Eur. Phys. J. E (2014) 37: 37 DOI 10.1140/epje/i2014-14037-x

THE EUROPEAN PHYSICAL JOURNAL E

Regular Article

Free-energy landscapes of granular clusters grown by magnetic interaction Jorge Gonz´ alez-Guti´errez1 , J.L. Carrillo-Estrada1 , O. Carvente2 , and J.C. Ruiz-Su´ arez3,a 1 2 3

Instituto de F´ısica, Benem´erita Universidad Aut´ onoma de Puebla, A. P. J-48, Puebla 72570, M´exico Departamento de Ingenier´ıa F´ısica, Universidad Aut´ onoma de Yucat´ an, M´erida, Yucat´ an 97310, M´exico CINVESTAV-Monterrey, PIIT, Nuevo Le´ on, 66600, M´exico Received 15 October 2013 and Received in final form 14 March 2014 c EDP Sciences / Societ` Published online: 21 May 2014 –  a Italiana di Fisica / Springer-Verlag 2014 Abstract. We experimentally study the aggregation of small clusters made of non-Brownian dipolar beads in a vibro-fluidized system. The particles are paramagnetic spheres that add around a fixed magnetic seed inside a granular gas of glass beads. We observe that under appropriate physical conditions symmetric and asymmetric cluster configurations are created and, as the number of particles increases, the aggregation time obeys a power law. We use an ensemble statistics to evaluate the free-energies and entropies landscapes of the granular clusters. The correspondence between such landscapes shows that, even if the system is of macroscopic scale and not in strict equilibrium, our approach to understand the relationship between the cluster structures and the interactions that create them is reliable.

1 Introduction The classical theory of nucleation and aggregation [1–3] has been successfully extended to describe crystallization phenomena in molecular [4], solutions [5], proteins [6] and colloidal systems [7]. However, despite several attempts to develop the out-of-equilibrium thermodynamics [8] and statistical mechanics [9] of small clusters, a complete theory, or a comprehensive experimental knowledge, has not yet been fully established. The intrinsic non-equilibrium nature of the processes involved in the nucleation and aggregation, and sometimes the high dissipation regime in which these processes occur, make the development of a fundamental theory a quite difficult task. Due to the small number of particles the fluctuations in the measured quantities are large, thus, the treatment requires the so-called fluctuation theorems [9]. Most of the attempts to develop the thermodynamics of growth for small clusters are addressed on the construction of an ensemble description based on the knowledge of the network of forces affecting the particles that form the clusters [10,11]. In this order of ideas, an interesting experiment on the thermodynamics of very small colloidal clusters at equilibrium was carried out by G. Meng et al. [12]. The authors aimed to answer important questions like: what cluster structures are favoured by entropy? Or how does the competition between potential energy and entropy evolve as the number of particles N approaches the bulk limit? They studied the formation of clusters made of colloidal a

e-mail: [email protected]

particles at each value of 6 ≤ N ≤ 10, and estimated the free-energy landscape from an ensemble statistics using Boltzmann distribution, ΔF = −kB T ln P ; where kB is the Boltzmann’s constant, T is a fixed temperature of the heat bath and P the probability of observing some given clusters, which thereafter were classified by comparing them to finite sphere packings. Their clusters, observed by optical microscopy, were aggregations induced by the competition of thermal fluctuations and short-range depletion forces. We report in this paper the aggregation of 3D small clusters grown in a granular gas and use a distribution of measured probabilities to calculate free-energy and entropy landscapes. However, an important difference with the Meng et al. physical system is that the “atoms” in our system are paramagnetic grains in the presence of a magnetic field due to a localized source, a magnetic particle that acts as a seed. Hence, the interactions that drive the aggregation are not depletion forces but dipolar induced forces. Depending on the strength of the magnetic field given to the seed, we found different symmetric and asymmetric cluster configurations. This result is interesting because even with the inherently anisotropic interaction, when the strength of magnetic field is low, we observe the signature of a van der Waals effective potential conjectured by P. de Gennes et al. [13] (where symmetric structures form). In our experiments we can follow the evolution of small clusters particle by particle, therefore the aggregation time for different strength magnetic fields and different number

