HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY Vol. 46(2) pp. 47–54 (2018) hjic.mk.uni-pannon.hu DOI: 10.1515/hjic-2018-0018 THE INITIAL MAGNETIC SUSCEPTIBILITY OF DENSE AGGREGATED DIPOLAR FLUIDS SÁNDOR NAGY *1 1Institute of Mechatronics Engineering and Research, University of Pannonia, Gasparich M. u. 18/A, Zalaegerszeg, H-8900, HUNGARY To correlate the dipole moment and density dependence of the initial magnetic susceptibility on the basis of the former related theories and the probability analysis of chain formation, physically based analytical correlation equation was derived. After the local magnetic field strength and the chaining probability between two particle have been determined the chain and particle distributions came from the geometric distribution. The initial magnetic susceptibility was resulted from the summation of Langevin initial susceptibility of k-length chains. Two particles were considered in a chain if the interaction energy between them was below a certain limit. By varying slightly this energy limit around 70–75 % good agreement has been obtained between the simulation and theoretical data. Monte Carlo simulations were used to calculate the initial magnetic susceptibility of dipolar hard sphere system at different dipole moments and densities. Keywords: dipolar fluids, initial susceptibility, Monte Carlo simulation 1. Introduction The investigation of dipolar fluids has been induced by the evolution of magnetorheology and electrorheology over the last two decades. The viscosity of electrorheo- logical (ER) fluids increases dramatically due to an ex- ternal electric field. ER fluids can be obtained by dispers- ing solid particles with dielectric permittivity εp in a fluid with dielectric permittivity εf , where εp > εf . The dis- persed particles are of between 0.1 mm and 100 mm in diameter. The polarized particles are organized into pairs and chains. The magnetic analogy of the phenomenon de- scribed above is the magnetorheological (MR) effect. If the magnetic permeabilities of the liquid and dispersed particles differ, then in an external magnetic field the par- ticles are also arranged in chains. The dispersing medium can be water, oil, an organic solvent, etc. while the dis- persed particles can be some kind of iron oxide or ferrit. In this paper, the magnetic terminology and centimetre- gram-second (CGS) system of units are used. In the fig- ures, the reduced quantities are applied. Electro- and magnetorheological fluids typically ex- hibit a reduced density of up to ρ∗ = 0.4 (where ρ∗ = ρσ3; ρ and σ are the concentration and diameter of the suspended particles, respectively). The magnetic proper- ties, e.g. magnetization curve and initial magnetic suscep- tibility, are well described by the various theories within this range of reduced density. The magnetization M can be obtained by summation of the dipole moments in the *Correspondence: sata123.sandor@gmail.com unit volume: M = 1 V ∑ i mi. (1) In the absence of any external magnetic field the fluid is isotropic and according to Eq. 1 the magnetization is zero. When any external magnetic field is present, the field-oriented components of dipole moments should be summarized as M = ρm 〈cos Θ〉 H0 H0 , (2) where H0 = |H0| and 〈cos Θ〉 is the ensemble aver- age of the cosine of the angle between mi and H0, and m = |mi|. Since the directions of H0 and M are identi- cal, vector notation can be omitted. The initial magnetic susceptibility is equal to the initial gradient of the mag- netization curve χ0 = ∂M ∂H0 ∣∣∣∣ H0=0 . (3) In practice, ER and MR fluids can be used for the trans- mission of torque or force, in vibration dampers and brak- ing systems, etc. The magnetic properties generally are calculated from Monte Carlo simulations because it is not necessary to know the velocity and acceleration of the particles nor the forces between them. The expressions of the related models are listed in Ta- ble 1. (One line belongs to one theory and the first line is the head of the table, e.g. Table 1: 2.4 refers to the 4th cell in the 2nd line within the 1st table.) Three different mailto:sata123.sandor@gmail.com 48 NAGY Table 1: The expressions of the effective magnetic field, the magnetization and the initial magnetic susceptibility of the related theories. Model He – Effective magnetic field M – Magnetization χ0 – Initial magnetic susceptibility 1 “Langevin” [1] H0 ρmL ( mH0 kBT ) ρm2 3kBT = χL 2 “Weiss” [2] H0 + 4π 3 M(He) ρmL ( mHe kBT ) χL 1− 4π 3 χL 3 “Pshenichnikov” [3] H0 + 4π 3 M(H0) ρmL ( mHe kBT ) χL ( 1 + 4π 3 χL ) 4 “Ivanov” [4] H0 + 4π 3 M(H0) ( 1 + 4π 48 ∂M ∂H0 ) ρmL ( mHe kBT ) χL ( 1 + 4π 3 χL + (4π) 2 144 χ2 L ) 5 “Tani” and “Szalai” [5, 6] M(H0) = ρmL+ 4π 3 ρ 2βm3LL ′ + 1 10ρ 