Introduction to polymer viscoelasticity 3rd edition




















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Introductory material on fundamental mechanics is included to provide a continuous baseline for. Linear viscoelasticity 1. Creep test of a viscoelastic rod A rod of a polymer is held at a constant temperature, stress free, for a long time, so that its length no longer changes with time. The rod is in a state of equilibrium. At time zero, we attach a weight to the. Download it once and read it on your Kindle device, PC, phones or tablets. Use features like bookmarks, note taking and highlighting while reading Introduction to Polymer cturer: Wiley.

The book takes a multidisciplinary approach to the study of the viscoelasticity of polymers, and is self-contained, including the essential mathematics, continuum mechanics, polymer science and statistical mechanics needed to understand the theories of polymer : Kwang Soo Cho. Description: This book provides a unified mechanics and materials perspective on polymers: both the mathematics of viscoelasticity theory as well as the physical mechanisms behind polymer deformation processes.

Introductory material on fundamental mechanics is included to provide a continuous baseline for readers from all disciplines. Linear viscoelasticity is an extension of linear elasticity and hyperelasticity that enables predictions of time—dependence and viscoelastic flow.

Linear viscoelasticity has been extensively studied both mathematically [19], and experimentally [20] and can be very. A molecular approach to the fundamentals of viscoelastic behavior in polymers, bridging the gap between introductory accounts and advanced research level monographs.

This second edition includes new coverage of the theory of reptation, the kinetic theory of rubber elasticity, and an entirely new chapter on dielectric relaxation. Presents all derivations in detail, and treats concepts and. Viscoelasticity is a subject of great complexity fraught with conceptual difficulties. It is possible to distinguish two basic approaches to the subject which we shall designate as the continuum mechanical approach and the molecular approach.

We can now replace the integral in 7. The contribution from the Brownian forces is only an isotropic stress, as can be shown either by using the expression for the smoothed-out Brownian forces, or by integrating the Langevin equations directly. It can be converted into surface integral at infinity. The second integral on the right 7 Polymer Solutions of 7. Here we care only about the end-to-end vector of the polymer chain, and all interactions between the solvent and the chain are localised at two beads, located at the chain ends, r1 and r2.

The model is also called the linear elastic dumbbell model to emphasise the linear force law being used. Although the general equations have been developed in the previous section, it is instructive to write down all the equations again, for this particular case. The fluctuation-dissipation theorem 7.

Otherwise the flow is weak. Since we are more interested in the end-to-end vector, the process R c can now be discarded.

The Fokker—Planck equation for the density distribution function for R reads 7. First, from 7. Secondly, R and F b have widely different time scales. Thus, from 7.

The model 7. In some numerical applications, 7. The Oldroyd-B constitutive equation qualitatively describes many features of the so-called Boger fluids. In an unsteady state shear flow, the stresses increase monotonically in time to their steady values, without stress overshoots which are sometimes observed with some dilute polymer solutions. It was derived from a network of polymer strands, a model meant for concentrated polymer solutions and melts.

It is remarkable that two models for two distinct microstructures, a dilute suspension of dumbbells and a concentrated network of polymer strands, share a common constitutive framework. Start-Up Shear Flow. This reflects the linear spring in the model that allows the end-to-end vector of the dumbbell to grow without bound in a strong flow.

The prediction of an infinite stress at a finite elongational rate is not physically realistic. It is due to the linear dumbbell model being allowed to stretch infinitely.

Constraining the dumbbell to a maximum allowable length will fix this problem e. The linear elastic dumbbell model is also inadequate in oscillatory flow: it predicts a shear stress proportional to the amplitude of the shear strain, irrespective of the latter magnitude.

With multiple relaxation times, the Rouse model, Fig. The Boger fluids show little shear thinning over a large range of shear rates, but this is no doubt due to the high solvent viscosity that completely masks the contribution from the polymer viscosity; any amount of shear-thinning from the polymer contribution would hardly show up on the total fluid viscosity.

In general, dilute polymer solutions usually show some degree of shear thinning. The fix is to adopt a more realistic force law for the chain. Problems Problem 7. This is the Stokes—Einstein relation, relating the diffusivity to the mobility of a Brownian particle. Problem 7. Chapter 8 Suspensions Particulates Suspension is a term used to describe an effective fluid made up of particles suspended in a liquid; examples of such liquids abound in natural and man-made materials: blood, milk, paints, inks.

