incidental time lag in measurement). "*" indicates the complex conjugate and "T" the transpose (this is how you square a complex vector). The angle q is usually converted to wavevector, K = 2 sin(q/2)/l, or momentum transfer vector q = 2p k. q is related to size r by r = 2p/q = 1/K.

G₁(K, t) = <(DC(K, 0))²> exp(-D_mK²t)

where D_m is the mutual diffusion coefficient and DC is the concentration change as a function of time and wavevector, K. Normalizing by <(DC(K, 0))²> yields the electric field correlation function, g⁽¹⁾,

The mutual diffusion coefficient is dependent on the wavevector (scattering angle) since at different size scales, r = 1/K, the mechanisms of diffusion differ. In the limit of K goes to 0, i.e. large size scales, the mutual diffusion coefficient, D_m, is defined by the Stokes-Einstein relationship,

D_m = (1/N_Af_m) (dp/dc)_T,m'

where f_m is the mutual friction factor, (dp/dc)_T,m' is the osmotic susceptibility. At infinite dilution (dp/dc)_T,m' = N_AkT, so at infinite dilution (no interactions) and K = 0 (size scales much larger than the particles of interest), D_m = kT/f_m, as previously discussed.

A cursory comparison of dynamic mechanical, creep and stress relaxation measurements would lead one to consider that there is some relationship between these time dependent mechanical observations. We can generically consider that for any force or field (here stress) applied to a sample that leads to a displacement (here strain) some work is performed associated with the displacement, dx, caused by the field, y. The work per unit volume is given by dW/V = ydx. For example, the area under a stress strain curve yields the work per volume. The derivative in this equation is with respect to time, for instance, (dW(t)/dt)/V = y(t)(dx(t)/dt).

In a tensile creep experiment we have dW(t)/V = y(t) dx = s₁₁de₁₁(t) = D(t) s₁₁ ds₁₁. This relationship yields the instantaneous differential work performed in the creep measurement. The total work performed is obtained by integration over time. The response of the material to a force or field is related only to the material so it is possible that a single response function can be obtained that describes the response of the material to any force, or sequence of forces. This function is called the primary response function. The primary response function, m(t), describes the response of a material to an infinitely short pulsed force or field, y(t) = y₀ d(t). Such a short pulse, y(t), is mathematically described by the delta function, d(t), times the magnitude of the force or field, y₀. The function, d(t), has a value of 1 for a given time and a value of 0 for all other times. The primary response function describes the response, x(t), to this pulsed force or field, x(t) = y₀ m(t). You can think of y(t) as the kind of force that is applied when you strike a bell. The response function relates a perturbation to a response just as the simple constitutive equations given above related a perturbation to a response. The advantage of the response function is that it describes the response to a "base-unit" of perturbation, i.e. a delta function, from which any other perturbation can be constructed by integration, as described below.

Strobl elucidates the features of the response function, m(t), by looking as several classic examples of response, figure 5.4 below. A Hookean elastic displays an instantaneous response and an instantaneous relaxation (c), a Newtonian fluid displays and instantaneous response and no relaxation (b). A relaxing system displays an exponential decay (d), which is quite different than the response function for a harmonic oscillator (a). If the peak values for the harmonic oscillator were considered only, there is some similarity between an exponential decay and the harmonic oscillator.

Using the primary resonse function, m(t), any displacement from an arbitrary force can be calculated,

Following Strobl, p. 200, consider a creep experiment where a stress, y(t), is applied to a sample at t=0, for t>0; y(t) = y₀. The creep, x(t), is given by

For this constant stress experiment the displacement, x(t) can be directly normalized by the force, y₀, to yield the susceptibility, a(t), or the integral of the response function, or the cumulative response function.

In the stress relaxation experiment the strain is fixed, x(t) = x₀, while the stress relaxes, y(t),

It is natural to reduce the stress, y(t), by the fixed strain, x₀, to yield a time dependent modulus, a(t) = y(t)/x₀. Then,

where u = y(t') and dv = (da/dt') dt', so v = a(t') and du = dy(t'), then,

The force at time t, y(t), is composed of incremental steps of strain, dx, times the time dependent modulus, a(t).