2β2m5ζ ′ Idd∆ − 16π2 27 ρ3β2m5LL ′ + 1 3ρ 3β2m5LL ′ Idd∆ χL ( 1 + 4π 3 χL + (4π)2 144 χ2 Lf(ρ) ) magnetic fields will be used. The applied external mag- netic field is denoted by H0 and the sum of the external and generated magnetic fields by He. He is always par- allel to H0. The third one is the local magnetic field Hl which is of chain-parallel orientation and its formation is due to dipole-dipole interactions between the particles. The well-known Langevin function is applied from the initial theory [1] in the magnetization formula (Ta- ble 1: 1.3), where L(α) = cothα − 1/α. The magnetic dipole moment of the particles is denoted by m and the applied external magnetic field by H0, while the Boltz- mann constant is represented by kB. The expression of magnetic susceptibility can be written as in Table 1: 1.4. This is known as Langevin susceptibility which is indi- cated by χL as well. In this approach the effective mag- netic field He exerted on the given particle is equal to the external magnetic field (Table 1: 1.2). According to the more accurate model by Weiss [2] the effective magnetic field is equal to the sum of the ex- ternal magnetic field and the magnetic field induced by the magnetization (Table 1: 2.2). The formula of the mag- netization (Table 1: 2.3) is similar to the previous one but H0 is substituted by He. Due to the iterative nature of the magnetization expression the initial magnetic suscep- tibility (Table 1: 2.4) exhibits divergence at χL = 3/4π, therefore, overestimates the real values. Above this initial magnetic susceptibility limit, when χL ≥ 3/4π, the zero- field magnetization is not equal to zero: M(H0) 6→ +0, ifH0 → +0. Moreover, in weak external magnetic fields, one H0 value belongs to three equilibrium magnetization values. The effective magnetic field has been substituted for the external magnetic field in the expression of the ef- fective magnetic field (Table 1: 3.2) in the theory by Pshenichnikov et al. [3]. The magnetization formula (Ta- ble 1: 3.3) is the same as in Weiss’ theory. The initial magnetic susceptibility (Table 1: 3.4) is in good agree- ment with the simulations but underestimates them at higher densities or higher dipole moments. That is why it seems to be a good method to extend the expression of the effective magnetic field (Table 1: 4.2) by Ivanov et al. [4]. The magnetization formula is once again identical (Table 1: 4.3) but a new term is in- troduced in the initial magnetic susceptibility (Table 1: 4.4). Although at higher densities it yields higher val- ues than in Pshenichnikov’s model, it underestimates the simulation data as well. The factor of the third term is (4π)2/144 = 1.0966 and perhaps it could be higher, but in this case at low densities the initial magnetic suscepti- bility overestimates the simulations. The perturbation theory by Tani et al. [5] is worth mentioning because a density-dependent correction was used to complete the third term of the susceptibility (Ta- ble 1: 5.4), where f (ρ) = 9Idd∆/π 2 − 16, and Idd∆ = 17π2 9 [ 1− 0.93952ρ∗ + 0.36714(ρ∗) 2 1− 0.92398ρ∗ + 0.23323(ρ∗) 2 ] . The formula of the magnetization curve for this pertur- bation theory was calculated by Szalai et al. The ex- pressions that are not mentioned in Table 1: 5.2 can be found in Ref. [6]. The values of the susceptibility more closely resemble the simulation data but still underesti- mate those. It is worth mentioning the study by Huke and Lücke [7] who introduced the so-called “dipolar coupling con- stant” into the second term of the initial magnetic sus- ceptibility, but the third term was ignored in expressions in Table 1: 4.4 and 5.4. Thereby their theory at higher densities underestimates and at lower densities overes- timates the simulation data. Furthermore, the theory of mean-spherical approximation (MSA) [8–10] should also Hungarian Journal of Industry and Chemistry THE INITIAL MAGNETIC SUSCEPTIBILITY OF DENSE AGGREGATED DIPOLAR FLUIDS 49 be mentioned which provides a formula for initial mag- netic susceptibility and magnetization as well, but the va- lidity of these are within the range of up to m∗ < √ 1.5. With regard to the distribution of chain aggregates in ferrofluids [11, 12], it has been found that in the ab- sence of any external magnetic field, the chain size dis- tribution is proportional to pk exp(−ε), where the chain length is denoted by k and the dimensionless energy pa- rameter ε is a function of the maximum dipole interaction energy but independent from the density, therefore, it is a constant here and the probability of bond formation be- tween two adjacent particles in a chain is p. Subsequently, the chain size distribution decreases according to an ex- ponential function because pk = exp(−k/k0), where