Article pubs.acs.org/IECR
On Thermodynamic Consistency of a Stochastic Constitutive Model for Glassy Polymers Grigori A. Medvedev and James M. Caruthers* School of Chemical Engineering Purdue University West Lafayette, Indiana 47907, United States ABSTRACT: A full understanding of the thermomechanical behavior of materials in the glassy state remains one of the outstanding challenges in condensed matter physics. A recently developed stochastic constitutive model (SCM) that explicitly acknowledges nanoscale dynamic heterogeneity has had success in describing a number of nonlinear mechanical and relaxation experiments. The model employs stochastic differential equations to describe evolution of the local thermodynamic variables of entropy and the stress tensor, where the magnitude of the fluctuations is itself a strong nonlinear function of these variables. Notwithstanding the successes of the model, there is a legitimate question whether the formulation is thermodynamically consistent because, unlike the SCM, the stochastic systems treated in the literature satisfy the fluctuation dissipation theorem and assume constant magnitude of fluctuations. We propose expression for the free energy that results in fulfilment of the H-theorem statement and develop the second law in the Clausius−Duhem form for the SCM, where the expression for the power of the external driving forces is modified in the stochastic case as compared to that in standard continuum mechanics.
■
INTRODUCTION Recently, a stochastic constitutive model (SCM) has been developed for describing the thermomechanical behavior of glassy polymers.1 The SCM is a three-dimensional, tensorially correct constitutive model that extended an earlier onedimensional, nonlinear, viscoelastic volume relaxation model that included fluctuations developed by Medvedev, Starry, Ramkrishna, and Caruthers.2 The SCM has been able to predict several experimentally observed features of the glassy behavior that have proven difficult to describe by the traditional constitutive models for glassy polymers.1,3 The SCM is different from the standard models of continuum mechanics in that it describes material at a nanoscale, where the fluctuations of the thermodynamic variables are large and, hence, have to be explicitly taken into account. Thus, the governing equations describing the evolution of the local thermodynamic variables in the SCM are stochastic differential equations (SDE).4 The complete description of the material response is in terms of the distribution density function satisfying the Fokker−Planck equation (FPE)4 and the macroscopically measured quantities are obtained as ensemble averages. At a first glance, the SCM approach can be thought of as a particular case of “continuum mechanics with fluctuations”, a field that lately has been getting considerable attention.5 One of the major concerns in this field is the question of the thermodynamic consistency for the situation when the fluctuations are not just a small perturbation or the system is far from equilibrium. These issues can be legitimately raised as well in the case of the SCM notwithstanding its successes in describing experimental data. At this point, one possible approach would be to wait until the general theory has been worked out in the hope that it will be applicable to the SCM. However, examining the literature, the SCM has unique features currently not being considered. Hence, this communication attempts to address the question of thermodynamic consistency of the SCM. © XXXX American Chemical Society
Standard continuum mechanics is built upon the assumption that the medium is homogeneous and that all physical quantities are sufficiently smooth for mathematical operations like differentiation to be well defined. Once the continuum approximation has been made, thermodynamic consistency can be assured using the rational thermodynamics methodology of Truesdell, Coleman, and Noll,6 where the second law of thermodynamics in the form of the Clausius−Duhem inequality7 is imposed as a constraint on the various thermodynamic field variables. The result of this approach is that the constitutive equations for the stress, entropy and other thermodynamic field variables are all constructed from the Helmholtz free energy. This approach has been used to construct thermodynamically consistent constitutive models for elastic materials,7 viscoelastic materials,8 and for viscoelastic materials that relax on a material time scale9the latter being of particular significance for the description of behavior of glassy polymers. A classic example of a thermodynamically consistent theory of a system with fluctuations is fluctuation hydrodynamics (see for example Landau and Lifshitz10), where the quantities experiencing fluctuations are the local fields such as velocity, pressure, and so forth. In this analysis, the standard momentum and energy balance equations of continuum mechanics, respectively, acquire an additional stochastic stress and stochastic heat flux terms, where the stochastic contributions are small perturbations of the underlying continuum mechanic equations. The magnitudes of the Langevin sources are set by requiring that the small fluctuations in thermodynamic variables be normally distributed when the system is in equilibrium. The Special Issue: Doraiswami Ramkrishna Festschrift Received: April 9, 2015 Revised: July 15, 2015 Accepted: July 16, 2015
A
DOI: 10.1021/acs.iecr.5b01347 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research key here is that the fluctuation hydrodynamic equations can be derived from the more fundamental equations of statistical mechanics for which the thermodynamic consistency had already been established (although for simplicity the defining equations are often postulated in textbooks).11 In contrast, situations when the fluctuations are not small or the underlying dynamic equations are highly nonlinear are an active area of research where general results are yet to emerge. A particular case that has lately gathered significant attention is biological/cellular systems with an emphasis on the molecular motors.5b,12 These are driven systems far from equilibrium (which is yet another way of saying that these are systems strongly interacting with the environment), where fluctuations are important because of the small system size. Another example of a fluctuating system far from equilibrium and subject to external driving is the soft glass rheology model, for which the thermodynamic consistency has recently been established.13 The SCM as formulated in ref 1, which is a theory describing a nanometer size domain in a glassy material, is in some ways similar to these systems, although there are important distinctions that will be discussed later. Operationally, we would like (i) to propose a meaningful expression for the free energy of the SCM that satisfies the Htheorem in the absence of external driving and (ii) to establish a relation that is equivalent to the Clausius−Duhem inequality in case of the externally driven system. This program is different (and in a way opposite) than that of rational thermodynamics.7 In the rational thermodynamics, an expression for free energy is postulated and the restrictions imposed by the Clausius− Duhem inequality serve to obtain the resulting constitutive equations for entropy and stress. In the case of the SCM considered here, the free energy is unknown, the constitutive equations are postulated, and a free energy is constructed that is consistent with the second law. The paper is organized as follows. In the next section, we will briefly review the standard case of Brownian particles in order to introduce the relevant stochastic machinery. Subsequently, in the section on the SCM, we first consider the nonfluctuating macroscopic limit of the SCM and then treat the fluctuation case. Finally, we conclude with a discussion of implications and questions that need to be addressed in the future.
