Microcanonical Entropy and Mesoscale Dislocation Mechanics and Plasticity Amit Acharya Carnegie Mellon University Abstract A methodology is devised to utilize the statistical mechanical entropy of an isolated, constrained atomistic system to define the dissipative driving-force and energetic fields in continuum thermomechanics. A thermodynamic model of dislocation mechanics is discussed. One outcome is a definition for the mesoscale back-stress tensor and the symmetric, polar dislocation density-dependent, Cauchy stress tensor from atomistic ingredients. Dedication: This paper is dedicated to the memory of Professor Donald E. Carlson, teacher and friend to me. I owe a great debt for all I learned from him, in particular continuum mechanics. Don was a scholar and a gentleman, with a kind heart and a tremendous sense of humor. I miss him. 1. Introduction This work is an attempt at examining to what extent material-specific atomistic information can be incorporated into defining material response in continuum mechanics at ‘slow’ time scales (³ microseconds) with respect to the fast time scale of atomic vibrations (~femtoseconds). We rely on classical equilibrium statistical mechanics of isolated atomistic assemblies as our microscopic theory, e.g. (Berdichevsky 1997); the meso-/macro scale models are intended to be non-standard continuum mechanical models of defects in solids. The bridge is assumed to be the two laws of thermodynamics, as enunciated in the Classical Field Theories of Mechanics (Truesdell and Toupin 1960) in particular, the global form of the Clausius-Duhem Inequality as the embodiment of the Second law of thermodynamics. The essential idea is to give one possible operational form and meaning to the specific internal energy and entropy response functions of continuum thermomechanics, resorting to the atomistic nature of all solids and the equilibrium statistical mechanical microscopic theory of assemblies of such atoms. The nonstandard continuum models of relevant defects have to be developed based on a consideration of fundamental defect kinematics that give rise to balance/conservation laws for geometrically rigorous densities. The notion of ‘local thermodynamic equilibrium’ also has to be made operational, but this becomes somewhat easier, if only formally, due to a reliance on a microscopic finite dimensional Hamiltonian system for which the existence of a phase-space invariant measure (probability density function) can at least be considered plausible. We rely on statistical mechanics in the microcanonical ensemble primarily because continuum mechanics requires the notion of a meso/macro point-wise temperature that has to be defined without relying on the fact that it has a surrounding that has the same temperature. For a discussion and comparison of the microcanonical and canonical definitions of the entropy, see  email: [email protected]; Tel. (412) 268 4566; Fax. (412) 268 7813. (Berdichevsky 1997), and for an independent viewpoint on Berdichevsky’s development, (MacKay 1999). Moreover, it seems conceptually natural in the setting of continuum mechanics to have the ‘interaction with the bath’ of an atomistic subsystem to be handled by the macroscopic partial differential equations (PDEs) of the continuum model. The main contributions of this work are in 1) making precise how continuum field values at points in space-time can be utilized to describe well-defined local Hamiltonian systems, whose entropy then defines the continuum specific entropy field, and 2) application of the overall scheme to a field description of the mechanics of dislocations. The model makes no restrictions on geometric or material nonlinearities. The model of dislocation mechanics develops prior work (Acharya; 2001, 2004), (Acharya and Roy, 2006) which in turn builds on the theory of continuously distributed dislocations pioneered by (Kroner 1981), (Mura 1963), (Fox 1966), and (Willis 1967) and extends it to account for dissipative dislocation transport and nonlinearity due to geometric and crystal elasticity effects. Here, the main contribution is a theoretical treatment that produces a completely defined constitutive equation for a (generally non-symmetric) back stress tensor and a symmetric Cauchy stress that includes a dependence on the polar dislocation density tensor. Temperature dependence of the back stress and the Cauchy stress are automatic. This work does not make any fundamental statement about constitutive kinetic relations beyond uncovering the driving forces for dissipative mechanisms. However, even with the assumption of linear kinetic relations, the governing nonlinear partial differential equations for dislocations coupled to stress are very rich, and it is reasonable to expect them to be capable of predicting complex microstructure. Balance of energy in standard form and the global form of the Clausius- Duhem Inequality are assumed to be valid without question, and this may be construed as a major shortcoming of this work. Given the complexity of the venture as is, this may perhaps be considered acceptable with some justification, see e.g. (Man 1995). There are natural avenues for considering these extremely difficult questions of nonlinear