On the competition between interface energy and temperature in phase transition phenomena

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Submitted on 5 Oct 2020

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On the competition between interface energy and temperature in phase transition phenomena

Luca Bellino, Giuseppe Florio, Stefano Giordano, Giuseppe Puglisi

To cite this version:

Luca Bellino, Giuseppe Florio, Stefano Giordano, Giuseppe Puglisi. On the competition between

interface energy and temperature in phase transition phenomena. Applications in Engineering Science,

2020, 2, pp.100009. �10.1016/j.apples.2020.100009�. �hal-02947935�

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On the competition between interface energy and temperature in phase transition phenomena

Luca Bellino

Politecnico di Bari, (DMMM) Dipartimento di Meccanica, Matematica e Management, Via Re David 200, I-70125 Bari, Italy

Giuseppe Florio

Politecnico di Bari, (DMMM) Dipartimento di Meccanica, Matematica e Management, Via Re David 200, I-70125 Bari, Italy and INFN, Sezione di Bari, I-70126, Italy

Stefano Giordano

Institute of Electronics, Microelectronics and Nanotechnology - UMR 8520, Univ. Lille, CNRS, Centrale Lille, ISEN, Univ.

Valenciennes, LIA LICS/LEMAC, F-59000 Lille, France

Giuseppe Puglisi

Politecnico di Bari, (DICAR) Dipartimento di Scienza dell’Ingegneria Civile e dell’Architettura, Via Re David 200, I-70125 Bari, Italy

Abstract

Phase nucleation and propagation phenomena can be characterized by a cooperative behavior regulated by non local interactions between the multistable domains and with the loading device. Cooperativity is often macroscopically witnessed by a stress-peak, distinguishing the nucleation from the propagation stress, and by a larger size of the first nucleated domain. When low dimensional scales are considered, both in nanostructures or single molecule behaviors, the interfacial energy can compete with entropic effects, leading to the experimental observation of a temperature dependent phase transition strategy. We propose a fully analytical model, in the framework of Statistical Mechanics, measuring such energetic competition and temperature dependent behavior, that well reproduces important experimental evidences. The effectiveness of the model is successfully tested in predicting the temperature dependent phase transition behavior of shape memory nanowires.

Keywords:

phase transition, interfacial energy, non local interactions, size dependence, temperature dependence

1. Introduction

The description of finite microstructure domains evolution for multistable materials represents an important topic in material science that has been longly at the center of research activities for both its theoretical and technological importance [1]. Indeed, multi-states systems are of interest in several fields such as material science, biology, medicine, and engineering, both for natural and artificial materials. More recently,

5

a large effort in this field has been devoted to the design of new high-performing metamaterials. In this

Email addresses: luca.bellino@poliba.it(Luca Bellino),giuseppe.florio@poliba.it(Giuseppe Florio), stefano.giordano@univ-lille.fr(Stefano Giordano),giuseppe.puglisi@poliba.it(Giuseppe Puglisi)

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case a successful new inverse approach is adopted, with a multistable microstructure designed to obtain the desired macroscopic behavior [2,

3].

In the thermomechanical three-dimensional case, the problem is quite complex and it asks for a description not only of the local structure of an interface, but also of its curvature and topological properties,

10

necessary for the deduction of its excess energy [4]. The hypothesis of describing the microstructure evolution on the base of energy minimization was already proposed in the field of linear elasticity by Khachaturyan [5]

and coworkers. Later, Ericksen pioneering work [6] proposed a variational energetic approach in non linear elasticity theory with non convex energy densities. Grounded on this work, important theoretical advances in the field of modern continuum mechanics [7] and calculus of variation [8] successfully described basic as-

15

pects of phase transitions and of associated effects at the microstructure level. In particular, the analysis of the complex problem of minimization of the non (quasi) convex energy allowed important results regarding the phase fraction, the orientation of the interfaces and the stress corresponding to the phase evolution.

Nonetheless, continuum variational approaches minimizing the elastic energy density are not able to capture important aspects unless they introduce interfacial energy effects and non-local interactions. As a matter

20

of fact, these energetic contributions are crucial for a detailed description of the observed microstructure evolution [9] and they are fundamental not only from a theoretical point of view, but also for a correct portrayal of the material behavior and for the design of new materials [10]. In particular, the minimization of the (non-convex, bulk) elastic energy only cannot describe the existence of internal length scales regulating the effective observed phenomena leading to discrete processes of transition with nucleation and

25

propagation of finite domains (see Refs. [11,

12] and references therein). Based on these observations, dif-

ferent ‘augmented’ energetic models have been proposed. Schematically, on one side higher gradient [13,

14]

(antiferromagnetic) energy terms, representing surface energy contributions, have been added to the energy density, possibly deduced as a result of compatibility effects as in the case of polymer necking in the classical work presented in Ref. [15]. On the other side, non local (ferromagnetic and antiferromagnetic) energy

30

terms have been considered, thus introducing the desired length scale needed to describe the insurgence of periodic microstructures tipically observed in solid-solid martensitic phase transitions (see Refs. [8,

10] and

references therein).

Starting from the pioneering work in Ref. [16], a parallel framework in the field of discrete mechanics was proposed. In the simpler setting, one can consider a lattice of elements with non convex energy densities and

35

the presence of an intrinsic discreteness length-scale [17]. As a result, it is possible to describe fundamental properties such as energy barriers, metastable states, quasi-plastic and pseudo-elastic behaviors [18,

19]. On

the other hand, also this approach was not able to distinguish between different microstructures with the same phase fraction and, therefore, important details of the nucleation and phase fronts evolution phenomena could not be described. The model has been generalized by considering non local energy terms in Refs. [20–

40

22] where the authors were able to describe the important effect of a stress peak in order to distinguish

the different phenomena of nucleation and propagation. In analogy with higher gradient approaches in elasticity, only single wall solutions with a fully cooperative phase transition could be described. On the other hand, in Refs. [21,

22] a more detailed analysis of the influence of boundary conditions let the author

energetically distinguish between internal and boundary nucleation, with possible two phase fronts. These

45

features are often observed for example in polymer necking localization phenomena. Moreover, the model has been extended to the case of configurational transitions in protein macromolecules to study the successive unfolding events observed in protein stretched under single molecule force spectroscopy experiments [23,

24].

In this case the two energy wells correspond to two different configurations of the protein domains, with a folded

→

unfolded transition. Finally, based on multiscale approaches this class of models has been adopted

50

to deduce the macroscopic features of protein materials in Refs. [25,

26], showing a significant predictivity

of the experimental behavior.

In this paper we are interested in a description of the possibility of entropic effects delivering a temperature dependence of the stress peak, of the size of nucleation domain and, in general, of the cooperative nature of the transition. Schematically, temperature effects can have crucial effects in the behavior of multistble

55

materials for three different reasons. First, increasing or decreasing temperature may result in a change of the potential energy of the different possible configurations such as in the case of Shape Memory Alloys (SMA) [27]. Second, temperature effects regulate the capability of overcoming energy barriers, thus influenc-

2

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ing the key role of metastability, hysteresis and rate effects (see

e.g.

Refs. [27–29]). Third, when rate effects are excluded and the system is able to relax to the global minimum of the free energy, the entropic term can

60

favor solutions with multiple interfaces [30]. This is particularly relevant when the transition is controlled by low enthalpies such as in the case of conformational transitions of units with weak bonds in biological materials or when the scale of the system is very small. In this case, we have a competition between the interfacial energy (favoring solutions with low numbers of interfaces) and entropic energy terms (favoring solutions with a high number of interfaces). As a matter of fact, under these hypotheses, the behavior of the

65

system is described by the expectation values of relevant observables with respect to the thermal fluctuations and, therefore, the correct framework for the analysis is given by a Statistical Mechanics approach.