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of particles is easily measured. We found that such aggregation obeys a power law. We must remark, however, that our system is athermal, and this condition poses a fundamental problem: if our intention is to use an ensemble statistics ` a la Boltzmann, how to deal with kB ? Many efforts have been made to develop, based on thermal analogies [14–21], a thermodynamic formalism to describe granulates in the context of dynamic systems and non-equilibrium kinetic theory. Yet, thus far there are obscure issues. For example, it is a common practice in the field of granular matter to drop kB when one defines the temperature. However, this issue of the dropped units of kB has never been properly considered. Instead, we propose to use the very appealing although a bit irreverent ideas of H.S. Leff [22] and Arieh Ben-Naim [23], who claim, specially the second author, that the involvement of energy and temperature in the definition of entropy is a historically accident, a relic of the pre-atomistic era of thermodynamics. Had the kinetic theory of gases preceded Carnot, Clausius and Kelvin, Ben-Naim argues, the change in entropy would have been dimensionless, facilitating the identification of thermodynamic entropy with Shannon’s entropy [24]. Thus, following this argument (very convenient for granular systems) the granular temperature is simply an energy and there is no need to take kB as one and magically make its units disappear.

2 Experimental details and results The experimental procedure is as follows: we introduce a certain number of paramagnetic (neodymium) and nonparamagnetic (glass) particles inside a three-dimensional cell (15 × 15 × 2 cm) which is fixed on a vibrating table. The vertical vibration is characterized by a parameter Γ = (Aω 2 )/g, where A and ω = 2πf are the amplitude and angular frequency. In our experiments f = 45 Hz and Γ = 3. The paramagnetic particles (D = 5.01 mm and mass m = 1.7 g) are introduced in the cell fully demagnetized. The magnetic interaction is introduced by a magnetized seed fixed at the center of the base of the cell (the seed was composed by two spheres, with the same diameter than the paramagnetic particles, one paramagnetic glued to the bottom of the cell and one magnetic tied to it). The magnetization of the second bead is varied between 6 and 12 kG and the magnetic dipole is oriented perpendicular to the base. We used 144 glass particles with two different diameters: 4 and 6 mm (0.074 and 0.27 g), to introduce stochasticity in the system and ensure that as the neodymium particles condense, the granular temperature of the system does not go to zero. Image data are acquired through a digital camera (DRS Lightning RDT Plus) at 100 fps and a Pixelink PL-B742V at 1fps. The experiments begin by turning on the vibrator to form a gas of paramagnetic and glass particles at the chosen Γ . Since the fluxes are well compensated, the system immediately reaches a steady state. The aggregation process occurs when the paramagnetic particles, that move in random walks, are attracted to the seed by the action

Eur. Phys. J. E (2014) 37: 37

Fig. 1. (a) The probability distributions P of the super symmetric cluster for different magnetizations (N = 11). Inset: the corresponding aggregation time τ . Three asymmetric clusters (b-d) and a super symmetric cluster (e)

of induced dipolar interactions. In these conditions, we assume that correlation effects can be neglected and seize on the conclusions of D. A. Egolf, who states that the system recovers equilibrium and ergodicity conditions under the chaotic dynamics produced by the driven system [20]. At any moment, a particle of the bath can exchange energy with the structure that is forming and modifies it. When no change in the formed cluster (after the last particle aggregates), is observed on a time scale of several minutes (empirically, we have found that 30 min is enough), the system is said to be at equilibrium. First, at each value of the magnetizations 6 ≤ M ≤ 12 kG, we repeat the aggregation experiment 50 times to generate the same number of clusters with 11 paramagnetic particles plus the seed. When the strength of the magnetic field is high, we found a large number of asymmetric configurations (fig. 1(b)-(d)). This means that most of the particles get trapped in local minima. The maximum lateral size of the structure is about 4 times the diameter of the neodymium particles and the maximum height is 3.5 diameters. Structures where a single layer of paramagnetic particles completely covers the seed have symmetry 5 (considering that the spheres add around the three beads in the z axis, see fig. 1(e)). On the other hand, the aggregation time for different magnetizations τ (the time at which the aggregation process occurs) is shown in the inset of fig. 1. As expected, the longer it is the higher is the probability to produce super symmetric clusters. This is because paramagnetic particles can explore the entire phase diagram to find a deep potential energy minimum. We extended our observations to 7 ≤ N ≤ 10 using the particular seed magnetization of M = 6 kG. We generated 50 clusters at each value of N to measure their occurrence frequencies P and the aggregation time τ (for N = 7 we generated 100 clusters and obtained a similar distribution probability. Therefore, we consider the statistics with 50 repetitions to be reasonable for each value of N ). We found that τ increases as N ∗ − N decreases, see fig. 2(a). This means that under these conditions the

