Measurement of anisotropic energy transport in flowing polymers by using a holographic technique

Transport Phenomena in Complex Fluids

It has long been known that microstructural materials do not obey the Newtonian constitutive equation (2). Correspondingly, there exist decades of research, both experimental and theoretical, for finding an appropriate relation (or relations) to replace the Newtonian constitutive relation. Toward this end, it has been necessary to introduce new variables (or order parameters) to describe microstructural orientations. These new variables are then assumed to obey evolution equations of their own. The resulting theories are compared with a relatively large body of experimental work measuring stresses in well controlled flows, and the comparisons have led to increasingly sophisticated theories, such that reasonable quantitative predictions are now possible for stress in flowing complex fluids (4–6). Experimental progress eventually led to theoretical advance.

Less appreciated is the need for a modified energy equation in complex fluids. Nonetheless, it is straightforward to see how the equilibrium internal energy u and Fourier's law might be expected to fail. First of all, utilization of the specific internal energy u(ρ,T) requires the well known “local equilibrium approximation” (7). Physically, this means that a small packet of fluid in flow relaxes sufficiently rapidly to be in its equilibrium thermodynamic state at the local temperature T(r,t) and density ρ(r,t) (and composition, if multicomponent). Clearly, microstructural fluids with relaxation times of seconds cannot satisfy this restriction even in moderately rapid flows. Second, the orientation of microstructure, which gives rise to anisotropic stress, might also give rise to anisotropic heat flow. Such anisotropy has already been measured in deformed crosslinked rubber (8) and quiescent liquid crystals (9).

Correspondingly, a body of theoretical work has recently arisen to address these problems (10–12). From a practical point of view, the motivation to find an appropriate energy equation is great; many entangled polymers, for example, have viscosities 10⁶ times greater than water and significantly smaller thermal conductivities. Hence, it is estimated that it is possible to generate temperature rises as rapid as 100°C per s simply by fiber spinning (13). Although agreement is growing about the necessary mathematical structure of nonequilibrium thermodynamics of complex fluids, there is almost no existing body of experimental data with which to test these theories, aside from early, crude attempts. In other words, in contrast to the picture for stress predictions, the theories are far ahead of experiments for thermal properties in complex fluids.

In this article we present experimental evidence to test two main assumptions about thermal evolution in complex fluids. We examine both the proposed generalized energy density and the possibility for anisotropic thermal conductivity in flowing polymer melts. We show that our data are consistent with these ideas and confirm the existence of anisotropic heat flow in deforming polymers.

Generalized Energy Equation

The essential ideas for nonequilibrium thermodynamics are conceptually diverse, and sometimes hotly contended. However, there are points of agreement among most workers. For the sake of clarity, we use here a simple description, wherein the microstructural orientation of the fluid is assumed to be described by a conformation tensor c. This tensor c might represent, for example, the second moment 〈uu〉 of a unit vector u, which describes the orientation of a rod-like molecule, and a distribution of such orientations exist in the fluid. Nonetheless, the results are generally applicable to different sets of variables besides c, or for alternative interpretations.

We begin with the generally valid (but not-yet-closed) energy equation (3) which is essentially a generalization of the typical closed-system energy balance from equilibrium thermodynamics: dU = dQ + dW. In this equation, we use u = ε – 1/2ρv ², where ε is the total energy density, K is the absorptivity, and I is the intensity of radiation used to heat the sample. Compressibility has been neglected here. Roughly speaking, the left side is the accumulation of energy in a packet of fluid; the first term on the right side is the heat flow in; the second term is the rate of work on the packet from the surrounding fluid per unit volume; and the third term is the conversion of radiation into heat. We have used the scalar product of the stress tensor τ and the velocity gradient. The variables r and t have the usual meanings of spatial position and time.