spatio-temporal ‘homogenization,’ e.g., (Tartar a 2008), (Tartar 2009), (Artstein and Vigodner, 1996), but precise answers for the context at hand, even at a theoretical level, remain elusive despite outstanding work1, giving a sense of the arduous road ahead on this issue. To give an example of the thorny issues involved, in the conservative approach adopted here (and in all of practical equilibrium statistical mechanics), we declare by fiat the existence of a unique invariant measure for time-averaging an atomistic Hamiltonian system. Running time-averages of phase functions can be appended to the original ODE Hamiltonian dynamics to produce a singularly perturbed system (Acharya 2010). The rigorous results of (Arstein and Vigodner, 1996) then clearly indicate possibilities different from averaging based on a unique invariant measure for defining the theory for the dynamics of the time-averages, and it may be expected that such considerations would play a fundamental role in understanding kinetic relations from the ground-up. This paper is organized as follows: Section 2 defines the specific internal energy field of the continuum theory; Section 3 defines the notion of local thermodynamic equilibrium; Section 4 deals with the definition of the specific entropy field; Section 5 discusses the probabilistic 1 Such work is also technically very difficult for the general worker in mechanics (including myself, of course) to comprehend in its entirety. This seems to be an unfortunate barrier for progress, without an easy solution. 2 interpretation of the microcanonical entropy; Section 6 deals with the mesoscale model of dislocation mechanics. There are three Appendices. In Appendix A, a Helmholtz free energy density is defined from the entropy field, and it is shown how thermodynamic models may be recast in terms of this energy and temperature. Appendix B describes the detailed geometric reasoning behind the conservation law for the dislocation density tensor. Appendix C shows the connections of the dislocation kinematics presented here with that in (Acharya 2004) and Acharya and Roy (2006). To our knowledge, the approach proposed herein to make a connection between continuum and atomistic dynamics is new; there is a connection in spirit, with significantly different details, with the finite-temperature quasicontinuum ideas of Kulkarni et al. (2008). There exists a large literature on thermodynamics for dissipative response of solids at finite strain, beginning with the pioneering works of Coleman and Gurtin (1967) and (Rice 1971). The emphasis here is on making the statistical mechanical connections to define some of the fundamental ingredients (internal energy, entropy) of any continuum thermodynamic framework. As mentioned earlier, resorting to a finite dimensional Hamiltonian microscopic model representing atomistic assemblies has a distinct advantage in making standard (nonequilibrium) thermodynamic formalism less abstract, especially ideas related to thermodynamic processes consisting of ‘constrained equilibrium states.’ Thus, a primary goal of this paper is to make clear how a real equilibrium state of an appropriate, constrained microscopic system can be made plausible, what this equilibrium exactly means and how such equilibria naturally form a process at macroscopic time scales, and why the resulting thermodynamics can actually be applied to continuum dynamics with inertia in many circumstances. Rice (1971) considers a thermodynamic model of plasticity arising from dislocations; the model here contains a more detailed consideration of dislocation kinematics (that accommodates the standard model of crystalline slip as one ingredient) leading to the representation of dislocation transport through wave-propagative effects and length-scale effects in mechanical response. 2. Specific internal energy of the continuum theory Let x be a point in space occupied by a material point of a deforming body at time t. Consider a spatial volume ,x of fixed volume V around x. Let the continuum mass density x,t at the space-time location x,t be a piece of physical information available to us, for the moment from an unspecified source. Similarly, let us assume that the value of the continuum velocity vx,t is also known. For simplicity we consider atoms of a single species with individual mass m. The first objective is to state the physical assumptions behind defining a continuum specific internal energy (per unit mass) field. We now think of a collection of x,tV Nx,t: m atoms indexed by I . Let V be the velocity of the Ith atom. Let the mean velocity and I fluctuations be defined as 3 N 1      V  V ; V  V V . N I I I I1 Then N N N 1 mV V  1 mV VVV VV  mV V + 1 mV V . N 2 I I N 2  I I  2 N 2 I I I1 I1 I1 This implies that the total kinetic energy of the N atoms is given by N Nm m V V + V V . (1) 2 2 I I I1 As for the ‘potential’ energy, let us assume that the entire body, viewed as a atomistic system, is endowed with a potential for generating interatomic forces which can be characterized as a function of number of atoms, say M (typically M  N), and the positions r , j 1 to M , of the j M atoms. Let this potential energy function be    U M, r , j 1 to M . j This prescription