When this is not met, we simply have a collection of discrete individual particles suspended in a liquid. Most progress has been made with Newtonian suspensions, i. The review paper by Metzner [57] contained most of the relevant information on the subject. Consequently the microdynamics are also linear and instantaneous in the driving forces; only the present boundary data are important, not their past history.

This does not imply that the overall response will have no memory, nor does it imply that the macroscaled Reynolds number is small. The linearity of the micromechanics implies that the particle-contributed stress will be linear in the strain rate; in particular all the rheological properties shear stress, first and second normal stress differences, and elongational stresses will be linear in the strain rate.

Several investigators have indeed found Newtonian behaviour in shear for suspensions up to a large volume fraction. Shear-thickening behaviour, and indeed, yield stress and discontinuous behaviour in the viscosity-shear-rate relation have been observed, e. This behaviour cannot be accommodated within the framework of hydrodynamic interaction alone; for a structure to be formed, we need forces and torques of a non-hydrodynamic origin.

Furthermore, from the equations of motion in the absence of inertia and the body force, 8. The particle contribution can be decomposed into a symmetric part, and an antisymmetric part. The terms remaining can be identified with the force F p , and the torque T p imparted by the particle p to the fluid. In this case, in a representative volume V we expect to find only one sphere.

Thus, the microscale problem consists of a single sphere in an effectively unbounded fluid; the superscript p on the generic particle can be omitted, and the coordinate system can be conveniently placed at the origin of the sphere. The far-field boundary condition must be interpreted to be far away from the particle under consideration, but not far enough so that another sphere can be expected.

From 8. A similar theory has been worked out for a dilute suspension of spheroids by Leal and Hinch [51], when the spheroids may be under the influence of Brownian motion, using the solution for flow around a spheroid due to Jeffery [41]. If the particles are large enough so that Brownian motion can be ignored, then the last term, as well as the angular brackets, can be omitted in 8. The asymptotic values of the shape factors are given in Table 8. In essence, the viscosity is shear-thinning, the first normal stress difference is positive while the second normal stress difference is negative, but of a smaller magnitude.

The precise values depend on the aspect ratio and the strength of the Brownian motion. The predictions of the constitutive equation 8. As the concentration increases, we get subsequently into the semi-dilute regime, the isotropic concentrated solution, and the liquid crystalline solution. The reader is referred to Doi and Edwards [18] for more details.

Here, we simply note that the concentration region 1 1 is called concentrated, where the average distance between fibres is less than a fibre diameter, and therefore fibres cannot rotate independently except around their symmetry axes.

Any motion of the fibre must necessarily involve a cooperative motion of surrounding fibres. They found that if shearing is stopped after a steady state has been reached in a Couette device, the torque is reduced to zero instantaneously. If shearing is resumed in the same direction after a period of rest, then the torque would attain its final value that corresponds to the resumed shear rate almost instantaneously. However, if shearing is resumed in the opposite direction, then the torque attains an intermediate value and gradually settles down to a steady state.

How would you classify the memory of the liquid, zero, fading or infinite? Problem 8. Clearly close form solutions for more complex models may be very limited and are difficult to find.

In these cases, a more suitable method is required that can handle both the constitutive modelling and the flow problems. In the method, the fluid is modelled by a system of particles called DPD particles in their Newton 2nd law motions. In the original technique, these DPD particles are regarded as clusters of molecules, undergoing a soft pairwise repulsion, in addition to other dissipative and random forces designed to conserve mass and momentum in the mean.

The microstructure of the DPD fluid can be made as complicated as we like: DPD particles can be connected to form strings to model polymer solutions Kong et al. In BDS, the bulk flow field kinematics are specified a-priori; then the effects of the fluid microstructure evolution on the flow field are taken into account by coupling the BDS to the kinematics in an iterative manner.

Furthermore, BDS conserves particles mass , but not momentum. In contrast, DPD method conserves both the number of particles and also the total momentum of the system; its transport equations are of the familiar form of mass and momentum conservation.

In this point of view, DPD can be regarded as a particle-based solver for continuum problems — DPD particles are regarded as fictitious constructs to satisfy conservation laws. We are especially attracted to the DPD method because of the ease and flexibility of its modelling of a complex-structure fluid. The method conserves mass and momentum in the mean, and therefore is not only restricted to mesoscale problems — it is also applicable to problems of arbitrary scales and therefore it may be regarded as a particle-based method for solving continuum flow problems.