Specifically, by extending the results in Refs. [21,

22] and including temperature effects, in this paper

we consider a prototypical example constituted by a chain of elements with bistable potentials, describing a material with two distinct phases (i.e folded

→

unfolded configurations in a macromolecule or austenitic

70

and martensitic phases in SMA). In analogy with the continuous model with higher order gradient energy terms [14], here we mimic the important role of interfacial energy by considering next to nearest neighbor (NNN) interactions (see Figure

1). The zero temperature (mechanical limit) behavior of this system was

studied in Refs. [20–22]. The effectiveness of Statistical Mechanics approach in describing thermal effects observed in multistable systems has been recently shown in a series of papers [31–39]. In these papers several

75

theoretical and applied problems regarding entropic effects and boundary conditions have been analyzed. In particular, the authors have focussed on the analysis of the behavior of protein macromolecules represented as a system of bistable domains undergoing conformational folded

→

unfolded transition. In the same spirit, in recent papers [40,

41], these models have been used to consider the important temperature effects to

study a peeling behavior such as in geckoes pad, thus extending the approach in Ref. [42]. In this case,

80

the bistable elements are breakable so that the second state (detached state) is characterized by zero force and stiffness. Such type of models have been widely adopted also in the literature of DNA denaturation in analogy with the Peyrard-Bishop model [43].

The paper is organized as it follows. In Section

2

we introduce the model. In Section

3

we study the purely mechanical (zero temperature) case and obtain the stress strain relation at equilibrium. In Sections

85

4

and

5

we consider the effects of temperature in the cases of applied stress (Gibbs ensemble, soft device) and strain (Helmholtz ensemble, hard device), respectively. In Section

6

we compare the predictions of our model with results obtained using Molecular Dynamics (MD) simulations on shape memory nanowires. In Section

7

we discuss the results and draw the conclusions with possible perspectives.

2. Model

90

Following the approaches of Refs. [20–22], let us consider a chain of

n

+ 1 points, schematized in Fig.

1,

linked by nearest-neighborhood (NN) bistable units with reference length

l

and next-to-nearest-neighbor harmonic springs with natural length 2l. Each NN link is characterized by a two wells energy, corresponding to two different material states. They can represent two different phases [18], as in solid-solid phase transition, or two different conformational (folded and unfolded) states as in the case of unfolding phenomena in

95

protein molecules [23]. Besides, in the limit case when the second well degenerates to a constant energy we can describe the behavior of breakable bonds as in Ref. [42].

After introducing the “spin” variable

Sj

, that defines the phase of each unit as

Sj

=

(−

1 phase one

+1 phase two

,

(1)

we may write the potential energy of the NN springs (see Fig.

1(c)) as it follows:

φNN

(S

j, εj

) = 1 2

k l

"

(ε

j

+

ε0

)

²

(1

−Sj

)

2 +

ζ

(ε

j−ε0

)

²

+ 2η

(1 +

Sj

) 2

#

, j

= 1, ..., n, (2)

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d

<latexit sha1_base64="uwKnOwbaOJPfZAaR6AXaqCyJ4sY=">AAACCXicbVDLTgJBEJzFF+IL9ehlIjHxRHaRRI8kXjxCIkgCG9I7NDhh9pGZXhOy4Qu46od4M179Cr/DH3B33YOCdapUdae6y4uUNGTbn1ZpY3Nre6e8W9nbPzg8qh6f9EwYa4FdEapQ9z0wqGSAXZKksB9pBN9T+ODNbjP/4Qm1kWFwT/MIXR+mgZxIAZRKnfGoWrPrdg6+TpyC1FiB9qj6NRyHIvYxIKHAmIFjR+QmoEkKhYvKMDYYgZjBFAcpDcBH4yb5oQt+ERugkEeouVQ8F/H3RgK+MXPfSyd9oEez6mXiv16maDMxK/k0uXETGUQxYSCyeJIK83gjtEx7QT6WGokg+we5DLgADUSoJQchUjFOi6qkLTmrnayTXqPuXNUbnWat1Sz6KrMzds4umcOuWYvdsTbrMsGQLdkze7GW1qv1Zr3/jJasYueU/YH18Q2A15rG</latexit>

(a)

<latexit sha1_base64="BFtzL5+5OSfgBbmWaRl2G8egBC4=">AAACC3icbVDLTgJBEJzFF+IL9ehlIjHBC9lFEz2SePGIUR4JbEjv0OCE2Udmek0I4RO46od4M179CL/DH3B33YOCdapUdae6y4uUNGTbn1ZhbX1jc6u4XdrZ3ds/KB8etU0Ya4EtEapQdz0wqGSALZKksBtpBN9T2PEmN6nfeUJtZBg80DRC14dxIEdSACXSfRXOB+WKXbMz8FXi5KTCcjQH5a/+MBSxjwEJBcb0HDsidwaapFA4L/VjgxGICYyxl9AAfDTuLDt1zs9iAxTyCDWXimci/t6YgW/M1PeSSR/o0Sx7qfivlyrajMxSPo2u3ZkMopgwEGk8SYVZvBFaJs0gH0qNRJD+g1wGXIAGItSSgxCJGCdVlZKWnOVOVkm7XnMuavW7y0qjnvdVZCfslFWZw65Yg92yJmsxwcZswZ7Zi7WwXq036/1ntGDlO8fsD6yPb0tPmyY=</latexit> (b)<latexit sha1_base64="5beK1Ga+gRXrk1OI89Y77d2vxrM=">AAACC3icbVDLTgJBEJzFF+IL9ehlIjHBC9lFEz2SePGIUR4JbEjv0OCE2Udmek0I4RO46od4M179CL/DH3B33YOCdapUdae6y4uUNGTbn1ZhbX1jc6u4XdrZ3ds/KB8etU0Ya4EtEapQdz0wqGSALZKksBtpBN9T2PEmN6nfeUJtZBg80DRC14dxIEdSACXSfdU7H5Qrds3OwFeJk5MKy9EclL/6w1DEPgYkFBjTc+yI3BlokkLhvNSPDUYgJjDGXkID8NG4s+zUOT+LDVDII9RcKp6J+HtjBr4xU99LJn2gR7PspeK/XqpoMzJL+TS6dmcyiGLCQKTxJBVm8UZomTSDfCg1EkH6D3IZcAEaiFBLDkIkYpxUVUpacpY7WSXtes25qNXvLiuNet5XkZ2wU1ZlDrtiDXbLmqzFBBuzBXtmL9bCerXerPef0YKV7xyzP7A+vgFM9Zsn</latexit>

F

<latexit sha1_base64="WW/rpV2JkCbfCAdp6aJio946DBg=">AAACCXicbVDLSgNBEJyNrxhfUY9eBoPgKezGgB4DgnhMwDwgWULvpBOHzD6Y6RVCyBfkqh/iTbz6FX6HP+DuugdNrFNR1U11lxcpaci2P63CxubW9k5xt7S3f3B4VD4+6Zgw1gLbIlSh7nlgUMkA2yRJYS/SCL6nsOtNb1O/+4TayDB4oFmErg+TQI6lAEqk1t2wXLGrdga+TpycVFiO5rD8NRiFIvYxIKHAmL5jR+TOQZMUChelQWwwAjGFCfYTGoCPxp1nhy74RWyAQh6h5lLxTMTfG3PwjZn5XjLpAz2aVS8V//VSRZuxWcmn8Y07l0EUEwYijSepMIs3QsukF+QjqZEI0n+Qy4AL0ECEWnIQIhHjpKhS0pKz2sk66dSqzlW11qpXGvW8ryI7Y+fskjnsmjXYPWuyNhMM2ZI9sxdrab1ab9b7z2jByndO2R9YH99PgZqo</latexit>