Eur. Phys. J. E (2014) 37: 37

Fig. 2. (a) The aggregation time τ obeys a power law for different number of paramagnetic particles 7 ≤ N ≤ 11 and M = 6 kG (see inset). Note that N ∗ = 12. (b) shows a time sequence of photographs with the aggregation of 7 paramagnetic particles to the magnetic seed. The photographs were taken at t = 1, 3, 8, 12, 15, 17, 19, 68, 87, 174 s.

formation of structures with 12 particles, although physically feasible, takes a very long time (indeed, we waited at least five hours and never observed the aggregation of the 12th particle). τ diverges because there are no magnetic bondings large enough to counteract the granular “thermal” energy. A time sequence of snapshots showing the aggregation processes are depicted in fig. 2(b). The obtained clusters are compared with each other and their occurrence frequencies P of each one for different N are shown in fig. 3(a). To simplify the identification of the structures, we used a nomenclature governed under completes rules. For example, all structures tagged as Cba (where a = 1, . . . , 5 and b = 1, . . . , 5) form the five symmetry group. This means that if we somehow extract a lateral particle of C55 , we can obtain C45 or C54 depending if the particle belongs to the upper or lower pentagon, respectively. With this classification, we find that about 40% of the structures can be assembled from C55 . The Hba group is formed by structures which have a particle at the center of one of the side faces of the two pentagons (here a = 1, . . . , 3 and b = 1, . . . , 3). Note that the difference between H22 and H22∗ is that the upper tetrahedron is rotated (see fig. 3(a)). The Jba group is formed by a variant of the H group (here, a particle of the upper pentagon of the characteristic face of the Hba group is extracted). Again a and b take values from 1 to 3. The Bba group is formed by structures in which a particle is attached to one of the lateral edges. A minus sign means a parity transformation on the a or b plane. The values of a and b are 1-3. The structures E and T do not pertain to a particular group but are reproducible. Finally, the X group are amorphous structures that were not reproducible.

3 Free energy and entropy We determine the free energy and entropy of the clusters using an ensemble statistics. Classically, a physical system

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represented by a canonical ensemble (NVT) is a system that can exchange energy via thermal contact with an external reservoir. Due to this, the system remains at a specified temperature. In a canonical ensemble, the number of particles N and the volume V remain constants, conditions that are satisfied in our case (V is the volume of the structures and once there are not rearrangements inside them it does not change). The granular temperature is the average of the velocity fluctuations in a fluidized granular system and is analogous to the thermodynamic temperature [25]. This temperature has units of energy as Ben-Naim suggests, allowing to express the Helmholtz free energy in terms of T (instead of kB T ). The granular temperature has already been successfully used to describe aggregation phenomena in a granular gas where the particles interact via dipolar hard sphere potentials like the one schematically shown in fig. 3(b) [26]. Following the same steps as [26], we take a sequence of images and found that the granular gas has a well-defined granular temperature with no significant gradients. However, in this Ben-Naim scheme [23], there is no need to measure T since we use experimental probabilities to estimate the landscapes of free energy and entropy. Due to the fact that our system is conceptually analogous to a canonical ensemble (see appendix A), and using the idea of Ben-Naim [23], we determine the free energy considering that the probability of a state s, Ps , is given by the Boltzmann distribution [27] Ps ∝ e(−Fs /T ) , where Fs is the Helmholtz free energy of state s. Our structures are the macro states (C55 , C33 , etc.) while each micro state is a particular form to ensemble a given cluster permuting particles. We are aware that this relation is strictly valid only in equilibrium. However, assuming that the energy fluxes are finely compensated, and at the spatial scales of the order of the cell size the system does not exhibit significant temperature gradients, we use this simplified Boltzmann distribution to calculate the free energy as Fs = −T ln (Ps /P0 ),

(1)

where P0 is the less probable state (i.e. the state with the highest free energy). The free-energy curves are shown in fig. 4(a). We observe the emergence of a complex freeenergy landscape with many local minima and ground states. Each structure represents a local free-energy minimum, the depth of which is proportional to the probability. Since symmetric structures are the most probable states in our system, such structures have an energy depth larger than any other (i.e. more cohesion). In particular, the structure C55 has the maximum energy depth (ground state). Formally speaking, in an out-of-equilibrium system there is not an a priori principle indicating that the states with the slower value of the free energy would be the most probable ones. However, if the energy fluxes are well compensated and if in the system there are no present significant inhomogeneities, in the attained steady state the assumption that the aforesaid principle exists is a plausible hypothesis. In order to give a stronger support to this framework, we make some numerical calculations.