First, consider the purely mechanical system of a massless particle in a potential field U(x) in a viscous medium. The equations of motion are given by ς
dxi ∂U (x) =− + Fi(t ) ∂xi dt
(1)
where x = (x1,x2,x3) is the vector of the particle position, ς is the coefficient of friction, and F(t) is the external/driving force that does not result from a potential. Assume now that the particle is Brownian, that is, the particle is small enough to sense the fluctuations in molecular bombardment by the surrounding medium. Then the force F(t) has both a systematic and random components, where the latter is of the Langevin type. Then eq 1 is replaced with an SDE given by T
dxi = −
1 ∂U (x) 1 dt + Fi(t )dt + ς ∂xi ς
D dWi
(2)
where dW is Wiener process and D is the intensity of the fluctuations. For simplicity, we will use the same notation for both the stochastic and nonstochastic position vector x. The FPE corresponding to eq 2 is ∂ ∂ p = −∑ J ∂t ∂ xi i i Ji = −
(3a)
1 ∂U (x) 1 ∂p 1 p− D + Fi(t )p 2 ∂xi ς ∂xi ς
(3b)
where p(x,t) is the probability density distribution for the stochastic variable x and J is the probability flux vector. In the absence of the driving force, eqs 3 has an equilibrium solution pe(x) satisfying J = 0, namely pe (x) =
⎡ 2 ⎤ 1 exp⎢ − U (x)⎥ ⎦ Z ⎣ Dς
(4)
where Z is the normalization constant. Equation 4 becomes the Boltzmann distribution if the Einstein relations apply, specifically
D=
■
2kBθ ς
(5)
where θ is the temperature and kB is the Boltzmann constant. Equation 5 is an example of the fluctuation dissipation theorem (FDT).15 FDT is essentially a linear response statement conveying the idea that in case of small perturbations the system near equilibrium relaxes similarly whether said perturbation is caused by external force or by an internal spontaneous fluctuation. Taking into account eqs 4 and 5, the expression for the flux (eq 3b) can be rewritten as
BROWNIAN PARTICLE IN A VISCOUS MEDIUM Historically, the first (as well as the most straightforward conceptually) fluctuating system to be studied was the Brownian motion of a colloidal particle in a viscous medium as treated by Einstein, Smoluchowski, and Langevin.14 For a Brownian particle, the origin of the stochastic Langevin force appearing in the dynamic equation (i.e., momentum balance) is unambiguous and its magnitude is prescribed by the fluctuations dissipation theorem (FDT). Also, the thermodynamic quantities such as free energy, entropy, and so forth for an ensemble of Brownian particles in a thermostat are calculated according to the standard statistical mechanic expressions. As a result, the thermodynamic consistency in the form of the H-theorem and the Clausius−Duhem inequality is readily established. In briefly rederiving these classical results here, we have a 3-fold goal: to introduce a consistent notation used throughout the paper, to establish a technique for treating systems with fluctuations of the Langevin type, and to have a reference for the case of the SCM considered in the next section.
Ji = Jiint + Jiext = −
kBθ ∂ ⎡ p ⎤ 1 p ln⎢ ⎥ + Fi(t )p ς ∂xi ⎣ pe ⎦ ς
(6)
The first term in eq 6 is internal and vanishes at both equilibrium and as θ → 0. The second term is a consequence of the external driving force. Equation 6 again emphasizes the similarity of the probability flux due to external and internal perturbations. When considering an ensemble of Brownian particles described by the probability density p(x,t), the free energy of the system is obtained according to the standard statistical mechanical recipe as B
DOI: 10.1021/acs.iecr.5b01347 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research 1 U (x) = −ln[pe (x)] + ln[Z ] kBθ
Ψ(t ) = E(t ) − θS(t ) =
∫ dxp(x, t )U(x) + kBθ ∫ dxp(x, t )ln[p(x, t )]
Equation 13 is a consequence of eqs 4 and 5. Substituting eq 13 into eq 7, we have
(7)
where E is the average internal energy and S is the average entropy for which the Gibbs formulation has been used, that is
∫ dxp(x, t )ln[p(x, t )]
S(t ) = − k B
Ψ(t ) = Ψ e + kBθ
∂p dΨ = dx{U + kBθ(ln[p] + 1)} dt ∂t ∂ = − dx{U + kBθ(ln[p] + 1)} ∑ Ji i ∂xi
∫
∫
⎧ ∂U
∑ ∫ dxJi ⎨
⎩ ∂xi
i
+ kBθ
⎫ ∂ ln[p]⎬ ∂xi ⎭
(9)
It follows from eq 3b by rearranging terms that J ∂U 1 ∂ = −ς i − ςD ln[p] + Fi(t ) ∂xi p 2 ∂xi
(10)
THERMODYNAMIC CONSISTENCY OF THE STOCHASTIC CONSTITUTIVE MODEL 1. Deterministic Limit of the SCM. We begin by postulating that a thermodynamic system with memory is described by the Helmholtz free energy expression that is quadratic
⎧ J ∑ d xJi ⎨−ς i + ⎜⎛⎝kBθ − 1 ςD⎟⎞⎠ ∂ ln[p] 2 ∂xi ⎩ p i ⎫ + Fi(t )⎬ ⎭
∫
= −ς
∫ dx 1p ∑ Ji2 + ∑ Fi(t ) ∫ dxJi i
i
ψ (t ) = ψ e +
=
∑ Fi(t ) i
i
= −∑ Fi(t ) i,j
=
i
⎛
t
⎞ dHi(ξ)
∫−∞ exp⎜⎝− t* −τ ξ* ⎟⎠
M i (t ) =
0
dξ
dξ (16)
where H(t) is a seven-component vector of temperature and deformation history introduced for compactness of notation such that H0 ≡ θ and the components H1 through H6 are the six components of the symmetric strain tensor (see ref 1 for details). Formally, this is the structure of the thermoviscoelastic constitutive model of Caruthers and co-workers,9 where just a single relaxation time is assumed as opposed to a spectrum typically employed in that earlier work. The nonlinearity in the system is due to the material time t* in eq 16 given by
∂p ∂t
∫ dxxi ∂∂x Jj
∑ Fi(t ) ∫ dxJi
(15)