Soft device ~ Gibbs ensemble Hard device ~ Helmholtz ensemble

n

<latexit sha1_base64="jf0Ts7m5g2RpViR0hqTbK9o5D2k=">AAAB83icbVC7TsNAEDyHVwivACXNiQiJKrIDEhQUkWgoE4k8pMSKzpdNOOV8tu72kCIrX0ALFR2i5YMo+Bds4wISphrN7GpnJ4ilMOi6n05pbX1jc6u8XdnZ3ds/qB4edU1kNYcOj2Sk+wEzIIWCDgqU0I81sDCQ0Atmt5nfewRtRKTucR6DH7KpEhPBGaZSW42qNbfu5qCrxCtIjRRojapfw3HEbQgKuWTGDDw3Rj9hGgWXsKgMrYGY8RmbwiClioVg/CQPuqBn1jCMaAyaCklzEX5vJCw0Zh4G6WTI8MEse5n4nzewOLn2E6Fii6B4dgiFhPyQ4VqkDQAdCw2ILEsOVCjKmWaIoAVlnKeiTSuppH14y9+vkm6j7l3UG+3LWvOmaKZMTsgpOSceuSJNckdapEM4AfJEnsmLY51X5815/xktOcXOMfkD5+MbdnORdQ==</latexit>

n 1

<latexit sha1_base64="C2tOqES8yku1g92crekpa2MHi0I=">AAAB9XicbVA9TwJBEN3DL8Qv1NJmIzGxkdyhiRYWJDaWGOUjAULmlgE37O1dduc05MJPsNXKztj6eyz8Lx4nhYKvenlvJvPm+ZGSllz308ktLa+sruXXCxubW9s7xd29hg1jI7AuQhWalg8WldRYJ0kKW5FBCHyFTX90NfWbD2isDPUdjSPsBjDUciAFUCrd6hOvVyy5ZTcDXyTejJTYDLVe8avTD0UcoCahwNq250bUTcCQFAonhU5sMQIxgiG2U6ohQNtNsqgTfhRboJBHaLhUPBPx90YCgbXjwE8nA6B7O+9Nxf+8dkyDi24idRQTajE9RFJhdsgKI9MOkPelQSKYJkcuNRdggAiN5CBEKsZpKYW0D2/++0XSqJS903Ll5qxUvZw1k2cH7JAdM4+dsyq7ZjVWZ4IN2RN7Zi/Oo/PqvDnvP6M5Z7azz/7A+fgGU+2R5w==</latexit>

. . .<latexit sha1_base64="hS9sOeGjL84+ezfBHD9tCRGfb1A=">AAAB93icbVC7TsNAEDyHVwivACXNiQiJKrIDEhQUkWgog4STSIkVnS+bcMr5bN2tkSIr30ALFR2i5XMo+BfOxgUEphrN7Gh3J0ykMOi6H05lZXVtfaO6Wdva3tndq+8fdE2cag4+j2Ws+yEzIIUCHwVK6CcaWBRK6IWz69zvPYA2IlZ3OE8giNhUiYngDK3kD8cxmlG94TbdAvQv8UrSICU6o/qnzfE0AoVcMmMGnptgkDGNgktY1IapgYTxGZvCwFLFIjBBVhy7oCepYRjTBDQVkhYi/ExkLDJmHoV2MmJ4b5a9XPzPG6Q4uQwyoZIUQfF8EQoJxSLDtbAtAB0LDYgsvxyoUJQzzRBBC8o4t2Jqa6nZPrzl7/+SbqvpnTVbt+eN9lXZTJUckWNySjxyQdrkhnSITzgR5JE8kWdn7rw4r87b92jFKTOH5Bec9y+jLpNF</latexit>

1<latexit sha1_base64="D6aG9XmJAD1eG2wmX58o7UaNrDs=">AAAB83icbVC7TsNAEDyHVwivACXNiQiJKrIDEhQUkWgoE4k8pMSKzpdNOOV8tu72kCIrX0ALFR2i5YMo+Bds4wISphrN7GpnJ4ilMOi6n05pbX1jc6u8XdnZ3ds/qB4edU1kNYcOj2Sk+wEzIIWCDgqU0I81sDCQ0Atmt5nfewRtRKTucR6DH7KpEhPBGaZS2xtVa27dzUFXiVeQGinQGlW/huOI2xAUcsmMGXhujH7CNAouYVEZWgMx4zM2hUFKFQvB+EkedEHPrGEY0Rg0FZLmIvzeSFhozDwM0smQ4YNZ9jLxP29gcXLtJ0LFFkHx7BAKCfkhw7VIGwA6FhoQWZYcqFCUM80QQQvKOE9Fm1ZSSfvwlr9fJd1G3buoN9qXteZN0UyZnJBTck48ckWa5I60SIdwAuSJPJMXxzqvzpvz/jNacoqdY/IHzsc3F2CROA==</latexit>

0<latexit sha1_base64="lNRIvV52CD4YK//UC5zRqYFB/D8=">AAAB83icbVC7TsNAEDyHVwivACXNiQiJKrIDEhQUkWgoE4k8pMSKzpdNOOV8tu72kCIrX0ALFR2i5YMo+Bds4wISphrN7GpnJ4ilMOi6n05pbX1jc6u8XdnZ3ds/qB4edU1kNYcOj2Sk+wEzIIWCDgqU0I81sDCQ0Atmt5nfewRtRKTucR6DH7KpEhPBGaZS2x1Va27dzUFXiVeQGinQGlW/huOI2xAUcsmMGXhujH7CNAouYVEZWgMx4zM2hUFKFQvB+EkedEHPrGEY0Rg0FZLmIvzeSFhozDwM0smQ4YNZ9jLxP29gcXLtJ0LFFkHx7BAKCfkhw7VIGwA6FhoQWZYcqFCUM80QQQvKOE9Fm1ZSSfvwlr9fJd1G3buoN9qXteZN0UyZnJBTck48ckWa5I60SIdwAuSJPJMXxzqvzpvz/jNacoqdY/IHzsc3FdGRNw==</latexit>

. . .

<latexit sha1_base64="hS9sOeGjL84+ezfBHD9tCRGfb1A=">AAAB93icbVC7TsNAEDyHVwivACXNiQiJKrIDEhQUkWgog4STSIkVnS+bcMr5bN2tkSIr30ALFR2i5XMo+BfOxgUEphrN7Gh3J0ykMOi6H05lZXVtfaO6Wdva3tndq+8fdE2cag4+j2Ws+yEzIIUCHwVK6CcaWBRK6IWz69zvPYA2IlZ3OE8giNhUiYngDK3kD8cxmlG94TbdAvQv8UrSICU6o/qnzfE0AoVcMmMGnptgkDGNgktY1IapgYTxGZvCwFLFIjBBVhy7oCepYRjTBDQVkhYi/ExkLDJmHoV2MmJ4b5a9XPzPG6Q4uQwyoZIUQfF8EQoJxSLDtbAtAB0LDYgsvxyoUJQzzRBBC8o4t2Jqa6nZPrzl7/+SbqvpnTVbt+eN9lXZTJUckWNySjxyQdrkhnSITzgR5JE8kWdn7rw4r87b92jFKTOH5Bec9y+jLpNF</latexit>

F<latexit sha1_base64="WW/rpV2JkCbfCAdp6aJio946DBg=">AAACCXicbVDLSgNBEJyNrxhfUY9eBoPgKezGgB4DgnhMwDwgWULvpBOHzD6Y6RVCyBfkqh/iTbz6FX6HP+DuugdNrFNR1U11lxcpaci2P63CxubW9k5xt7S3f3B4VD4+6Zgw1gLbIlSh7nlgUMkA2yRJYS/SCL6nsOtNb1O/+4TayDB4oFmErg+TQI6lAEqk1t2wXLGrdga+TpycVFiO5rD8NRiFIvYxIKHAmL5jR+TOQZMUChelQWwwAjGFCfYTGoCPxp1nhy74RWyAQh6h5lLxTMTfG3PwjZn5XjLpAz2aVS8V//VSRZuxWcmn8Y07l0EUEwYijSepMIs3QsukF+QjqZEI0n+Qy4AL0ECEWnIQIhHjpKhS0pKz2sk66dSqzlW11qpXGvW8ryI7Y+fskjnsmjXYPWuyNhMM2ZI9sxdrab1ab9b7z2jByndO2R9YH99PgZqo</latexit>