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Fig. 3. (a) Shows the occurrence frequencies P of the stable clusters for 7 ≤ N ≤ 11. (b) The magnetic potential between the seed and the beads follows the well-known dependence 1/r3 as the curve indicates.

The magnetic energies of the clusters shown in fig. 3(a) can be estimated summing the idealized interaction between dipolar hard spheres [26] Ui,j =

μ2 μi · μ ˆj − 3(ˆ μi · rˆij )(ˆ μj · rˆij )] , 3 [ˆ rij

(2)

where μ is the magnitude of the dipole, rij = |rj − ri | is the particle separation distance taking into account the hard core of the spheres, μ ˆi,j , rˆij are unit vectors in the direction of the two dipole moments. Of course, assuming that the induced dipoles have approximately the same magnitude, we need to know the dipole orientations that led to a given structure. In order to do this, arbitrary orientations are put at the center of the particles and then allowed to rotate in all directions to find the configuration that minimizes the magnetic energy. Take for example the structure C55 obtained with N = 11 (see fig. 3). Figure 5 shows the obtained dipole orientations that minimize

the energy of such cluster (for the minimization procedure we used the conjugate gradient minimization method in Mathematica [28]). In fig. 4(a) we note that the calculated magnetic energies for the clusters with N = 11 drop as expected. Entropy S was derived by Shannon in 1948 [24] for information theory and is a function of the probability Ps of occurrence of different states in a system. It is defined as n  S=− Ps ln Ps . (3) s=1

The most general interpretation of Shannon theory is that entropy is a measure of our ignorance of the system. If Shannon’s entropy is the sum of Ps ln Ps , then each term does carry some entropy information. In a system with a uniform distribution of probabilities, each Ps ln Ps term would be the same, therefore, the difference between

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Fig. 4. (a) Free-energy landscape. The reference states of free energy are chosen to be the highest free-energy states at each value of N . The red squares for N = 11 are calculated magnetic energies using eq. (2). (b) Shannon’s entropy landscape. The reference states of entropy are chosen to be the states with the lowest entropy at each value of N .

Fig. 5. Dipole orientations that minimize the magnetic energy of the C55 structure. The magnetic sphere at the center has a given magnetization and the others 10% of it.

any two terms, (our ΔSs ) would be zero as there is no preferential state the system prefers to go. Of course, the total Shannon’s entropy would be maximum (for a system of N equiprobable states, this total entropy would be ln(N )). However, in our system of clusters where we do not know a priori the frequencies and the probability distribution is not uniform, following ΔSs reveals more information than the total entropy. The same occurs for the free energy. S ≈ 0 when either Ps ≈ 0 or ≈ 1, and it is maximum when Ps ≈ 0.37 (indeed, when a state never or always appears, we know the system perfectly; but when

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it occurs a bit less that half of the times, we ignore it so much that we would never bet anything for it). Some measures of the complexity of powder structures have been used to statistically characterize aggregates in different physical contexts. Most of these works address the discussion on the construction of statistical ensembles based on the calculation of the compactivity and forces networks that affect the particles [29–32]. Also, more or less in the same vein than the work here reported, recent studies have been carried out to understand the kinetic trapping of colloids trapped in small clusters [33,34]. Stochasticity and fluctuations dominate the thermal behaviour in small systems. Since the force is a fluctuating quantity, then: energy, heat flows, and work processes will fluctuate as well, and the amount of work converted into heat and exchanged with the thermal bath will fluctuate in magnitude and even in sign. Small systems out of equilibrium are characterized by irreversible heat losses between the system and the environment. Pioneering developments towards a unified treatment of arbitrarily large fluctuations in small systems are embodied in the so-called fluctuation theorems. These fluctuations theorems relate the probabilities of a system exchanging a given amount of energy with the properties of the thermal bath in a non-equilibrium process. In our case, the reproducibility of the resulting frequencies of appearance for the different configurations, given a value of the Γ parameter, assures that the energy balance necessary to reach a steady state condition is well established. Shannon’s entropy has been used to investigate a wide variety of phenomena in many different branches of science: earthquake events [35], binary hard-disk glasses [36], protein folding [37], etc. It is easy to show that the thermodynamic entropy in the canonical ensemble,   ∂ ln Q + kB ln Q, (4) S = kB T ∂T N,V  (where Q = n e−En /kB T ) is equal to Shannon’s entropy when kB = 1. By using the probabilities obtained from the frequencies of appearance of the different configurations, we calculate Shannon’s entropy landscapes from eq. (3), they are shown in fig. 4(b). Note that, despite being a granular system not in strict equilibrium but in a steady state, there is a correspondence between changes in free energy and entropy in the way thermodynamics predicts (ΔS > 0), except for N = 11, where ΔS ≈ 0 (note that the probability of appearance of state C55 is very large and for state X is very small, so their entropies are similar). Nevertheless, due to the fact that ΔU < 0, see fig. 4(a), the reduction of free energy is guaranteed. To fully understand the physical condition of the granular fluid with N = 11 and 12, in which the aggregation presents large fluctuations, it would require the use of a proper fluctuation theorem [9], something is beyond the scope of this work. In our experiments, we found the symmetric cluster C55 to be the most probable state for these physical conditions. Since the aggregation is driven by the magnetic interactions, not available sites with deep potential wells