With the new “tensor order parameter” for complex fluids c, we write the energy density as We also assume that a generalized Fourier's Law describes the heat flow where k is now a thermal conductivity tensor. Eqs. 2 and 3 represent general assumptions for microstructural modeling. Simple fluids are recovered by removal of the conformation tensor c from Eq. 2 and making the thermal conductivity a scalar. Under these general assumptions, however, Eqs. 1-3 lead to a closed set of equations (if an evolution equation for c is specified). The structure of nonequilibrium thermodynamics is more sophisticated than this simple example suggests; however, the picture here does give the necessary general idea.

Note that an evolution equation for c is necessary to complete the set of equations, making this equation highly model specific. Our experiment is designed, however, to obviate its need. In the experiment described below, we will test only the first term on the left side and the first term on the right side of Eq. 1. Therefore, since the evolution of the new variables drops out, the use of more sophisticated models, involving more detailed order parameters, will not change the analysis: the experiments are model independent. Nevertheless, our goal is to provide experimental measurements that help construct models that might then be used to predict temperature and velocity fields for complex fluids.

Thermal Holographic Grating Technique

The basic idea in thermal holographic gratings (14) is to induce a temperature modulation δT in the sample by light. This temperature modulation induces a density modulation δρ, which, in turn, yields a refractive-index modulation δn. This δn modulation can then be followed optically to monitor the temperature dynamics. We now explain the details.

To create the thermal grating, crossed coherent beams from a “writing” (3-W Ar⁺) laser form an interference pattern in the sample having intensity I(r,t) = I ₀(t)[2 + cos(2πg·r)], where g is the grating vector with period Λ = 1/|g|. The magnitude of the grating period is determined by the angle of intersection of the crossed beams and the wavelength of the incident light. A tiny amount of dye (quinizarin, in our case) is added to the sample to give it prescribed absorptivity K. This radiative pumping provides the temperature gradient driving energy transport.

What is unique here is that a homogeneous shear flow is also applied to the sample, which is sandwiched between two glass prisms. The prisms allow the light from the writing laser to pass through the sample in several directions, providing many different possible grating vectors g. The prisms are driven by a stepper motor, which controls the imposed flow field. Immediately after stopping the flow, the temperature evolution Eq. 1 becomes This equation includes the general assumptions of Eqs. 2 and 3, as well as energy conservation. We designed our experiment so that the temperature modulations do not exceed 10 mK. Since the flow field is homogeneous, the conformation tensor c is a function of time, but not position. It is then convenient to separate the temperature field into a bulk term T_b(t), and a modulation term T(r,t) = T _b(t) + δT(t)cos(2πg·r). The resulting evolution equation for the temperature modulation is If the intensity of the writing laser is temporally a square-wave pulse, then the temperature modulation following the pulse, found from solving Eq. 5, is δT(t)∼exp(–t/τ_g), where the thermal grating relaxation time is defined as

Note that changing the orientation of the grating vector g probes varying components of the conductivity, when we measure τ_g (as described next). Also, for a given grating orientation, τ_g is expected to depend quadratically on the grating period Λ = 1/|g|.

In the holographic grating technique, the temperature field is probed by a second (reading) laser, which is diffracted by the resulting modulation of the refractive-index tensor of the sample. In our setup, we use a low-power (≈10 mW) HeNe laser, which is not absorbed by the dyed sample. This HeNe beam is incident at the Bragg angle, and a rapid-response photodetector is used to follow the decay of the diffracted intensity (also at the Bragg angle). Because of the small amplitude of the temperature modulation, the refractive index is also modulated sinusoidally. Hence, if the proposed energy equation is correct, the voltage at the photodetector output should be described by The first term is the signal, the second term arises from coherent, scattered light from the sample, and the third from incidental, incoherent light striking the photodetector. The incoherent term C ² can be measured at long times, but generally A and B are not known with sufficient accuracy and must be fit to the data. Nevertheless, with good sample preparation, and clean optics, it is usually possible to keep B ² « A ².

To be consistent with the proposed energy equation, the intensity of the diffracted beam should obey Eq. 7, and the grating relaxation time τ_g should grow quadratically with the grating period Λ for a given orientation. To test these ideas, we subjected a polyisobutylene melt to a steady shear flow and measured the thermal diffusivity after cessation.