We review the technique here in details, together with some test problems of interest, to complete our adventure in the constitutive modelling of complex-structure fluids. The angular brackets denote an ensemble average with respect to the distribution function of the quantity concerned. Of course, there is no need to separate the dissipative and random forces strength into a scalar and a configuration-dependent weighting function in the manner indicated — this is done only to conform with existing DPD notation.

Langevin Equation Since the random force has only well-defined statistical properties, the so-called Langevin equation 9. Specifying an initial state, as done in 9. The velocity space v, or the configuration space r , may be integrated out of f r, v, t , in which case we have a configuration-space, or a velocity-space description of the stochastic system 9. The equation governing the probability is sometimes known as the Fokker—Planck or Smoluchowski, or simply the diffusion equation of the process.

Fluctuation-Dissipation Theorem There are three time scales in the stochastic differential equation 9. Equation 9. Note that the requirement 9. For the stochastic system 9. From the phase space description 9. Configuration Space: Fokker-Planck-Smoluchowski Equation Sometimes, it is more convenient to deal directly with the configuration-space distribution; the velocity-space can be integrated out from 9.

In the limit of small inertia, the stochastic system 9. We continue our discussion of DPD in a 3D setting. In the original viewpoint, a DPD particle may be thought of as a cluster of fluid molecules; alternatively, it may be regarded as a fictitious construct to solve a flow problem of a complex-structure fluid — we will return to these different viewpoints later.

Outside a certain cutoff radius rc , the interactions are zero. Here we may allow the cutoff radius to be different for different type of forces. There are some key ideas in the DPD theory, which we will go through in details. In fact, the system 9. When the conservative forces are chosen appropriately, the associate system is the basis for molecular dynamics MD simulation.

Key Idea 2: Conservative Force is Soft Repulsion The second key idea is that the conservative force is chosen to be a soft repulsion. This allows a much larger time step to be employed easily by a factor of 10 , as compared to a much smaller time step, when a standard MD molecular potential, such as Lennard-Jones potential, is used Keaveny et al. We may consider lumping these connector forces in the conservative forces FiCj as a matter of convenience. The conservative forces together with any connector forces determine the rheology of the system.

Key 3: Dissipative Force is Centre-to-Centre The dissipative force slows down the particle, extracts parts of the kinetic energy injected by the random force.

The fluctuations of the random force inject kinetic energies into and heat up the system, which is then taken away and cooled down by the dissipative force. This balance allows a thermodynamic temperature to be defined. It is to be noted that A is an [N , N ] matrix, with second-order tensors as elements. It is also a symmetric matrix of zero row and column sums; it has zero as one eigenvalue, with corresponding unit eigen-vector.

Likewise, B is an [N , 1] matrix, with three-dimensional vectors as elements. A formal solution of the linear system 9. This, when used in 9. The requirement 9. When the detailed balance, 9. The dissipative and random forces act like a thermostat in the conventional molecular dynamics MD system. Clearly, the addition of the dissipative and random forces to a conservative system may appear to be artificial, but the versatility of DPD lies in its ability to satisfy conservation laws in the mean to be shown later.

It is possible to think of the DPD system as a coarse-grained model of a physical model. Therefore we could construct models of complex-structure fluids by endowing the simple DPD particles with features in a manner similar to modelling, for example, polymeric solution by a suspension of Rouse chains.

Sometimes, a full statistical description f X, t is not required, since the quantities of interest only involve a subset of the state variable, the rest of the state variable can be integrated out. The velocities can be integrated out of 9. First, we re-write 9. Although the second and the third terms are similar in form, the conservative force in second term involves only the configuration whereas the dissipative force in the third term involves both the configuration and the velocity; different treatments will be necessary.

Kinetic Pressure Tensor Although the first term on the right of 9. We first define the peculiar velocity as 9. The first term on the right of 9. Irving and Kirkwood [40] showed how to recast the right side of 9. Conversely, flow problems for a complex-structure fluid may be solved by this particle-based method; the stress i. Of the total stress 9. Stress from Conservative Forces We first re-write the conservative-forcecontributed stress 9. Next the sum can be taken inside the integral: 9.