where the matrix of the coefficients Δij is positive-definite and symmetric. The memory function is given by
dxi dt
∑ Fi(t ) ∫ dxxi
∑ ΔijMi(t )MJ(t ) i,j
(11)
The Einstein relation eq 5 has been used during the second step of deriving eq 11. The first term on the right-hand-side of eq 11 is entropy production and because of the “minus” sign it is clearly nonpositive. Also, as expected the entropy production term becomes zero at equilibrium, that is, when Ji = 0. The second term is the power of the external/driving forces. To see this, we calculate dW ext = dt
(14)
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and when substituted into eq 9, this gives rise to dΨ = dt
⎡ p(x , t ) ⎤ ⎥ pe (x) ⎦
∫ dxp(x, t )ln⎢⎣
The quantity Ψ(t) defined according to eq 14 satisfies the Clausius−Duhem statement without reference to the potential. All that is required is that the equilibrium distribution exists. Moreover, as shown in Appendix A, the H-theorem holds in case of a system described by an SDE (and hence FPE), where both the drift and the noise magnitude are functions of the stochastic variable x. This is more general than the case described by eq 2, where the noise magnitude √D is a constant at a given temperature. When the free energy expression given by eq 14 is used, the H-theorem follows (see eq A.11). Critically, this result does not require the Einstein relations (i.e., the FDT) to hold. To summarize the conclusions of this section, for a system described by an FPE, assuming that it has an equilibrium solution, it is possible to construct a “free energy” according to eq 14 such that it satisfies the H-theorem. The FDT is not required for this result.
(8)
Differentiation of eq 7 with respect to time and using eq 3a results in
=
(13)
j
t* = (12)
∫0
t
1 dξ a (ξ )
(17)
where the dependence of the shift factor a on time is consequence of its dependence upon the time dependent thermodynamic variables, which will be specified shortly. The constitutive equations for entropy and the components of the stress tensor are obtained as derivatives of eq 15 in conjunction with the symmetry of the Δij matrix
Therefore, eq 11 is the second law statement in the Clausius− Duhem form for the isothermal case. In the absence of external driving, that is, Fi = 0, eq 11 becomes the H-theorem for the free energy defined according to eq 7. Although eq 11 establishes the thermodynamic consistency of the system of Browninian particles experiencing the potential U (x), the following generalization is useful. It is possible to eliminate the potential from the expression in eq 7 using
xi = C
∂ψ = ∂Mi
∑ 2ΔijMi j
(18) DOI: 10.1021/acs.iecr.5b01347 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research ⎡ ⎞ ∂ 1 ⎛ ⎜⎜xi + σi2 ln a(x)⎟⎟ + dxi = ⎢ − ⎢⎣ a(x)τ0 ⎝ ∂xi ⎠
where x0 is the entropy and x1 through x6 are the components of the stress tensor, which is conjugated to the chosen strain tensor. Equation 18 is inverted as Mi =
∑ j
1 −1 (Δ )ij xj 2
+ σi (19)
(20)
the constitutive equations can be cast in the closed differential form dxi 1 xi + =− dt a(x)τ0
∑ 2Δij
dHj(t )
j
dt
(21)
In eq 21, the main assumption of the SCM model is given; specifically, that the shift factor is a function of the x variables, that is, entropy and the six components of the stress tensor. The free energy can be expressed in terms of the x variables as ψ (t ) = ψ e +
∑ i,j
1 −1 (Δ )ij xixj 4
∑ 1 (Δ−1)ij xi i,j
2
1 =− a(x)τ0
(22)
∑ i,j
∑ xi
dt
i
dHi 1 =− dt a(x)τ0
∑ i,j
(26a)
1 1 ∂ xip(x , t ) − σi2 p(x , t ) a(x)τ0 a(x)τ0 ∂xi dHj p (x , t ) + ∑ 2Δij dt j
dH ∑ xi i dt i
1 −1 (Δ )ij xixj ≤ 0 2
(26b)
where p(x,t) is the probability density distribution for the stochastic variable x. Note that the form of eqs 26 implies that the Ito interpretation4 of the stochastic integral has been used which may have real physical consequences in case of the multiplicative noise as in eq 25. An important caveat, however, is that it can be shown using the derivation given in ref 1 that the requirement that the equilibrium distribution (see eq 27 below) be Gaussian results in the FPE that is invariant of the interpretation of the stochastic integral in the underlying SDE (e.g., even if the Stratonovich interpretation4 is used). Comparing eq 26 to eqs 3 above, we see that the main difference is in the probability flux, which in case of eq 26b does not come from a potential. In the absence of the “driving force”dH/dt eq (26) has an equilibrium solution pe(x) satisfying J = 0, namely
(23)
Upon rearranging terms in eq 23, we arrive at dψ − dt
(25)
Ji = −
dxj 1 −1 (Δ )ij xixj + 2
2 dWi a(x)τ0
∂ ∂ p = −∑ J ∂t ∂ xi i i
Taking the time derivative, we find dψ = dt
j
dHj ⎤ ⎥d t dt ⎥⎦
Two comments are in order: 1. As compared to the deterministic eq 21, the underlined quantities appear in the stochastic version, where σi is the magnitude of fluctuations in xi when the system is at equilibrium. For notational brevity, we are using the same notation for macroscopic and mesoscopic variables. The first underlined quantity in the drift term is required in order that the equilibrium distribution be Gaussian as explained in ref 1. 2. As compared to the Brownian particle case eq 2, the restoring force in the drift term is not due to a potential, that is, it does not have the form of a derivative of some U(x). Also the fluctuation magnitude, that is, the factor in front of the Wiener process is itself x-dependent. The Fokker−Planck equation corresponding to eq 25 is given by1