k

<latexit sha1_base64="g3xfZIjS3VvAEZPqtZFgA84AwGA=">AAACCXicbVDLTgJBEOzFF+IL9ehlIjHxRHaRRI8kXjxCIkgCGzI7NDhh9pGZXhOy4Qu46od4M179Cr/DH3B33YOCdapUdae6y4uUNGTbn1ZpY3Nre6e8W9nbPzg8qh6f9EwYa4FdEapQ9z1uUMkAuyRJYT/SyH1P4YM3u838hyfURobBPc0jdH0+DeRECk6p1JmNqjW7budg68QpSA0KtEfVr+E4FLGPAQnFjRk4dkRuwjVJoXBRGcYGIy5mfIqDlAbcR+Mm+aELdhEbTiGLUDOpWC7i742E+8bMfS+d9Dk9mlUvE//1MkWbiVnJp8mNm8ggigkDkcWTVJjHG6Fl2guysdRIxLN/kMmACa45EWrJuBCpGKdFVdKWnNVO1kmvUXeu6o1Os9ZqFn2V4QzO4RIcuIYW3EEbuiAAYQnP8GItrVfrzXr/GS1Zxc4p/IH18Q2MWprN</latexit> ⇣k<latexit sha1_base64="8MC/6JorVB2dVhoBNqfjJdRNw9A=">AAACD3icbVC7TsNAEDyHVwivACXNiQiJKrJDJCgj0VAGiTykxIrWl0045fzQ3RopWPmItPAhdIiWT+A7+AFs4wICU41mdjW740VKGrLtD6u0tr6xuVXeruzs7u0fVA+PuiaMtcCOCFWo+x4YVDLADklS2I80gu8p7Hmz68zvPaA2MgzuaB6h68M0kBMpgFKpN3xEAj4bVWt23c7B/xKnIDVWoD2qfg7HoYh9DEgoMGbg2BG5CWiSQuGiMowNRiBmMMVBSgPw0bhJfu6Cn8UGKOQRai4Vz0X8uZGAb8zc99JJH+jerHqZ+K+XKdpMzEo+Ta7cRAZRTBiILJ6kwjzeCC3TdpCPpUYiyP5BLgMuQAMRaslBiFSM07oqaUvOaid/SbdRdy7qjdtmrdUs+iqzE3bKzpnDLlmL3bA26zDBZmzJntiztbRerFfr7Xu0ZBU7x+wXrPcvDC2dOQ==</latexit>↵k<latexit sha1_base64="g5TpiUkiw6LAyCv60w1VpKT3apg=">AAACEHicbVDLSgNBEJz1GeMr6tHLYBA8hd0Y0GPAi8cI5gHJEnonnWTI7IOZXiEs+Ylc9UO8iVf/wO/wB9xd96CJdSqquqnu8iIlDdn2p7WxubW9s1vaK+8fHB4dV05OOyaMtcC2CFWoex4YVDLANklS2Is0gu8p7Hqzu8zvPqE2MgweaR6h68MkkGMpgFKpNwAVTYHPhpWqXbNz8HXiFKTKCrSGla/BKBSxjwEJBcb0HTsiNwFNUihclAexwQjEDCbYT2kAPho3ye9d8MvYAIU8Qs2l4rmIvzcS8I2Z+1466QNNzaqXif96maLN2Kzk0/jWTWQQxYSByOJJKszjjdAyrQf5SGokguwf5DLgAjQQoZYchEjFOO2rnLbkrHayTjr1mnNdqz80qs1G0VeJnbMLdsUcdsOa7J61WJsJptiSPbMXa2m9Wm/W+8/ohlXsnLE/sD6+Abp8nZU=</latexit>

2l

<latexit sha1_base64="KcOcLqZ4etRHhXUQJfiJvxP1xwI=">AAACCnicbVC7TsNAEDyHVwivACXNiQiJKrINEpSRaCgDIg8psaLzZR1OOZ+tuzVSZOUP0sKH0CFafoLv4AewjQtImGo0s6vZHT+WwqBtf1qVtfWNza3qdm1nd2//oH541DVRojl0eCQj3feZASkUdFCghH6sgYW+hJ4/vcn93hNoIyL1gLMYvJBNlAgEZ5hJ964c1Rt20y5AV4lTkgYp0R7Vv4bjiCchKOSSGTNw7Bi9lGkUXMK8NkwMxIxP2QQGGVUsBOOlxaVzepYYhhGNQVMhaSHC742UhcbMQj+bDBk+mmUvF//1ckWbwCzlY3DtpULFCYLieTwKCUW84VpkxQAdCw2ILP8HqFCUM80QQQvKOM/EJGuqlrXkLHeySrpu07louneXjZZb9lUlJ+SUnBOHXJEWuSVt0iGcBGRBnsmLtbBerTfr/We0YpU7x+QPrI9vBOWbCA==</latexit> l<latexit sha1_base64="xZQ/Gih02a9jldIZVoWJ5uGjMAI=">AAACCXicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY8BLx4TMA9IljA76cQhsw9meoWw5Aty1Q/xJl79Cr/DH3B33YMm1qmo6qa6y4uUNGTbn1ZpY3Nre6e8W9nbPzg8qh6fdE0Ya4EdEapQ9z1uUMkAOyRJYT/SyH1PYc+b3WV+7wm1kWHwQPMIXZ9PAzmRglMqtdWoWrPrdg62TpyC1KBAa1T9Go5DEfsYkFDcmIFjR+QmXJMUCheVYWww4mLGpzhIacB9NG6SH7pgF7HhFLIINZOK5SL+3ki4b8zc99JJn9OjWfUy8V8vU7SZmJV8mty6iQyimDAQWTxJhXm8EVqmvSAbS41EPPsHmQyY4JoToZaMC5GKcVpUJW3JWe1knXQbdeeq3mhf15qNoq8ynME5XIIDN9CEe2hBBwQgLOEZXqyl9Wq9We8/oyWr2DmFP7A+vgGNZZrM</latexit>

d

<latexit sha1_base64="uwKnOwbaOJPfZAaR6AXaqCyJ4sY=">AAACCXicbVDLTgJBEJzFF+IL9ehlIjHxRHaRRI8kXjxCIkgCG9I7NDhh9pGZXhOy4Qu46od4M179Cr/DH3B33YOCdapUdae6y4uUNGTbn1ZpY3Nre6e8W9nbPzg8qh6f9EwYa4FdEapQ9z0wqGSAXZKksB9pBN9T+ODNbjP/4Qm1kWFwT/MIXR+mgZxIAZRKnfGoWrPrdg6+TpyC1FiB9qj6NRyHIvYxIKHAmIFjR+QmoEkKhYvKMDYYgZjBFAcpDcBH4yb5oQt+ERugkEeouVQ8F/H3RgK+MXPfSyd9oEez6mXiv16maDMxK/k0uXETGUQxYSCyeJIK83gjtEx7QT6WGokg+we5DLgADUSoJQchUjFOi6qkLTmrnayTXqPuXNUbnWat1Sz6KrMzds4umcOuWYvdsTbrMsGQLdkze7GW1qv1Zr3/jJasYueU/YH18Q2A15rG</latexit>

(a)

<latexit sha1_base64="BFtzL5+5OSfgBbmWaRl2G8egBC4=">AAACC3icbVDLTgJBEJzFF+IL9ehlIjHBC9lFEz2SePGIUR4JbEjv0OCE2Udmek0I4RO46od4M179CL/DH3B33YOCdapUdae6y4uUNGTbn1ZhbX1jc6u4XdrZ3ds/KB8etU0Ya4EtEapQdz0wqGSALZKksBtpBN9T2PEmN6nfeUJtZBg80DRC14dxIEdSACXSfRXOB+WKXbMz8FXi5KTCcjQH5a/+MBSxjwEJBcb0HDsidwaapFA4L/VjgxGICYyxl9AAfDTuLDt1zs9iAxTyCDWXimci/t6YgW/M1PeSSR/o0Sx7qfivlyrajMxSPo2u3ZkMopgwEGk8SYVZvBFaJs0gH0qNRJD+g1wGXIAGItSSgxCJGCdVlZKWnOVOVkm7XnMuavW7y0qjnvdVZCfslFWZw65Yg92yJmsxwcZswZ7Zi7WwXq036/1ntGDlO8fsD6yPb0tPmyY=</latexit>

(b)