makes it clear that while the total potential energy of the body can be written down unambiguously, it is not so clear how one might define the potential energy of a certain subset of the atoms only in terms of the positions of that subset of atoms. In the setting of continuum mechanics, we write the total energy of arbitrary subparts of the body, say , as 1    vv+ dv,    2  where the first term in the parenthesis represents the kinetic energy per unit mass field and the second, the internal energy per unit mass field. It seems natural then to associate N  1vv+dv  NmV V +mV V U , (2)    2  2 2 I I c I1 where U is an atomic interaction energy term that needs to be defined. c Motivated by the form of (2), we associate the continuum velocity field with the local mean atomic velocity, so that 1 1 v V  vv  V V . (3) 2 2 Let the specific internal energy field (per unit mass) around x,t, be denoted by (x,t). We postulate that it corresponds to the total energy of a local, isolated, constrained, ergodic Hamiltonian system consisting of Nx,t atoms, with the potential of the original material. Thus for each t tt, given (x,t), Nx,t x,tx,tV  Ex,t:UNx,t,r , j 1 to Nx,t   mV V  j 2 I I (4) I1 subject to appropriate constraints to be defined. 4 The constraints are meant to represent the action of all the other atoms in the body on this set of Nx,t atoms beyond what can be represented through the specification of the value of the total energy, Ex,t. These constraints have the following generic form: let a point of the 6N dimensional phase space of the Hamiltonian system be denoted by Y . The constraints are then written as Y z x,t,i1 to 2k, k a positive integer, (5) i i where  are real-valued functions of Y and z are values of macroscopic fields at the point i i x,t. We discuss the specifics of these constraints in Section 6 in the context of a concrete example. The manner in which (4) may be physically interpreted (and tested) as a definition of specific internal energy at x,t is as follows: given the mass density x,t consider the Nx,t nearest atoms to the point x at time t. Now assume that the positions and velocity fluctuations of these atoms satisfy (5) for the time interval t,tt, given the values of z x,t; also, i assume that the velocity fluctuations and the positions of these atoms are such that the extreme right-hand-side of (4) evaluated for these arguments attains a constant value, Ex,t, over the time interval t,tt. Then x,t may be defined as Ex,t x,tV. Ergodicity is an abstract, but very useful, mathematical concept – for our purposes, we take it to be practically ‘equivalent’ to one of the most useful properties of an ergodic Hamiltonian system: consider a set A on the energy surface Y :HY E of the system. Consider the time ;A,Y spent by a trajectory of the system in the set A starting from initial condition Y , over a total time of evolution . If the system has the property that ;A,Y lim A   independent of almost all trajectories with energy E used to generate it, where  is a real- valued function on subsets A of the common energy surface, then we call the system ergodic. 3. Local equilibrium A primary assumption we make here is that the local Hamiltonian system evolves on a time scale (~ femtoseconds) that is much smaller than the time scale of evolution (e.g. ~ microsecond) of the continuum theory; in particular, the separation is large enough such that for E held constant, the local atomistic system equilibrates on the macroscopic time-scale. By this, we mean the following: consider the 6N2k dimensional set of points forming the accessible states (or phase space) of the constrained Hamiltonian system. Consider further the 6n2k1 dimensional subset of this constrained phase space, consistent with the prescribed value of the energy E, and consider an arbitrary subset A of it. Choose almost any trajectory of the constrained system with energy E and consider the ratio 5 ;A  of the time  spent by the trajectory in the set A, and the total time of evolution  of the trajectory. Let t be a minimum interval of time on the t scale below which the continuum theory shows no appreciable evolution – this is an important conceptual ingredient, and we think of this as the time resolution of the continuum theory. Then, as a definition of microscopic equilibrium, we require that, given anyA as defined above, there exists a constant (A) such that ;A for any  t A ,  where 01 is a (user-specified) threshold. Of course, we keep in mind that the set A depends on E by definition, so the function  really depends upon E for its definition. 