Considering the fact that the memory integrals eq 16 satisfy the differential relations dM i dHi(t ) 1 = − Mi + dt aτ0 dt
∑ 2Δij
(24)
The expression on the left-hand side is the Clausius−Duhem combination (i.e., x0 dH0/dt = S dθ/dt where S is the entropy, etc.) and the expression on the right-hand side is clearly nonpositive; therefore, the second law is established. This is not surprising since the initial expression for the free energy is a particular case for a single relaxation time of the thermoviscoelastic constitutive model of Lustig et al. for which the Clausius− Duhem inequality served as the point of departure.9a Unlike in the case of a particle (see eq 1 in the previous section), the question of thermodynamic consistency is meaningful here even in the absence of fluctuations because the system is described in terms of the thermodynamic variables. 2. Consequence of Fluctuations in the SCM. Now, assume that the system under consideration is small and hence subject to fluctuations. This would be the case if one were modeling a nanometer size region of material. The addition of fluctuations gives a new level of complexity to the single relaxation time constitutive model. The set of stochastic differential equations (SDE) describing evolution of local (entropy and stress) variables in the SCM is given by (see ref 1)
pe (x) =
⎛ 1 1 ⎜⎜ − exp ∏i (2πσi)1/2 ⎝ 2
∑ i
xi2 ⎞ ⎟⎟ σi2 ⎠
(27)
The fundamental difference between the stochastic and deterministic cases is that unlike the deterministic eq 21, the stochastic eq 25 does not result from differentiation of a Helmholtz free energy and is essentially a postulate. In fact, obtaining a physically reasonable expression for the free energy corresponding to ensemble of domains described by eq 25 would constitute a significant step toward the overall goal of establishing the thermodynamic consistency of the SCM. ̈ A “naive” way of going about constructing a free energy would be to consider an expression similar to eq 7 for the Brownian particles; specifically D
DOI: 10.1021/acs.iecr.5b01347 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research Ψ′ = ⟨ψ ⟩ − θS =
V kBθ L3
∫ d x p ( x , t )ψ ( x ) +
d V Ψ′ = 3 kBθ ∑ dt L i
∫ dxp(x, t )ln[p(x, t )]
The term ⟨ψ⟩ in eq 28 is just the ensemble average of the free energy of the domains and the second term is the entropic contribution, where the factor V/L3 is the number of domains of size L per unit mass and V is the specific volume. Now, we evaluate the time derivative of Ψ′ in an attempt to establish the H-theorem. First we take the derivative of the average free energy, that is, the first term in eq 28 = =
d dt
d V Ψ′ = − 3 kBθ ∑ dt L i
(36)
Π=
i,j
∫
∂p xixj dx ∂t
Π=
1 = ∑ (Δ−1)ij xi Jj dx i,j 2 1 ∂ = −∑ (Δ−1)ij Jj σi2 ln pe dx 2 xi ∂ i,j
∫
(30)
d V kθ p ln pdx 3 B dt L ∂p V = 3 kBθ (ln p + 1) dx ∂t L V ∂ = 3 kBθ ∑ Ji ln pdx ∂xi L i
∑
=− (31)
∫ Ji σj2 ∂∂x ln pe dx + j
V kB L3
∂ Ji ln pdx ∂xi
(32)
We would now like to proceed in the same way as in the development of eqs 10−11. First, we notice that eq 26b can be rearranged as follows ⎤ 1 ⎡ ∂ ⎢xi + σi2 ln p⎥ + ∂xi a(x)τ0 ⎣ ⎦
=−
1 ∂ ⎛ p⎞ σi2 ln⎜ e ⎟ + a(x)τ0 ∂xi ⎝ p ⎠
∑ 2Δij j
∑ 2Δij j
dHj dt
dHj dt
(33)
Now, if it were true that the following equation (which can be called the “generalized Einstein relations” or the FDT) holds δijσi2 = 2Δij
i
⎤ ∂ ln pe ⎥dx ∂xi ⎦
∫ pxidx
dHj 1 V k θ 2 2Δij ⟨xi⟩ 3 B dt L σi
(38)
V kBθ L3
dHi ⟨xi⟩ dt
(39)
∑ i
dHi ⟨xi⟩ dt
V kBθ ∑ L3 i
∫
⎡ ∂ ⎛ p ⎞⎤ 2 p σi2⎢ ln⎜ e ⎟⎥ dx a(x)τ0 ⎢⎣ ∂xi ⎝ p ⎠⎥⎦
(40)
Equation 40 is the Clausius−Duhem form of the second law, which should be compared to the deterministic case given in eq 24. Unfortunately, eq 34 is a critical step in arriving at eq 40, but eq 34 is not fulfilled in case of the SCM developed in ref 1. This is because the equilibrium distribution given in eq 27 does not come from the Helmholtz free energy eq 22. This is very different from the case of the Brownian particle considered above where the equilibrium distribution eq 4 is fully determined by the potential U(x). Specifically, consider the Helmholtz free energy as a function of entropy and stress given by eq 22. This expression in turn stems from the Helmholtz free energy being a functional of temperature and strain histories given by eq 15. The matrix Δij has off-diagonal terms representing the so-called thermal stress effects, which are well known experimentally. To omit these terms would not be true to the physics of glassy polymers. On the other hand, the matrix in the left-hand side of eq 34, which is the matrix of the equilibrium correlators ⟨xixj⟩e, is in fact diagonal. This is because the equilibrium cross correlator of entropy and stress is zero as is well known from the statistical mechanics.16 The fact that eq 34 does not hold demonstrates that the fluctuation dissipation theorem is violated in case of the SCM, which complicates its mathematical treatment; however, as the above
Combining eqs 29 and 31
p
−p
which is the power of the external driving. Substituting eq 39 in eq 36, we obtain d Ψ′ − dt
∫
d 1 Ψ′ = −∑ (Δ−1)ij dt i,j 2
∑ i
∫
=−
⎡ ∂p
∫ ⎢⎣ ∂x
=
Π=
∫
Ji
dHj V 2Δij kθ 3 B ∑ dt L i,j
i,j
Now, we take the time derivative of the second, i.e., the entropic, term in eq 28. We have