Irving and Kirkwood pointed out that all the terms inside the curly brackets in 9. Problem 9. The third term on the right of 9. This limit is thought to be relevant to the general DPD case. This limit is not equivalent to a steady-state flow assumption - it only guarantees that the inertial terms as represented by Reynolds number in the momentum equations are zero, but any other time-dependent behaviour inherited, for example from the boundary conditions, has not been eliminated.

Taking the dissipative force to the left side, 9. Below are two schemes proposed to deal with this singularity. Scheme 1 We first record a physical constraint for the DPD system. The present DPD system can be solved by a direct solver such as the one based on the LU factorisation with partial pivoting, with a high computational cost at large number of particles. Scheme 2 The system matrix A in 9. Therefore solution of 9. If the previous spectral radius before deflation is one, it is now less than one and iterative numerical implementation to 9.

Furthermore, it is straightforward to implement the iterative scheme in parallel as 9. For both Schemes 1 and 2, the Euler algorithm is employed to advance the position of the particles 9. The standard weighting function 9.

Boltzmann temperature is also normalised to unity, k B T. One could provide an estimate for this effective size of a DPD particle by various means. Role of the Conservative Force The inclusion of the conservative force repulsive force into the DPD formulation is to provide an independent mean of controlling the speed of sound compressibility to the number density and the temperature of the DPD system [56].

Here, we limit our attention to the case where the compressibility of the system is matched to that of the water at room temperature [33].

If the weighting function wC is fixed at its linear form, the size of solvent particles induced by the conservative forces will be controlled by means of the repulsion parameter ai j. A larger value of ai j results in a larger size of the particle and vice versa. From expressions 9. This can be equated to the diffusivity in 9. Note that expression 9. The DPD particles representing the solvent phase are assumed not to be clustered, and are of a size considerably less than that 9. This latter condition is of course not satisfied here and the expression 9.

However, the solvent particle size approaches zero as any one of the number density n, the cutoff radius rc or the thermodynamic temperature k B T approaches infinity. Although expression 9. Figure 9. However, in Fig. Special care is thus needed if one tries to control the solvent particle size via k B T.

It is noted that i reducing k B T makes ai j smaller which can result in the clustering of particles; and ii the particle size, defined in 9.

These observations imply that i controlling particles size via n and rc is clearly more effective than via k B T ; and ii increasing rc results in a faster decrease in the particle size than increasing n. From this approximate analysis, we can see that there are many possible combinations of n and rc that reduces ae f f.

Therefore the Schmidt number is about unity. For a real fluid of physical properties like those of water, the Schmidt number is O , and therefore there is the need to improve on the dynamic behaviour of the DPD system. A simple way to do this has been proposed Fan et al. From 9. With the standard choice of DPD parameters, a simple change Table 9. Clearly, the most efficient way to increase Sc is to increase rc. We prefer to combine the modified weighting function with a moderate increase in the cutoff radius for dissipative weighting function, so that a physical level of Sc can be reached with a moderate increase in the computation cost.

In this section, we will explore the response of a DPD fluid with respect to its parameter space, where the model input parameters can be chosen in advance so that i the ratio between the relaxation and inertia time scales is fixed; ii the isothermal compressibility of water at room temperature is enforced; and iii the viscosity and Schmidt number can be specified as inputs. These impositions are possible with some extra degrees of freedom in the weighting functions for the conservative and dissipative forces.

Substitution of 9. Small time steps are required for a numerical simulation, but particles are well distributed because of a low Mach number. The low-mass DPD system will be further discussed in Sect. Also, as rc increases from 1 to 2. To do so, apart from ai j , there is a need for having another free parameter. Expression 9. Equations 9. Analytic solution to 9.

For 9. As shown in 9. We are interested in the case where the dissipative contribution is a dominant part, i. Results by the velocity-Verlet algorithm are also included. It can be seen that the first-order ETD algorithm works effectively for relatively-large time steps. Furthermore, for a given small time step, the ETD algorithm is much more accurate than the velocity-Verlet algorithm.

Velocity-Verlet algorithm fails to converge except at small time steps, and the associated errors are much larger than those produced by the ETD algorithm. Velocity, temperature and number density results obtained with the first-order ETD scheme are presented in Fig.

We obtain a parabolic velocity profile in the x direction, as expected. Percentage errors relative to the exact value obtained by 9. Convergent-Divergent Channel Flow Next we consider a more complex flow: a simple DPD fluid flows through a periodic channel with abrupt contraction and diffusion shown in Fig.

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Dover Publications, Inc. New York,. This book describes many boundary value problems.



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