4. Specific entropy field of the continuum theory In defining a continuum entropy per unit mass field, we adapt the developments in Berdichevsky (1997) for our purposes. For each x,t in the body, define Nx,t HY: UX   1mV V , (6) 2 I I I1 where Y is the list ( ) Y = X ,X ,,X ,V ,V ,,V 1 2 N(x,t) I 2 N(x,t) and we have used the shorthand UX:UNx,t,r , j 1 to Nx,t (7) j (cf. (4)). Next a phase space region Y :HY Ex,t,Y z x,t,i 1 to 2k (8) i i is defined. Furthermore, Ex,t:x,tx,tV , where  is the continuum internal energy per unit mass field. Let us assume that there is some invariant physical meaning that can be associated with volumes of regions in phase space. An important construct of the theory is the volume of the phase-space region defined by (8): Ex,t,zx,t;Nx,t: volY :HY Ex,t,Y z x,t,i1 to 2k (9) i i In writing z, we mean the entire array z ,i 1 to 2k . i Note that volumes in the 6N dimensional phase space of the above atomistic Hamiltonian system have physical units of momentumposition3N Energytime3N. We now assume that our microscopic measurements can only resolve an action scale, say h , and above. Then, 6 following Berdichevsky (1997), and all attendant assumptions therein (nondegeneracy of constraints, incompressibility of phase flow of the constrained system and ergodicity of the constrained system being the main ones), given E,z,N it makes sense to define the entropy of the constrained Hamiltonian system as  1 2k  SE,z;N:ln E,z;N. (10) h3Nx,t z z  1 2k S(E,z,N) is simply a measure of the 6N2k dimensional volume of the constrained Hamiltonian system bounded by the energy surface E. The objective of this paper until now has been to establish a procedure for defining point-wise values of the quantities E,z,N based on evaluations of the continuum fields ,,z. Thus, we define the continuum entropy per unit mass field from the entropy of the constrained Hamiltonian system defined above as C  1 2k  x,t,x,t,zx,t: ln Ex,t,zx,t;Nx,t,(11) x,tV h3Nx,t z z  1 2k where C is a constant with units of energy/absolute temperature. This expression for the specific entropy is simply a measure of the 6N2k dimensional volume of the constrained Hamiltonian system bounded by the energy surface E. The motivation behind this definition is as follows; in the case of an unconstrained, ergodic, Hamiltonian system with Hamiltonian quadratic in the momenta, it is a principal result that the long-time average of the kinetic energy of any given particle along almost any trajectory with fixed energy is a constant (equipartition) and, by definition, this common value is called the absolute temperature, say T . This further definition is motivated from macroscopic thermodynamics where the result 1 S  (12)  E where  is the macroscopic temperature (i.e. the perceived level of hotness), S is the entropy of the system, and E is the energy. For the unconstrained Hamiltonian system, defining the phase space volume bounded by the E-energy surface as EvolY :HY E (13) it can be shown (e.g. Berdichevsky, 1997) that E TE . (14)  E E Therefore, defining entropy in microscopic terms (utilizing ideas going back to Boltzmann, Gibbs and Hertz, according to Berdichevsky, 1997) as E SEln     0 where  is a constant required on dimensional grounds, implies TE. 0 In our case of the constrained Hamiltonian system, equipartition is not a derived result. However, macroscopic thermodynamics yields a result similar to (12), as we subsequently show. 7 Moreover, in the (unconstrained) case where there is a formal proof, a natural geometric/physical interpretation of entropy arises as the 6N dimensional phase space volume of the system bounded by the energy surface E. These three facts motivate the definition (11). We agree to pose the constraints in non-dimensional form by definition, so the logarithm makes sense. In general, a physically dimensional constraint, say   z, may be normalized arbitrarily so as to be stated in dimensionless form  z. Finally, it should be carefully noted that the dependence of the specific entropy field on the continuum mass density field goes far beyond what is apparent through its explicit appearance in the formula (11). Turning to prospects for making the theory practical, it is clear that if the function S can be determined then the response function for the specific entropy of the material, ,,z, is completely defined. There exists a great deal of expertise in the physics literature for calculating various energies and entropies of molecular systems, e.g. Frenkel and Smit (2002), mostly in the context of calculations with the canonical ensemble. It is conceivable that these strategies can be gainfully adapted to the specific problems mentioned herein. Practical approximation of the entropy function S , while an important subject in its own right, is beyond the scope of this paper. In the following section we mention a probabilistic interpretation of the entropy S that suggest some naïve methods for approaching the calculation of S(z,E) that involve data collection from the unconstrained Hamiltonian system to yield information on the entropy of the constrained system. 