This setup allows the measurement of all four nonzero components of the thermal diffusivity tensor as a function of time after an arbitrary shearing flow. Additional details are provided in Supporting Text and Figs. 5–8, which are published as supporting information on the PNAS web site.

Results

Fig. 1 shows the photodetector response of the diffracted reading beam 45 ms after the cessation of the steady shear flow. The shear rate is 10 s^–1, and the total strain is 25 (more details about the polymer are in Supporting Text). In this experiment, the grating vector is in the flow (x) direction. Also shown is the best fit of Eq. 7 to the data. The quality of the fit, the lack of a pattern in the residuals (Fig. 1a Inset), and the Gaussian distribution of the residuals (Fig. 1b) indicate that the energy equation passes the first test. The second test is made by keeping our experimental setup the same, but changing the size of the grating period Λ by varying the angle of intersection of the crossed writing beams. Fig. 2 shows that the grating relaxation time τ_g indeed grows quadratically with the grating period, and we conclude that our data are in agreement with the proposed energy equation.

We point out that our analysis exploits the separation of time scales in the experiment. The relevant time constants are (i) the polymer relaxation time τ_p ≅ 1 s, (ii) the thermal grating relaxation time τ_g ≅ 1 ms, and (iii) the time for propagation of sound μs, where c ₀ is the adiabatic speed of sound in the sample, using the classical Newtonian estimation (i.e., κ_s is the isentropic compressibility). From these estimates, we see that, on the time scale of the scattering experiment, the polymer effectively acts as a solid, elastic rubber, and that the density modulation forms instantaneously on the time scale of the thermal relaxation. The latter is important, because it is the density modulation that yields the refractive-index modulation δn being probed: δT → δρ → δn. Previous thermal holographic grating measurements on (nonflowing) glass formers have shown decay that is not in agreement with Eq. 7 when the relaxation time of the fluid (or the conformation tensor c) is comparable to τ_g (15).

The slope of the line in Fig. 2 can be used to find the xx component of the thermal diffusivity tensor D_xx = k_xx/ρĉ^P. The number thus obtained is the instantaneous value at 45 ms after cessation of a shear flow of 10 s^–1. Because the scattering measurement is much more rapid than the polymer relaxation times (≈20–1,000 s, as measured by small-amplitude oscillatory shear, shown in Table 1, which is published as supporting information on the PNAS web site), multiple measurements of the thermal conductivity can be taken after the step. The result yields the time dependence shown in Fig. 3. Repeating the experiments for the same flows, but different grating-vector orientations, allows us to obtain the full tensor D_xx, D_yy, D_zz, and D_xy, where y is the shear direction.

Fig. 3 shows all nonzero components as a function of time after the shear cessation. The fact that the diagonal components show different deviations from their equilibrium values, and that there is even a nonzero off-diagonal component to k, indicates that the generalized Fourier's Law, Eq. 3, is correct.

It is interesting to note that the anisotropy of the thermal conductivity is similar in shape to that seen for the stress tensor. In fact, a linear relationship between the two tensors τ ∼ k has been proposed by van den Brule (16). Step strain experiments on polyisobutylene performed in our laboratory for a simpler flow cell (which allowed gratings in the x and z directions after step strains only), agreed with such a linear relation (17). However, the average orientation of the chain segments, as measured by the birefringence extinction angle, is constant for relaxation after a step strain (Lodge–Meissner rule). Hence, the flow field here provides a much stricter test. In Fig. 4 we plot thermal diffusivity versus stress in cessation of steady shear, by making time parametric. Within experimental uncertainty, we find a linear relationship between the two tensors.

Clearly, more experiments are necessary to test theories for nonequilibrium fluids with microstructure. Many interesting questions remain. For example, does thermal transport remain diffusive for time scales comparable to polymer relaxation times? Does the heat capacity depend on polymer orientation under flow? Is the linear relationship between stress and thermal conductivity universal? The experiment introduced here offers the possibility to begin attacking these important questions.