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are left when the shell is completely formed (explaining why the sticking coefficient of the 12th particle is insignificant). The situation is different for entropy driven aggregation, where as the cluster size increases the available sites increases too. Moreover, being the C55 cluster the most favourable state (with the minimum energy), for a reduced number of beads more structures appear around the incomplete C55 . Due to the fact that paramagnetic particles are able to fully explore the phase diagram, the greater the aggregation time the more the number of super symmetric clusters, indicating that the beads are trapped in local minima. For high magnetizations, the aggregation times are short and non-symmetric clusters are obtained (analogous to a fast cooling where vitrified structures are formed).

Now, we can rewrite the first term of the equation in ∂ ∂ = − T12 ∂(1/T terms of 1/T, so ∂T ) . Thus  kT −k QT



∂ ln Q ∂T 

∂Q ∂1/T

N,V

In conclusion, the novelty of this work is that a driven granular system has been studied within the framework of Shannon’s theory of information to quantify the entropy and free-energy participation in the cluster formation processes. This approach allows us to use the statistical mechanics canonical ensemble formalism to find freeenergies and entropies landscapes. Despite the anisotropy of the magnetic potential and the non-strict equilibrium of the system, symmetric structures are found and free energies, estimated only by probabilities, are in accordance to changes in entropy (i.e. while ΔF < 0, ΔS > 0). Also, we have shown that the average cluster size and its symmetry are determined by the nature of the potential and the kinetic energy of the particles of the bath. Paraphrasing Edwards et al. [10]: the actions exerted by shaking our system do not act on grains individually, yet, welldefined cluster states result. It is worth remarking that geometrical constraints, observed at the scale of few particles, could explain why structural arrest occurs in granular compaction or crystallization [28,38,39]. This work has been supported by Conacyt, Mexico, under Grants 101384 and 104616, and VIEP-BUAP grant CAEJEXC11-1. J.G.G. acknowledges a fellowship from CONACyT.

  kT ∂Q = Q ∂T N,V N,V   −k  ∂e−En /kT = = QT n ∂1/T N,V =

 En e−En /kT  En P n k  En e−En /kT =k =k , QT n k kT Q kT n n 1 −En /kT . Qe

where the probability Pn =  kT

4 Conclusions



∂ ln Q ∂T

 =k N,V

Therefore

 En P n n

kT

.

(A.3)

Now we can rewrite the equation for S as    En P n ∂ ln Q S = kT + k ln Q = + k ln Q = k ∂T kT N,V n  En P n  k +k Pn ln Q, kT n n  given that n Pn = 1. Therefore S=k



 Pn

n

 En + ln Q . kT

(A.4)

Since  ln Pn = ln

1 −En /kT e Q

Finally S = −k





=−

En − ln Q. kT

Pn (ln Pn ).

(A.5)

(A.6)

n

If k = 1 (as Ben-Naim suggests), we get Shannon’s entropy  S=− Pn (ln Pn ). (A.7) n

Appendix A. Shannon entropy in the canonical ensemble We begin with the well-known expression for the differential of Helmholtz energy in the canonical ensemble  dF = −SdT − P dV + μn dNn , (A.1) n

from where we can identify S as − ∂F ∂T . Since F = −kT ln Q  (where Q = n e−En /kT is the partition function),   ∂ ln Q S = kT + k ln Q. (A.2) ∂T N,V

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Free-energy landscapes of granular clusters grown by magnetic interaction.

We experimentally study the aggregation of small clusters made of non-Brownian dipolar beads in a vibro-fluidized system. The particles are paramagnet...
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