5. Probabilistic interpretation of Entropy We review some basic assumptions of equilibrium statistical mechanics in the microcanonical ensemble, following Berdichevsky (1997). Consider the unconstrained Hamiltonian system (6)- (7), assumed to be ergodic. Given phase functions Y,i 1 to P and a trajectory Y with i energy E we first define the amount of time , out of a total time of evolution , during which all the following relations z Y z z , i1 to P i i i i are satisfied. Let  z z z . We now assume that the limit 1 2 P 1  lim lim : f z ,,z ,E 0  1 P exists and, as shown in the notation, depends only on the z ,i 1 to P, and E, and is i independent of the trajectory used to define it. The function f is called the probability density of the characteristics , i1 to P. i We recall the definition (13) and define z ,,z ,E:volY :HY Ex,t,Y z x,t,i 1 to P. i P i i Then a fundamental result (under several strong assumptions) (Berdichevsky, 1997) is that  P1 f z ,,z ,E E z ,,z ,E. (15) 1 P E Ez z 1 P 1 P 8 Now, (15) and (10) imply   f z,E Eh3N expSE,z;N ; z:z ,i 1 to 2k, (16) E E i where it is understood that the left hand side now refers to the unconstrained Hamiltonian system defined by (6)-(7); in particular, that the system contains N atoms. Thus, if the probability density function of the 2k characteristics can be evaluated/approximated by some means (e.g. experimentally), then (16) provides a method of determining the entropy S of the local discrete system. We note here that given the number of strong assumptions that have to hold to define the constrained Hamiltonian system and its entropy (11), it could just as well be as effective to simply define the entropy of a certain set of characteristics of an isolated system by the formula (16), thus bringing it closer in line with some other approaches to defining the entropy based on purely probabilistic grounds (Swendsen 2006). This would also then get rid of what appears to be a somewhat artificial constraint of having to deal with only an even number of characteristics (arising from the fact that the constrained system has to be Hamiltonian). Molecular Dynamics (MD) based evaluation of f z,E;N By definition, f z,E;N can, in principle, be computed by integrating along representative MD trajectories for different values of E. However, as this may require long-time MD evolution, it is not clear how promising such an approach might be for practical evaluation of the probability density function for further utilization in (16) to determine SE,z;N. Evaluation of f z;E based on experimental observations in the large-N limit Presumably, bodies with arbitrarily fixed values of energy E and a large number of atoms N  admit the possibility of being obtained. Note that even with a large number of atoms, such bodies can serve as the isolated Hamiltonian systems discussed earlier. Consider one such value of E and let there be A bodies forming a sample space. We now make the following strong assumption: observing each body in the sample over a macroscopic time interval, say t, is equivalent to making A observations on a single trajectory with energy E, each observation consisting of a t time interval. Let us index the specimens by I 1 to A. Each has a set of values z ,i = 1 to k associated with i it, say z ,i 1 to 2k,I 1 to A. Suppose B specimens were found to correspond to values of z*, iI i i1 to 2k, for the constraints; i.e. z = z*,i=1 to 2k, j=1 to B, with each I Î{1,2,,A}. iI i j j Under the stated assumptions, an approximation of the value of the probability density function f at the argument z;E may be written as Bt B f hz;E  . At A 9 The same procedure can then be repeated for all values of z;E of interest. Maximum Entropy states and most probable states In closing, we note the following associations of the probabilistic interpretation of entropy with Gibbs’s thermodynamic postulate about macroscopic equilibrium. Let us rewrite (16) in the form f =expæççççS éêê1+ln{(¶ ¶E)-1h3N(¶S ¶E)}ùúúö÷÷÷÷÷. (17) çç ê S ú÷÷ çèç êë úû÷÷ø When the fraction in the above expression is much smaller in magnitude w.r.t. unity, we obtain Boltzmann’s expression for the entropy ln f (z,E)=S(z,E) (18) which expresses the relationship of the probability of observing the macroscopic state z with system energy E to the entropy of the system. In particular, Gibbs’s postulate that for fixed energy, the macroscopic state (here z) that is observed (equilibrium) corresponds to the state that maximizes the entropy would suggest, according to (18) that Gibbs’s maximum entropy equilibrium also corresponds to the most probable state(s), under the assumption leading to (18) from (17) (e.g. for large N , as shown by heuristic arguments in Berdichevsky (1997). 6. Example: Mesoscale Dislocation Mechanics Up to the specification of constitutive equations, we assume that the following field equations are valid for describing the mechanics of a body containing dislocations, including resolving the mechanics of single dislocations (i.e. ‘microscopic theory’):  divv 0, divv (Balance of Mass) (19) t vdivT v, divT vv (Balance of Linear Momentum) (20) t div0 (Dislocations cannot end in the material) (21) curlW  (Fundamental statement of elastic incompatibility and dislocations) (22)  curlV, curlvV t (Balance of Burgers vector content) (23)  divv LT :  W WLV . (24) Here, as in classical continuum mechanics, we assume that mass of arbitrarily small sets of particles in the body is conserved for all motions. In the above,  is the mass density, v is the material velocity, T is the Cauchy stress,  is the dislocation density (Nye) tensor, W is the inverse elastic distortion tensor, V is the dislocation velocity field relative to the material, and L is the material velocity gradient. A superposed dot represents a material time derivative, whereas a subscript comma followed by a t represents differentiation with respect to time of the Eulerian representation of the field in question. All spatial derivative operators are defined on the 10