<latexit sha1_base64="5beK1Ga+gRXrk1OI89Y77d2vxrM=">AAACC3icbVDLTgJBEJzFF+IL9ehlIjHBC9lFEz2SePGIUR4JbEjv0OCE2Udmek0I4RO46od4M179CL/DH3B33YOCdapUdae6y4uUNGTbn1ZhbX1jc6u4XdrZ3ds/KB8etU0Ya4EtEapQdz0wqGSALZKksBtpBN9T2PEmN6nfeUJtZBg80DRC14dxIEdSACXSfdU7H5Qrds3OwFeJk5MKy9EclL/6w1DEPgYkFBjTc+yI3BlokkLhvNSPDUYgJjDGXkID8NG4s+zUOT+LDVDII9RcKp6J+HtjBr4xU99LJn2gR7PspeK/XqpoMzJL+TS6dmcyiGLCQKTxJBVm8UZomTSDfCg1EkH6D3IZcAEaiFBLDkIkYpxUVUpacpY7WSXtes25qNXvLiuNet5XkZ2wU1ZlDrtiDXbLmqzFBBuzBXtmL9bCerXerPef0YKV7xyzP7A+vgFM9Zsn</latexit>

F

<latexit sha1_base64="WW/rpV2JkCbfCAdp6aJio946DBg=">AAACCXicbVDLSgNBEJyNrxhfUY9eBoPgKezGgB4DgnhMwDwgWULvpBOHzD6Y6RVCyBfkqh/iTbz6FX6HP+DuugdNrFNR1U11lxcpaci2P63CxubW9k5xt7S3f3B4VD4+6Zgw1gLbIlSh7nlgUMkA2yRJYS/SCL6nsOtNb1O/+4TayDB4oFmErg+TQI6lAEqk1t2wXLGrdga+TpycVFiO5rD8NRiFIvYxIKHAmL5jR+TOQZMUChelQWwwAjGFCfYTGoCPxp1nhy74RWyAQh6h5lLxTMTfG3PwjZn5XjLpAz2aVS8V//VSRZuxWcmn8Y07l0EUEwYijSepMIs3QsukF+QjqZEI0n+Qy4AL0ECEWnIQIhHjpKhS0pKz2sk66dSqzlW11qpXGvW8ryI7Y+fskjnsmjXYPWuyNhMM2ZI9sxdrab1ab9b7z2jByndO2R9YH99PgZqo</latexit>

Soft device ~ Gibbs ensemble Hard device ~ Helmholtz ensemble

n 1

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"

i

<latexit sha1_base64="u7injxvVSljLyDoc1m+ox8auugw=">AAACFXicbVDLTgJBEJzFF+IL9ehlIjHxRHaRRI8kXjxiIo8ENqR3aHDC7Ow600tCCN/BVT/Em/Hq2e/wB9xFDgrWqVLVnequIFbSkut+OrmNza3tnfxuYW//4PCoeHzStFFiBDZEpCLTDsCikhobJElhOzYIYaCwFYxuM781RmNlpB9oEqMfwlDLgRRAqeR3x2AwtlJFuid7xZJbdhfg68RbkhJbot4rfnX7kUhC1CQUWNvx3Jj8KRiSQuGs0E0sxiBGMMROSjWEaP3p4ugZv0gsUMRjNFwqvhDx98YUQmsnYZBOhkCPdtXLxH+9TDF2YFfyaXDjT6WOE0ItsniSChfxVhiZdoS8Lw0SQfYPcqm5AANEaCQHIVIxSUsrpC15q52sk2al7F2VK/fVUq267CvPztg5u2Qeu2Y1dsfqrMEEe2Jz9sxenLnz6rw57z+jOWe5c8r+wPn4Bpx5oEE=</latexit>

NN

<latexit sha1_base64="riQeTVndoUhJW0D18QiBAwubbt0=">AAACE3icbVDLTsJAFJ3iC/GFunQzSkxckRZJdEnixhXBRB4JJWQ6XGDCtJ3M3JqQhs9gqx/izrj1A/wOf8C2dqHgWZ2cc2/OvcdTUhi07U+rsLG5tb1T3C3t7R8cHpWPTzomjDSHNg9lqHseMyBFAG0UKKGnNDDfk9D1Znep330CbUQYPOJcwcBnk0CMBWeYSH1XTcUwbrrnzcWwXLGrdga6TpycVEiO1rD85Y5CHvkQIJfMmL5jKxzETKPgEhYlNzKgGJ+xCfQTGjAfzCDOTl7Qy8gwDKkCTYWkmQi/N2LmGzP3vWTSZzg1q14q/uulijZjs5KP49tBLAIVIQQ8jUchIYs3XIukIaAjoQGRpf8AFQHlTDNE0IIyzhMxSiorJS05q52sk06t6lxXaw/1SqOe91UkZ+SCXBGH3JAGuSct0iachGRJnsmLtbRerTfr/We0YOU7p+QPrI9vDeye0w==</latexit>

Si= 1

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Si= +1

<latexit sha1_base64="+8QWa4qdvkrWYp1iZjMXHs3XLUs=">AAACDnicbVDLSgNBEJyNrxhfUY9eBoMgCGE3CnoRAl48RjQPSELonXTimNkHM71CWPIPueqHeBOv/oLf4Q+4u+5Bo3Uqqrqp7nJDJQ3Z9odVWFpeWV0rrpc2Nre2d8q7ey0TRFpgUwQq0B0XDCrpY5MkKeyEGsFzFbbdyVXqtx9RGxn4dzQNse/B2JcjKYASqXU7kJcnzqBcsat2Bv6XODmpsByNQfmzNwxE5KFPQoExXccOqR+DJikUzkq9yGAIYgJj7CbUBw9NP86unfGjyAAFPETNpeKZiD83YvCMmXpuMukB3ZtFLxX/9VJFm5FZyKfRRT+WfhgR+iKNJ6kwizdCy6Qc5EOpkQjSf5BLnwvQQIRachAiEaOkrVLSkrPYyV/SqlWd02rt5qxSt/O+iuyAHbJj5rBzVmfXrMGaTLAHNmdP7NmaWy/Wq/X2PVqw8p199gvW+xdmiJxE</latexit>

k

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⇣k<latexit sha1_base64="a3CniMjQpxgABSrBctpF0szpVww=">AAACD3icbVC7TsNAEDyHVwivACXNiQiJKrIDEpSRaCiDRB5SYkXryyacfH7obo0UonxEWvgQOkTLJ/Ad/AC2cQEJU41mdjW748VKGrLtT6u0tr6xuVXeruzs7u0fVA+POiZKtMC2iFSkex4YVDLENklS2Is1QuAp7Hr+TeZ3H1EbGYX3NI3RDWASyrEUQKnUHTwhAfeH1Zpdt3PwVeIUpMYKtIbVr8EoEkmAIQkFxvQdOyZ3BpqkUDivDBKDMQgfJthPaQgBGneWnzvnZ4kBiniMmkvFcxF/b8wgMGYaeOlkAPRglr1M/NfLFG3GZimfxtfuTIZxQhiKLJ6kwjzeCC3TdpCPpEYiyP5BLkMuQAMRaslBiFRM0roqaUvOcierpNOoOxf1xt1lrWkXfZXZCTtl58xhV6zJblmLtZlgPluwZ/ZiLaxX6816/xktWcXOMfsD6+MbCvmdNQ==</latexit>