∫
(37)
Using eq 34 the eq 38 is converted to
∂ ln pe ∂xi
i
i
dHj 1 V 2Δij kθ 3 B ∑ 2 dt L σi i,j
(29)
where we have used the symmetry of the matrix Δ and the fact that
θ∑
⎛ p⎞
∫ p ∂∂x ln⎜⎝ pe ⎟⎠dx
=
∫
xi = −σi2
dHj V kθ 2Δij 3 B ∑ dt L i,j
Finally ∏ is evaluated as follows
i,j
∑
∫
⎡ ∂ ⎛ p ⎞⎤ 2 p σi2⎢ ln⎜ e ⎟⎥ dx + Π a(x)τ0 ⎢⎣ ∂xi ⎝ p ⎠⎥⎦
where
∫ ∑ 14 (Δ−1)ij xixjp(x, t )dx 1 −1 (Δ )ij 4
(35)
i
Using eq 33 again to express Ji, eq 35 becomes
(28)
d ⟨ψ ⟩ dt
⎛ p⎞
∫ Ji ∂∂x ln⎜⎝ pe ⎟⎠dx
(34)
then using eqs 33 and 34, eq 32 can be transformed to E
DOI: 10.1021/acs.iecr.5b01347 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research
friction coefficient remains independent of position; to have the FDT violated as to mimic the SCM situation is the goal of this exercise. The probability flux eq 3b becomes
discussion shows it appears a necessary price for capturing an important aspect of the glassy behavior, that is, presence of thermal stresses. It should be clear from the analysis in the preceding ̈ construction of the free energy Ψ′ as paragraph that the “naive” given in eq 28 is not appropriate. Instead, we propose to use the form of the free energy as given by eq 14. This form does not depend on the particular form of the potential or even on the existence of a potential. All that is required is the existence of the equilibrium distribution. Thus, the expression eq 14 can be used as is with the only difference being the number of domains given by the factor V/L3. By taking the time derivative of eq 14, we obtain V dΨ = 3 kBθ dt L =
⎡ ⎛ p⎞
Ji = −
2 ∂U (x) 1 ∂D(x) ∂ ln[pe (x)] = − − D(x) ∂xi ∂xi ςD(x) ∂xi
⎛ p⎞
dΨ = dt
(41)
i
Examining eq 41, it is the same as eq 35; however, unlike in the case of Ψ′ arriving at eq 41, its derivation did not require the use of eq 34, which does not hold for the SCM as explained above. Now, we can proceed by using eq 33 in the form of ⎧ ∂ ⎛ p ⎞ a(x)τ0 ⎪ Ji ⎨− + ln⎜ e ⎟ = ∂xi ⎝ p ⎠ σi2 ⎪ ⎩ p
⎫ dH ⎪ ∑ 2Δij J ⎪⎬ dt ⎭ j
=−
τ V k θ ∑ 02 3 B L i σi
(42)
E (t ) =
J2
1 ∂U (x) 1 dt + Fi(t )dt + ς ∂xi ς
(43)
D(x) dWi
⎩
⎫ 1 ∂ ς D(x)⎬ 2 ∂xi ⎭
(47)
∫ d x p( x , t ) U ( x)
(48)
Or alternatively, the entropy is not given by the standard Gibbs formula eq 8. This is despite the fact that the particle is subjected to the potential U(x)as before and the only difference is in the noise term where the average of the noise is still zero. At the same time, if eq 14 is used for the free energy instead of eq 7, the H-theorem readily follows because the SDE in eq 44 is a particular case of the general SDE as developed in Appendix A. Finally, when the driving force is not zero the Clausius−Duhem relations have the form
The eq 43 has the desired Clausius−Duhem like form, where the term on the right-hand-side is nonpositive; however, the power of the external driving term on the left-hand side clearly differs from the expression given in eq 39, which is a simple average of the deterministic formula eq 24. Note that this difference does not disappear in the deterministic limit, which implies that σ2i → 0 and L3 → ∞ in such a way that σ2i L3 is finite. Therefore, the driving power term in eq 43 does not reduce to the form of eq 24. We have just demonstrated that the SCM developed in ref 1 satisfies the H-theorem statement in the absence of external driving when the free energy is defined by eq 14. Also, when the external driving is applied, we obtained a Clausius−Duhem type relation as given in eq 43. The caveat is that the power of the driving force in eq 43 is nonstandard. To better understand the reason for this, it is instructive to consider the more familiar case of a Brownian particle, where modifications are made to render it similar to the SCM. To that purpose, consider the generalization of the eq 2 of the form dxi = −
i
⎛ ςD(x) ⎞ ∂ + ⎜kBθ − ⎟ ln[p] 2 ⎠ ∂xi p ⎝ Ji
Clearly, the right-hand side of eq 47 does not have the negative definite form. Thus, it can be stated that the energy of the ensemble of Brownian particles is not given by the simple expression
∫ a(x)Ji dx
∫ a(x) pi dx
⎧
∑ ∫ dxJi ⎨−ς −
Substituting eq 42 in the eq 41, we arrive at dHj τ0 V dΨ − 3 kBθ ∑ 2Δij dt σi2 dt L i,j
(46)
It is assumed that the combination of U (x) and D(x) satisfy the potential conditions given in eq A.5. It is straightforward to see that the free energy described by eq 7 no longer possesses the H-theorem property. Proceeding as in eqs 9−10 (and considering the F(t) = 0 case for simplicity), we have instead of eq 11
⎤ ∂p
∫ Ji ∂∂x ln⎜⎝ pe ⎟⎠dx
(45)
And the equilibrium distribution satisfies the equation
∫ ⎢⎢⎣ln⎜⎝ pe ⎟⎠ + 1⎥⎥⎦ ∂t dx
V kBθ ∑ L3 i
1 ∂U (x) 1 ∂ 1 (D(x)p) + Fi(t )p p− 2 ∂xi ς ∂xi ς
dΨ 2 − kBθ ∑ Fi dt ς i
∫
Ji D(x)
dx = −2kBθ ∑ i
∫
Ji2 pD(x)
dx (49)
Comparing with the previous case eq 11, the power of the external forces in the left-hand-side of eq 49 is not given by the simple expression of eq 12. These conclusions demonstrate that the case of the Brownian particle with the nonconstant fluctuation magnitude as given in eq 44 and the lack of the FDT results in the same unusual thermodynamic relations as the SCM. What is the source of these changes to the standard thermodynamic formulas? So far, we have not discussed a physical situation under which the x-dependent fluctuation magnitude may arise in case of Brownian particles. One possible experimental setup is to have a cell containing the liquid with the particles where opposite ends of the cell are maintained at different temperatures whereby creating a temperature gradient (and a heat flux) through the cell. However, the nonisothermal case has added mathematical complexity that we would like to avoid; besides, our above derivation only treated the isothermal case. So, instead of the temperature gradient, we postulate existence of unspecified