⌘kl

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"0

<latexit sha1_base64="Nj89BbHPx1Wj1feoQpC3Xxr4i5U=">AAACFnicbVDLTgJBEJzFF+IL9ehlIjHxItlFEz2SePGIiTwSQNI7NDhhdnYz00tCCP/BVT/Em/Hq1e/wB9xdOShYp0pVd6q7/EhJS6776eTW1jc2t/LbhZ3dvf2D4uFRw4axEVgXoQpNyweLSmqskySFrcggBL7Cpj+6Tf3mGI2VoX6gSYTdAIZaDqQASqTHi84YDEZWqlD33F6x5JbdDHyVeAtSYgvUesWvTj8UcYCahAJr254bUXcKhqRQOCt0YosRiBEMsZ1QDQHa7jS7esbPYgsU8ggNl4pnIv7emEJg7STwk8kA6Mkue6n4r5cqxg7sUj4NbrpTqaOYUIs0nqTCLN4KI5OSkPelQSJI/0EuNRdggAiN5CBEIsZJa4WkJW+5k1XSqJS9y3Ll/qpUdRd95dkJO2XnzGPXrMruWI3VmWCGzdkze3Hmzqvz5rz/jOacxc4x+wPn4xuu76A7</latexit> "0<latexit sha1_base64="O9gYyHstpKbTqcfKlTebsuhMv2g=">AAACFXicbVC7TsNAEDyHVwivACXNiQiJKrIDEpSRaCiDRB5SYkXryyaccj6bu3WkKMp3pIUPoUO01HwHP4BtUkBgqtHMrmZ3glhJS6774RTW1jc2t4rbpZ3dvf2D8uFRy0aJEdgUkYpMJwCLSmpskiSFndgghIHCdjC+yfz2BI2Vkb6naYx+CCMth1IApZLfm4DB2EoV6b7bL1fcqpuD/yXeklTYEo1++bM3iEQSoiahwNqu58bkz8CQFArnpV5iMQYxhhF2U6ohROvP8qPn/CyxQBGP0XCpeC7iz40ZhNZOwyCdDIEe7KqXif96mWLs0K7k0/Dan0kdJ4RaZPEkFebxVhiZdoR8IA0SQfYPcqm5AANEaCQHIVIxSUsrpS15q538Ja1a1buo1u4uK3V32VeRnbBTds48dsXq7JY1WJMJ9sgW7Ik9OwvnxXl13r5HC85y55j9gvP+BT2IoAQ=</latexit>

"

i

<latexit sha1_base64="6qDpdIS+jvu6b3rOvTgbSBP7g1k=">AAACEHicbVDLSgNBEJyNrxhfUY9eBoPgKezGgB4DXjxGMA9IltA76cQhsw9meoWw5Cdy1Q/xJl79A7/DH3B33YMm1qmo6qa6y4uUNGTbn1ZpY3Nre6e8W9nbPzg8qh6fdE0Ya4EdEapQ9z0wqGSAHZKksB9pBN9T2PNmt5nfe0JtZBg80DxC14dpICdSAKVSf2jk1IeRHFVrdt3OwdeJU5AaK9AeVb+G41DEPgYkFBgzcOyI3AQ0SaFwURnGBiMQM5jiIKUB+GjcJL93wS9iAxTyCDWXiuci/t5IwDdm7nvppA/0aFa9TPzXyxRtJmYlnyY3biKDKCYMRBZPUmEeb4SWaT3Ix1IjEWT/IJcBF6CBCLXkIEQqxmlflbQlZ7WTddJt1J2reuO+WWs1i77K7Iyds0vmsGvWYneszTpMMMWW7Jm9WEvr1Xqz3n9GS1axc8r+wPr4BjF6nd0=</latexit>

S<latexit sha1_base64="3S1K5NmMZbcbhxY63+fwMMWgIiA=">AAACDnicbVDLSgNBEJyNrxhfUY9eBoPgxbAbBb0IAS8eI5oHJCH0TjpxzOyDmV4hLPmHXPVDvIlXf8Hv8AfcXfeg0ToVVd1Ud7mhkoZs+8MqLC2vrK4V10sbm1vbO+XdvZYJIi2wKQIV6I4LBpX0sUmSFHZCjeC5Ctvu5Cr124+ojQz8O5qG2Pdg7MuRFECJ1LodyMsTZ1Cu2FU7A/9LnJxUWI7GoPzZGwYi8tAnocCYrmOH1I9BkxQKZ6VeZDAEMYExdhPqg4emH2fXzvhRZIACHqLmUvFMxJ8bMXjGTD03mfSA7s2il4r/eqmizcgs5NPooh9LP4wIfZHGk1SYxRuhZVIO8qHUSATpP8ilzwVoIEItOQiRiFHSVilpyVns5C9p1arOabV2c1ap23lfRXbADtkxc9g5q7Nr1mBNJtgDm7Mn9mzNrRfr1Xr7Hi1Y+c4++wXr/Qtp1JxG</latexit> i= 1 k

<latexit sha1_base64="moKrI1oUJ67Y44nxG8idpz77Ta4=">AAACCXicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY8BLx4TMA9IljA76cQhsw9meoWw5Aty1Q/xJl79Cr/DH3B33YMm1qmo6qa6y4uUNGTbn1ZpY3Nre6e8W9nbPzg8qh6fdE0Ya4EdEapQ9z1uUMkAOyRJYT/SyH1PYc+b3WV+7wm1kWHwQPMIXZ9PAzmRglMqtWejas2u2znYOnEKUoMCrVH1azgORexjQEJxYwaOHZGbcE1SKFxUhrHBiIsZn+IgpQH30bhJfuiCXcSGU8gi1Ewqlov4eyPhvjFz30snfU6PZtXLxH+9TNFmYlbyaXLrJjKIYsJAZPEkFebxRmiZ9oJsLDUS8ewfZDJggmtOhFoyLkQqxmlRlbQlZ7WTddJt1J2reqN9XWvaRV9lOINzuAQHbqAJ99CCDghAWMIzvFhL69V6s95/RktWsXMKf2B9fAOLJprJ</latexit>

Si= +1

<latexit sha1_base64="+8QWa4qdvkrWYp1iZjMXHs3XLUs=">AAACDnicbVDLSgNBEJyNrxhfUY9eBoMgCGE3CnoRAl48RjQPSELonXTimNkHM71CWPIPueqHeBOv/oLf4Q+4u+5Bo3Uqqrqp7nJDJQ3Z9odVWFpeWV0rrpc2Nre2d8q7ey0TRFpgUwQq0B0XDCrpY5MkKeyEGsFzFbbdyVXqtx9RGxn4dzQNse/B2JcjKYASqXU7kJcnzqBcsat2Bv6XODmpsByNQfmzNwxE5KFPQoExXccOqR+DJikUzkq9yGAIYgJj7CbUBw9NP86unfGjyAAFPETNpeKZiD83YvCMmXpuMukB3ZtFLxX/9VJFm5FZyKfRRT+WfhgR+iKNJ6kwizdCy6Qc5EOpkQjSf5BLnwvQQIRachAiEaOkrVLSkrPYyV/SqlWd02rt5qxSt/O+iuyAHbJj5rBzVmfXrMGaTLAHNmdP7NmaWy/Wq/X2PVqw8p199gvW+xdmiJxE</latexit>

⇣k<latexit sha1_base64="a3CniMjQpxgABSrBctpF0szpVww=">AAACD3icbVC7TsNAEDyHVwivACXNiQiJKrIDEpSRaCiDRB5SYkXryyacfH7obo0UonxEWvgQOkTLJ/Ad/AC2cQEJU41mdjW748VKGrLtT6u0tr6xuVXeruzs7u0fVA+POiZKtMC2iFSkex4YVDLENklS2Is1QuAp7Hr+TeZ3H1EbGYX3NI3RDWASyrEUQKnUHTwhAfeH1Zpdt3PwVeIUpMYKtIbVr8EoEkmAIQkFxvQdOyZ3BpqkUDivDBKDMQgfJthPaQgBGneWnzvnZ4kBiniMmkvFcxF/b8wgMGYaeOlkAPRglr1M/NfLFG3GZimfxtfuTIZxQhiKLJ6kwjzeCC3TdpCPpEYiyP5BLkMuQAMRaslBiFRM0roqaUvOcierpNOoOxf1xt1lrWkXfZXZCTtl58xhV6zJblmLtZlgPluwZ/ZiLaxX6816/xktWcXOMfsD6+MbCvmdNQ==</latexit>

"0

<latexit sha1_base64="Nj89BbHPx1Wj1feoQpC3Xxr4i5U=">AAACFnicbVDLTgJBEJzFF+IL9ehlIjHxItlFEz2SePGIiTwSQNI7NDhhdnYz00tCCP/BVT/Em/Hq1e/wB9xdOShYp0pVd6q7/EhJS6776eTW1jc2t/LbhZ3dvf2D4uFRw4axEVgXoQpNyweLSmqskySFrcggBL7Cpj+6Tf3mGI2VoX6gSYTdAIZaDqQASqTHi84YDEZWqlD33F6x5JbdDHyVeAtSYgvUesWvTj8UcYCahAJr254bUXcKhqRQOCt0YosRiBEMsZ1QDQHa7jS7esbPYgsU8ggNl4pnIv7emEJg7STwk8kA6Mkue6n4r5cqxg7sUj4NbrpTqaOYUIs0nqTCLN4KI5OSkPelQSJI/0EuNRdggAiN5CBEIsZJa4WkJW+5k1XSqJS9y3Ll/qpUdRd95dkJO2XnzGPXrMruWI3VmWCGzdkze3Hmzqvz5rz/jOacxc4x+wPn4xuu76A7</latexit> "0<latexit sha1_base64="O9gYyHstpKbTqcfKlTebsuhMv2g=">AAACFXicbVC7TsNAEDyHVwivACXNiQiJKrIDEpSRaCiDRB5SYkXryyaccj6bu3WkKMp3pIUPoUO01HwHP4BtUkBgqtHMrmZ3glhJS6774RTW1jc2t4rbpZ3dvf2D8uFRy0aJEdgUkYpMJwCLSmpskiSFndgghIHCdjC+yfz2BI2Vkb6naYx+CCMth1IApZLfm4DB2EoV6b7bL1fcqpuD/yXeklTYEo1++bM3iEQSoiahwNqu58bkz8CQFArnpV5iMQYxhhF2U6ohROvP8qPn/CyxQBGP0XCpeC7iz40ZhNZOwyCdDIEe7KqXif96mWLs0K7k0/Dan0kdJ4RaZPEkFebxVhiZdoR8IA0SQfYPcqm5AANEaCQHIVIxSUsrpS15q538Ja1a1buo1u4uK3V32VeRnbBTds48dsXq7JY1WJMJ9sgW7Ik9OwvnxXl13r5HC85y55j9gvP+BT2IoAQ=</latexit>

M ax

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"

i

NNN

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↵ <<latexit sha1_base64="yT+Sg7dIIjwshl01gdJFqwuXnTM=">AAACEHicbVC7TsNAEDyHVwivACXNiQiJKrIDEhQUkWgog0QeUmJF68smOeX80N0aKbL4ibTwIXSIlj/gO/gBbOMCEqYazexqdseLlDRk259WaW19Y3OrvF3Z2d3bP6geHnVMGGuBbRGqUPc8MKhkgG2SpLAXaQTfU9j1ZreZ331EbWQYPNA8QteHSSDHUgClUm8AKprCjT2s1uy6nYOvEqcgNVagNax+DUahiH0MSCgwpu/YEbkJaJJC4VNlEBuMQMxggv2UBuCjcZP83id+FhugkEeouVQ8F/H3RgK+MXPfSyd9oKlZ9jLxXy9TtBmbpXwaX7uJDKKYMBBZPEmFebwRWqb1IB9JjUSQ/YNcBlyABiLUkoMQqRinfVXSlpzlTlZJp1F3LuqN+8ta0y76KrMTdsrOmcOuWJPdsRZrM8EUW7Bn9mItrFfrzXr/GS1Zxc4x+wPr4xuGaZ1y</latexit> 0

(c) (d)

(e) (f)

NN

NNN

Figure 1: Scheme of the mechanical model in the two considered cases of (a) assigned forceFand (b) assigned total displacement d. Energies for the NN (c) and NNN (e) units and corresponding force-displacement diagrams are in (d) and (f), respectively.

where

−ε0

and

ε0

are the reference strains of the first and second well, respectively,

k/l

is the stiffness of

100

the first well (k has the dimension of a force),

ζk/l

is the stiffness of the second phase and

η

measures the transition energy with respect to the ground state.

Next, we consider the simplest case of non local interactions by introducing NNN harmonic oscillators of length 2l, with energy

φNNN

(ε

j, εj+1

) = 1

2

α k l εj

+

εj+12

, j

= 1, ..., n

−

1, (3)

where

α

measures the relative stiffness of the non local vs local springs. Observe that while in this paper we

105

are interested in the (concave) case with

α <

0, when non local energy terms penalize interfaces formation

4

(6)

(ferromagnetic hypothesis), the following calculations hold true also in the (convex) case with

α >

0, when the interfaces formation is favored (antiferromagnetic hypothesis). Moreover, while we deduce our theoretical relations for general values of

α, with the aim of getting analytic results we then focus on the case|α|

1, so that the NNN energy represents a perturbative term that let us distinguish between solutions with the

110

same phase fraction, but with different number of interfaces.

The total (adimensionalized with respect to

lk) energy of the system is then ϕ({Sj},{εj}

) = 1

2







n

X

j=1

"

(ε

j

+

ε0

)

²

(1

−Sj

)

2 +

ζ

(ε

j−ε0

)

²

+ 2η

(1 +

Sj

) 2

#

+

n−1

X

j=1

α

εj

+

εj+1

2





 .

(4) With the aim of simplifying the notation, we may introduce the following functions of the spin variable:

L(Sj

) =

(

1 if

Sj

=

−

1

ζ

if

Sj

= +1

, m(Sj

) =

(−ε0

if

Sj

=

−

1

ζε0

if

Sj

= +1

, q(Sj

) =

(ε²₀/2

if

Sj

=

−

1

ζε²₀/2 +η

if

Sj

= +1

.

(5) Accordingly, we may define the vectors

ε

=





 ε1

. . . εj

. . . εn







, s

=





 S1

. . . Sj

. . . Sn







, m

=





 m1

. . . mj

. . . mn







, q

=





 q1

. . . qj

. . . qn







, 1

=







1

. . . . . . . . .

1







,

(6)

where

mj

=

m(Sj

),

qj

=

q(Sj

) and each element of the vector

s

is the “spin” of the

j-th NN unit. In

115

addition, we may introduce the matrices

L

=







L1 0

. ..

Lj

. ..

0 Ln







, A

=







1 1

0

1 2 1

. .. ... ...

1 2 1

0

1 1







,

(7)

where

Lj

=

L(Sj

). Finally, by using (5), (6) and (7) we can compactly rewrite the energy of the system as

ϕ(s,ε) =

1

2

J ε·ε−m·ε

+

q,

(8)

where

J

=

J

(

s

) =

L

(

s

) +

αA

,

m

=

m

(

s

) and

q

=

q(s

) =

q

(

s

)

·1

.

Remark.

While previous description shows that the following analysis can be extended to the full general case, in order to focus on the physical results and get simply interpretable analytical formulas, in the

120

following we study the case of

ζ

= 1 and

η

= 0,

i.e.

we assume identical wells with no energy gap. Thus, with the aforementioned hypothesis we have

L

=

I, q

=

n ε²₀/2, m

=

ε0s.

(9) Observe that in the following we first obtain all the formulas without this assumption in terms of

J

and

m

and then we specialize them to the simpler case of (9).

3. Equilibrium

125

With the aim of getting physical insight into the problem of non local interactions and study their role in the proposed model, in this section we summarize the equilibrium configurations of the system when temperature effects are neglected. Here we study both the case of soft and hard devices, that, as demonstrated in Ref. [39], can be considered as limit regimes of a general problem in which the stiffness of the pulling device regulates the effective boundary conditions. We refer to Refs. [21,

22] for a more detailed

130

analysis when also the interaction with the loading experimental apparatus is considered.

(7)

3.1. Soft device

First, let us consider the case of soft device with applied force

F

(Fig.

1(a)). The potential energy of the

systems is

g(s,ε, σ) =ϕ(s,ε)−σ

n

X

j=1

εj

= 1

2

J ε·ε−

(m +

σ1)·ε

+

q,

(10) where

σ

=

F/k

is the adimensional load.

135

Equilibrium at assigned phase configuration

s

gives

∂g(s,ε, σ)

∂ε

=

J ε−

(m +

σ1) =0.

(11)

Therefore, the equilibrium strains are

εeq

=

J⁻¹

(m +

σ1),

(12)

corresponding to an average strain ¯

ε

:=

¹_nPn

j=1εj

and energy

¯

εeq

= 1

n εeq·1

= 1

n h

J⁻¹

(m +

σ1)·1i , geq

=

−

1

2

J⁻¹

(m +

σ1)·

(m +

σ1) +q.