(44)
The difference between eqs 2 and 44 is that the noise magnitude in eq 44 is no longer constant but rather dependent on the particle position. This in turn implies that the FDT in the form of eq 5 no longer holds, because by assumption the F
DOI: 10.1021/acs.iecr.5b01347 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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relaxation time Debye process for the stress and entropy plus a Wiener process that acknowledges the temporal fluctuations that are a clear feature of dynamic heterogeneity. The key nonlinearity is that the local relaxation time is a strong, asymmetric, nonlinear function of the local state variables, which is consistent with experimental observations of the glassy state. Another postulate, which is admittedly much less well justified, has also been used in developing the current version of the SCM: that the interactions between neighboring domains can be treated in a mean field sense.1 Although the local mobility/state postulate appears to be physically appealing, and perhaps even obvious, we are unaware of a rigorous continuum framework with analysis of thermodynamic consistency that is available for a nonlinear system with fluctuations of the type found in the SCM. The analysis reported in this paper is the first step toward the development of the needed framework, although we fully acknowledge that the work begun in this paper is far from complete. In the analysis of thermodynamics of the SCM there were three main assumptions/results: 1. In the SCM, it was assumed that the local/mesoscopic stress and entropy are described by the Langevin type SDE given by eq 25. This particular form of the equations is dictated by the requirement that the equilibrium distribution be Gaussian, that is, eq 27 in agreement with the requirements of statistical physics. This is a much stronger condition than simply requiring the variances of the small fluctuations of the thermodynamic variables to have values given by the equilibrium thermodynamics. In fact, it prescribes the equilibrium distribution for the fluctuations of arbitrary size. The resulting SDE given by eq 25 contains the fluctuation magnitude that is a strong function of the fluctuating variable. This particular form of the SDE is unusual in physical applications, although it is not unknown in the mathematical literature.4 2. The expression for the free energy Ψ was assumed to be of the form given by eq 14, which is shown to produce the H-theorem statement in case of a general SDE (see Appendix A). The fluctuation dissipation theorem is not required for this to be true. 3. Using 1 and 2 it was determined that a Clausius− Duhem-like form of the Second Law could be derived, that is, eq 43. The internal dissipation term (i.e., righthand side of eq 43) is nonpositive consistent with the expectations of thermodynamics. However, the detailed form of the power of external driving term is different from that in the deterministic Clausius−Duhem equation, that is, eq 39, and the difference between eq 43 and eq 39 does not disappear in the deterministic limit. It is shown that the nonstandard external driving power term is due to the violation of the FDT. The formulation of the SCM is quite different from other continuum formulations that include fluctuations. Specifically, in the SCM approach the FDT, that is, eq 34 is violated. As discussed above this is a result of two essential requirements: (i) that the equilibrium correlators of the fluctuations in entropy and stress are the ones dictated by statistical physics and (ii) that the instantaneous response to changes in externally controlled temperature and strain is the one observed macroscopically. Thus, the violation of the FDT is a price of
devices (i.e., Maxwell demons) installed in the cell with spatially varying concentration, where the role of the device is to provide a Brownian particle in its vicinity with a random push. This push is in addition to the random pushes already coming from the medium due to thermal motion. An important aspect of this reasoning is that preventing these devices from reaching equilibrium with the medium and from becoming uniformly distributed requires work that needs to be constantly supplied. This extra work, which is not present in the absence of the devices (i.e., the standard case) results in the changes to the overall energy and entropy in the system due to interaction between Maxwell demons and Brownian particles. Accounting for this extra work/energy, we believe is the source of the nontrivial modifications to the Clausius−Duhem relations in eq 49. The same type of reasoning applies to the SCM where the meaning of the variable x is different (local entropy and stress vs spatial location of a Brownian particle), but the form of the SDE and FPEs is similar.
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DISCUSSION In this communication, we have begun a systematic investigation of the thermodynamics of a stochastic constitutive model1 (SCM) that has recently been developed to describe the thermomechanical relaxation behavior of glassy materials. In traditional continuum mechanics, all quantities are first spatially and temporally averaged, where the averaged quantities are then assumed to be field variables that are sufficiently smooth that derivatives of these quantities at a mathematical point are meaningful. Using these smooth field variables, constitutive equations are then developed. The reason why this approach works is that (i) the magnitude of the fluctuations due to particulate nature is inversely proportional to the square root of the number of particles involved and hence to L−3/2, where L is a characteristic size and (ii) the characteristic time of the fluctuations usually scales with the size as τ = L/c, where c is the velocity of sound. As a result, the nanoscale fluctuations are large, but their characteristic times are so fastcompared with the time scales that are relevant for most experimentsthat the fluctuations average themselves out at least for situations sufficiently removed from absolute zero. Conversely, fluctuations on larger scales decay more slowly, but their magnitude is vanishingly small. The spatial/temporal averaged constitutive approach has been successfully used for a wide variety of materials; however, the traditional continuum mechanics approach has been unable to date to describe a number of important features of the thermomechanical behavior of glasses.1,3 One of the key features of the glass is dynamic heterogeneity,17 where there are orders of magnitude differences in the local mobility between neighboring nanoscale domains and these large mobility differences can persist for extremely long periods of time as compared to τ = L/c. A fundamental explanation for the causes and the mechanism of dynamic heterogeneity remains one of the major challenges of the modern condensed matter physics. However, once the existence of the heterogeneity is acknowledged, a much more physically appealing manner to consider describing the behavior of a system with dynamic heterogeneity is to postulate that the local mobility is given by the local stateof-the-system vs the traditional continuum constitutive postulate that the average mobility is given by the average state-of-the-system. This straightforward postulate has been employed in the SCM: the thermoviscoelastic behavior of a nanometer size domain in the glass is described just as a single G
DOI: 10.1021/acs.iecr.5b01347 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research keeping the SCM in a simple Langevin form, where the SDE turn into deterministic equations given in eq 21 in the limit of the domain size L → ∞. Violation of the FDT in glassy systems is a well-established result,18 and its origin is thought to be the fact that the equilibration/physical aging time is much longer than the experimental time. This is consistent with the SCM approach where this disparity of time scales is implied in the defining eq 25 in that the external driving term is instantaneous as opposed to the relaxation term which occurs on the characteristic time scale of a(x)τ0. Despite FDT violation being a familiar concept in glassy physics, we believe its implementation for a system described by an FPE is unique to the SCM. As the derivation in this paper shows, the consequence of the assumptions on which the SCM is based is the nonstandard form of the Clausius−Duhem relation in eq 43. This consequence can be resolved in two ways: perhaps for the SCM the new form of the power of the external driving is what should be used and there is no inconsistency. Or, perhaps the expression for the free energy given by eq 14 is not appropriate, that is, a different form may exist that would then lead to the traditional Clausius−Duhem equation. Although we wish that it was possible in this communication to provide a definitive answer, none of the above possibilities can be ruled out at this point. We believe that the current paper still provides considerable value in elucidating the challenges in fully formulating a thermodynamically consistent framework for continuum mechanics with large fluctuations. In particular, this paper has shown that the obvious choices for specifying the free energy in a fluctuating system like that in the SCM raise significant questions. Because of the considerable success of the SCM in describing nonlinear mechanical and relaxation phenomena that have heretofore evaded description by a significant effort over decades by a vigorous research community using traditional continuum models, we believe that there is motivation to pursue this significant modification of traditional continuum mechanics to now include large fluctuations, including a full understanding of the needed thermodynamic foundations.
Equation A.2 should be compared with eq 3b, where now the fluctuation magnitude, B(x), is a matrix and a function of x. If there is an equilibrium solution pe(x) satisfying J = 0, then it is given by ∂ ln[pe (x)] = ∂xi
As is well known, eq A.4 imposes additional conditions on the A(x) and B(x) functions (i.e., the potential conditions) resulting from the fact that the left-hand side of eq A.4 is a gradient. Hence, the right-hand side of eq A.5 must obey ∂ ∂xm =
⎤ ∂ Bkj (x)⎥ ⎥⎦ ∂xj
∂ ∂xi
⎣
k
j
⎡
−1 (x)⎢2Ak (x) − ∑ ∑ Bmk ⎢
⎣
k
j
⎤ ∂ Bkj (x)⎥ ⎥⎦ ∂xj
(A.5)
Provided eq A.5 is satisfied and thus the equilibrium distribution defined by eq A.4 exists, we define the free energy according to eq 14. Then its time derivative reads 1 d Ψ(t ) = kBθ dt = −∑ i
=
⎧ ⎡ p (x , t ) ⎤ ⎫∂ ⎥ + 1⎬ p(x , t ) e p (x) ⎦ ⎭ ∂t ⎪
⎪
⎪
⎪
∫ dx⎨⎩ln⎢⎣
⎧ ⎡ p (x , t ) ⎤ ⎫ ∂ ⎬ Ji (x , t ) + 1 ⎥ pe (x) ⎦ ⎭ ∂xi ⎪
⎪
⎪
⎪
∫ dx⎨⎩ln⎢⎣
∑ ∫ dxJi (x , t ) i
∂ ⎡ p (x , t ) ⎤ ln⎢ ⎥ ∂xi ⎣ pe (x) ⎦ (A.6)
It follows from eq A.2 that Ji (x , t ) p(x , t )
= A i (x) −
1 ∑ ∂ Bij(x) 2 j ∂xj
1 ∑ Bij(x) ∂ ln[p(x, t )] 2 j ∂xj
−
(A.7)
and from eq A.4 1 2
∑ Bik (x) k
∂ 1 ∂ ln[pe (x)] = Ai (x) − ∑ Bij (x) ∂xk 2 j ∂xj (A.8)
(A.1)
1 ∂ ∑ [Bij(x)p(x, t )] 2 j ∂xj
∂ ⎡ p (x , t ) ⎤ 1 ∑ Bij(x) ln⎢ e ⎥ ∂xj ⎣ p (x) ⎦ 2 j
(A.9)