(13)

Following Ref. [21], the inverse of the tridiagonal Hessian matrix

J

(non-singular in the hypothesis of small

α) can be expressed as

140

J⁻¹

=

∞

X

j=0

(

−α)^jL⁻¹

(AL

⁻¹

)

^j.

(14)

For small

α, taking into account that by using (9) we haveL

=

I

, gives

J⁻¹'I−αA.

(15)

Then, let us introduce the identities

s·s

=

n, s·1

= 2

p−n, 1·1

=

n,

As·s

= 4(n

−i−

1),

As·1

= 4(2

p−n)−

2(S

1

+

Sn

),

A1·1

= 4(n

−

1),

(16)

where

p

the

is number of elements in the second phase

and

i

is

the number of interfaces

,

i.e.

the number of times NN adjacent links have different phases,

S1

and

Sn

assign the phase of the boundary elements.

Finally, by using (9), (15) and (16), Eq. (13) can be rewritten as

145

¯

εeq

= (1

−

4α)

σ

+ 2χ

−

1

ε0

+ 2α

n

2σ +

S1

+

Sn ε0

, geq

n

=

−

(1

−

4α)

"

σ²

2 + 2χ

−

1

ε0σ

#

−

2α

n

σ²

+

S1

+

Sn

ε0σ

+

i ε²₀

+

n−

1

ε²₀

,

(17)

where

χ

:=

p/n

is the

phase fraction

.

Observe that the equilibrium configurations in (17), when NNN interactions are not considered (α = 0), only depend on the phase fraction

χ. On the contrary, when non local terms are introduced, the number of

interfaces distinguishes configurations with the same phase fraction. In particular, solutions with a larger number of interfaces are energetically penalized. It is easy to verify that the

global minimum

of the energy

150

is attained when all bistable units are in the first phase for

σ <

0 and in the second phase when

σ >

0, as shown in figure

2(a). Consequently, under the so called Maxwell hypothesis, when the configurations of the

system correspond always to the global energy minimum, we observe that as the force is increased the chain undergoes an “instantaneous” transition from the homogeneous state in the first phase to the homogeneous state in the second one with a fully cooperative phase transition and no interfaces.

155

6

(8)

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(a)

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(c)

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(d)

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Soft device Hard device

-10 -5 5 10

-0.5 0.5 1.0 1.5

-2 -1 1 2

-1.0 -0.5 0.5 1.0

-5 5

-0.5 0.5

-2 -1 1 2

-1.0 -0.5 0.5 1.0

-2 -1 1 2

-1.0 -0.5 0.5 1.0

-2 -1 1 2

-1.0 -0.5 0.5 1.0

-2 -1 1 2

-1.0 -0.5 0.5 1.0

-2 -1 1 2

-1.0 -0.5 0.5 1.0

-2 -1 1 2

-1.0 -0.5 0.5 1.0

¯

"_eq

g_eq/n '_eq/n

¯

"

¯

"

A B

-0.6 -0.4 -0.2 0.2 0.4 0.6

-6 -4 -2 2

eq

a b

= 0 =n

=n

= 0

=n

= 0

Figure 2: Mechanical equilibrium. Energies and stress-strain diagrams in the soft (a),(b) and hard (c),(d) device. Cyan curves represent the cases when only NN interactions are considered, orange curves show the behavior with the presence of non local terms. Dashed black lines are calculated with the Statistical Mechanics approach when the temperature goes to zero (see Sect.4,5). Parameter: n= 5,α= 0 in cyan curves andα=−0.05 in the orange ones, ˜β→ ∞in the black dashed lines,ε0= 1.

3.2. Hard device

Consider now the case of assigned total displacement

d, as schematized in Fig.1(b), and introduce its

dimensionless measure

δ

=

d l

=

n

X

j=1

εj

=

ε·1.

(18)

The equilibrium strain equations at fixed configuration

s

are again given by (12), where

σ

is the Lagrange multiplier measuring the force conjugate to the fixed total displacement in (18). By using Eq. (9), in the

160

case of small

α

(see Eqs (15)), Eq. (18) can be rewritten as

¯

ε

=

δ

n

= 1

n h

J⁻¹

(m +

σeq1)·1i '

1

n

(1

·1−αA1·1)σeq

+ (s

·1−αAs·1)ε0

,

(19)

where we indicate by

σeq

the equilibrium stress. Accordingly, we obtain the equilibrium energy

ϕeq

=

−

1

2

J⁻¹ m

+

σeq1

· m

+

σeq1

+

q

+

δ σeq ' ' −

1

2

1·1−αA1·1

σ_eq² − s·1−αAs·1−δ

ε0σeq

+ 1

2

α ε²₀As·s.

(20)

(9)

Eventually, by using (16) we obtain

σeq

=

¯

ε−

1

−

4α

2χ

−

1 + 2α

n S1

+

Sn

ε0

1

−

4α

n

(n

−

1)

,

(21)

and the equilibrium energy per element is given by

ϕeq

n

= 1 2

(h

(1

−

4α) (2

p/n−

1) +

^2α_n

(S

1

+

Sn

)

ε0−ε

¯

i2

1

−^4αn

(n

−

1) + 4α

n n−i−

1

ε²₀

)

.

(22)

In this case the

global minima

of the energy with respect to the assigned rescaled displacement

δ

are

165

given by solutions either homogeneous or with one single interface, as shown in Fig.

2(c). More in details,

in agreement with well known experimental observations, under the hypothesis of hard device the system changes state between the two homogeneous phase branches following a ‘sawtooth’ equilibrium path. Inter- estingly, the stress corresponding to the nucleation of the new phase (point

A

in Fig.

2(d)) is higher than

the stress corresponding to the propagation of the interface (e.g. point

B

in figure). It is important to

170

notice that this effect is not observed in absence of non local interactions (points

a

and

b). Moreover, one

may show that by increasing the value of

n

or of the non local energy

α

the nucleation can correspond to the cooperative transition of more elements. Such behavior is discussed in detail in Ref. [22], whereas here we are interested in the energetic competition of surface energy terms (measured by the parameter

α) and

entropic energy terms, so that we postpone the discussion to the following sections.

175

4. Applied stress: Gibbs ensemble

To describe the important effect of temperature, consider now our prototypical model with non local interactions in the framework of equilibrium Statistical Mechanics. Under isotensional conditions (i.e assigned stress

σ, soft device) we derive the canonical partition function in the Gibbs ensemble (denoted by

the subscript

G

), in order to study the system in thermal equilibrium [44]. By definition and by using the

180

energy in (10), we have

ZG

(σ) =

X

{S_j}

Z

Rⁿ

e⁻^βg^˜

d

ε

=

X

{S_j}

Z

Rⁿ

e⁻^β^˜

(

¹2J ε·ε−(m+σ1)·ε+q

)dε

,

(23) where ˜

β

=

βlk

and

β

= 1/(k

BT), with kB

the Boltzmann constant and

T

the absolute temperature. Here, we have summed over the discrete spin variables and we integrated the continuous strains.

Remark.

It is important to point out that by following [31,

40,41] in order to obtain analytical result we

assume that the two wells are extended beyond the spinodal point so that at the given configuration

s

we

185

may integrate all the strain all over

R

and avoid error functions. In [40] the authors numerically showed that in the temperature regimes of interest for real experiments this approximation does not influence the energy minimization.

A straightforward evaluation of the Gaussian integral leads to

ZG

(σ) =

X

{Sj}

s

(2π)

ⁿ

det(J )

e⁻^β^˜

(

⁻¹2J⁻¹(m+σ1)·(m+σ1)+q

)

,

(24) where, as expected, at the exponent we find the equilibrium energy in (13). Accordingly, preserving the

190

same first order approximation in (15) with

α

kept small and by using (9), we obtain

ZG

(σ)

'

s

(2π)

ⁿ

det(I

−αA)

X

{S_j}

e^β^˜{¹²^[(ε⁰^s+σ1)·(ε⁰^{s+σ1)−αA(ε}⁰^s+σ1)·(ε⁰^s+σ1)]−q}.

(25)

8