Jj (x , t ) ∂ ⎡ p (x , t ) ⎤ ln⎢ e ⎥ = −2∑ Bij−1(x) ∂xi ⎣ p (x) ⎦ p(x , t ) j
(A.10)
Ji (x , t )
The corresponding FPE given by eq 3a with the flux vector
p (x , t )
=−
And hence
(A.2)
where
k
⎡
∑ Bik−1(x)⎢⎢2Ak (x) − ∑
Combining eqs A.7 and A.8
dxi = Ai (x)dt + bij(x)dWj
∑ bik(x)bjk(x)
j
(A.4)
APPENDIX A. GENERAL FPE The purpose of this Appendix is to treat the case of a general SDE where both the drift term and the magnitude of the noise are functions of the stochastic variable. This situation arises in the case of a Brownian particle with the diffusion coefficient that is a function of position as well as in the case of the SCM. It clarifies which assumptions about a stochastic system are necessary to be able to establish the H-theorem, e.g. the FDT property is not required for the existence of the H-theorem, as will be shown below. Consider an SDE of the form
Bij (x) =
⎣
k
⎤ ∂ Bkj (x)⎥ ⎥⎦ ∂xj
4
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Ji (x , t ) = Ai (x)p(x , t ) −
⎡
∑ Bik−1(x)⎢⎢2Ak (x) − ∑
(A.3)
Substituting into eq A.6, we obtain H
DOI: 10.1021/acs.iecr.5b01347 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research 1 d Ψ(t ) = −2 ∑ kBθ dt i,j
∫ dx p(x1, t ) Bij−1(x)Ji (x, t )Jj (x, t )
(3) Medvedev, G. A.; Caruthers, J. M. Predictions of Volume Relaxation in Glass Forming Materials Using a Stochastic Constitutive Model. Macromolecules 2015, 48 (3), 788−800. (4) Gardiner, C. W. Handbook of Stochastic Methods; Springer− Verlag: Berlin, 1985; Vol. 13, p 442. (5) (a) Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 2012, 75 (12), 126001. (b) Klages, R.; Just, W.; Jarzynski, C. Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond; Wiley: Weinheim, Germany, 2013; p 450. (6) (a) Truesdel, C.; Toupin, R. A. The classical field theories; Springer: Berlin, 1960; p 632;. (b) Noll, W. A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal. 1958, 2, 197−226. (c) Coleman, B. D. Thermodynamics of materials with memory. Arch. Ration. Mech. Anal. 1964, 17, 1−46. (7) Truesdell, C. The Mechanical foundations of elasticity and fluid dynamics. J. Arch Ration Mech An 1952, 1, 125−300. (8) Coleman, B. D.; Noll, W. Foundations of linear viscoelsticity. Rev. Mod. Phys. 1961, 33, 239−249. (9) (a) Lustig, S. R.; Shay, R. M. J.; Caruthers, J. M. Thermodynamic constitutive equations for materials with memory on a material time scale. J. Rheol. 1996, 40 (1), 69−106. (b) Caruthers, J. M.; Adolf, D. B.; Chambers, R. S.; Shrikhande, P. A thermodynamically consistent, nonlinear viscoelastic approach for modeling glassy polymers. Polymer 2004, 45, 4577−4597. (10) Landau, L. D.; Lifshitz, E. M.; Pitaevskii, L. P. Statistical Physics, Part 2; Butterworth-Heinemann: Oxford, 1980; Vol. 9, p 388. (11) Mashiyama, K. T.; Mori, H. Origin of the Landau-Lifshitz hydrodynamic fluctuations in nonequilibrium systems and a new method for reducing the Boltzmann equation. J. Stat. Phys. 1978, 18 (4), 385−407. (12) Ritort, F. Nonequilibrium Fluctuations in Small Systems: From Physics to Biology. Adv. Chem. Phys. 2007, 137, 31−123. (13) (a) Sollich, P.; Cates, M. E. Thermodynamic interpretation of soft glassy rheology models. Phys. Rev. E 2012, 85, 031127. (b) Fuereder, I.; Ilg, P. Nonequilibrium thermodynamics of the soft glassy rheology model. Phys. Rev. E 2013, 88, 042134. (14) (a) Einstein, A. Investigations of the Theory of Brownian Movement; Dover: Mineola, New York, 1956;. (b) von Smoluchowski, M. Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Ann. Phys. 1906, 326 (14), 756−780. (c) Langevin, P. Sur la théorie du mouvement brownien. C. R. Acad. Sci. 1908, 146, 530−533. (15) Kubo, R. The fluctuation-dissipation theorem. Rep. Prog. Phys. 1966, 29, 255−284. (16) Landau, L. D.; Lifshitz, E. M. Statistical Physics, Part 1, 3rd ed.; Butterworth-Heinemann: Oxford, U.K., 1980; p 545. (17) Ediger, M. D. Spatially Heterogeneous Dynamics in Supercooled Liquids. Annu. Rev. Phys. Chem. 2000, 51, 99−128. (18) Cristani, A.; Ritort, F. Violation of the fluctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence. J. Phys. A: Math. Gen. 2003, 36 (21), R181−R290.
(A.11)
Because B−1(x) is positive-definite (due to eq A.3), the “free energy” Ψ(t) defined in eq 14 satisfies the H-theorem. Nomenclature
Ai (x) a(x), a(t) bij (x) Bij (x) D, D(x) E(t) Fi(t) Hi
drift vector in general SDE shift factor for relaxation time noise matrix in general SDE diffusion matrix in general FPE coefficient of diffusion of a Brownian particle average nonequilibrium energy external force on a Brownian particle vector of externally controlled thermodynamic variables Ji probability flux vector L characteristic size of a mesodomain Mi vector of memory integrals p(x,t) probability density function pe(x) equilibrium probability density function S(t) average nonequilibrium entropy t time t* material time U(x) potential energy of a Brownian particle V specific volume dWi vector of independent Wiener processes xi vector of fluctuating thermodynamic variables Z normalization constant Δij matrix of coefficient in the expansion of the free energy θ temperature σi magnitude of equilibrium fluctuations in variable xi ς friction coefficient of Brownian particle τ0 relaxation time in reference state ξ dummy variable for integration over time ψ, Ψ, Ψ′ nonequilibrium free energy
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work was supported by National Science Foundation grant number 1363326-CMMI. The authors are deeply indebted to Prof. Ramkrishna for introducing them to the field of stochastic processes and for many stimulating discussions that have led in part to this communication. It has been a delight to be a colleague and friend of Ramki for the past 38 years, where there is an anticipation of many more years of academic and personal friendship.
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REFERENCES
(1) Medvedev, G. A.; Caruthers, J. M. Development of a Stochastic Constitutive Model for Prediction of Post-Yield Softening in Glassy Polymers. J. Rheol. 2013, 57 (3), 949−1002. (2) Medvedev, G. A.; Starry, A. B.; Ramkrishna, D.; Caruthers, J. M. Stochastic Model for Volume Relaxation in Glass Forming Materials: Local Specific Volume Model. Macromolecules 2012, 45 (17), 7237− 7259. I
DOI: 10.1021/acs.iecr.5b01347 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