k0 = −1/ ln p. Based on some publications [13, 14], in the case of high dipole moments this exponential expres- sion turns into a power law: g(k) ∝ k−ι, with exponent ι ≈ 2.0− 2.5. Our investigation was performed in a dipolar hard- sphere (DHS) monodisperse system with a permanent magnetic dipole moment and of fixed density. It is sup- posed that the chains are perfectly straight and parallel to the local magnetic field. Furthermore, the average dis- tance between two neighbouring particles in a chain is the same as the distance between two neighbouring par- allel chains. The particles interact with each other only by the evolved mean magnetic field and the applied external magnetic field is superimposed on this, thus, the chains influence each other only by this mean magnetic field. 2. Theory 2.1 The appearance of probability analysis in the initial magnetic susceptibility The distribution of chains was calculated with the aid of probability analysis in a zero applied magnetic field. As was mentioned in the “Introduction”, Weiss’ theory states that the effective (now “local”) magnetic field converges to zero when χL < 3/4π and non-zero values when χL ≥ 3/4π. The central and surrounding particles are un- der the influence of this local magnetic field. The chain is oriented in the same direction as the local magnetic field. Let us denote the probability of chain formation between two particles whose direction relative to each other is par- allel to the local magnetic field by p. Now using this approach the exact distribution of chain length can be calculated because the probability that the chain length ought to be equal to k follows the geometric distribution with parameter q: gk = qpk−1, (4) where q = 1−p and the “chain distribution” is denoted by gk. According to its definition the geometric distribution shows the probability that a kth particle is connected to a chain of length k − 1 thus forming a chain of length k. A geometric sequence is described in Eq. 4, where the common ratio is denoted by p and the first term by q. The sum of the terms of a geometric sequence is S∞ = ft 1−cr , thus, ∑∞ k=1 gk = q 1−p = q q = 1. It is also important to calculate the so-called “particle distribution” that implies the number of those particles which are members of the chains of length k: hk = q2kpk−1. (5) The detailed deduction of hk and the sum of hk terms are described in Appendix A. The expected value of the geometric distribution with parameter q is 1/q, thus, here the average chain length is 1/q. The number of chains is equal to the number of par- ticles divided by the average chain length: n 1/q = nq, where the number of particles is denoted by n. Until now only the local magnetic field which arises from the strength of interaction energies between neigh- bouring particles and induces spontaneous magnetization in a random direction has been discussed, thus, the total magnetization of the system of volume V is equal to zero. When an infinitesimal external magnetic field H0 is switched on, non-zero total magnetization is formed. Since H0 is parallel to M, scalar notations are used in the following. As was observed from Pshenichnikov’s model the effective magnetic field is the sum of the external H0 and secondary (4π/3)M(H0) magnetic fields. The ques- tion arises why it is legitimate to use the expression of effective magnetic field from “Pshenichnikov” (Table 1: 3.2) instead of from “Weiss” (Table 1: 2.2). The answer is because H0 modifies infinitesimally the orientation of the chains but does not align them with its own direction, thus, the average angle between the local and external or even the effective magnetic fields is not equal to zero. When calculating the initial magnetic susceptibility, a chain of length k is considered as a particle with a dipole moment km, thus, in terms of magnetization the argu- ment of the Langevin function is kmHe kBT . The Langevin function is weighted by the distribution hk, and finally the gradient of magnetization in an infinitesimal external magnetic field is calculated as χ0 = ∂ ∂H0 ∣∣∣∣ H0=0 ρm ∞∑ k=1 hkL ( kmHe kBT ) . (6) An infinitesimally weak external magnetic field can be written as χ0 = 1 + p 1− p χL ( 1 + 4π 3 χL ) , (7) where the following infinite expression is used (|p| < 1): ∞∑ k=1 pk−1k2 = 1 + p (1− p)3 . (8) The detailed derivation of the initial magnetic suscepti- bility (Eq. 7) is given in Appendix B. 46(2) pp. 47–54 (2018) 50 NAGY Figure 1: The rates of the local magnetic field as a function of the density from Eq. 9 at three different dipole moments in the absence of any external magnetic field. 2.2 The numerical calculation of the probabil- ity of chain formation “p” The main challenge of our approach is the determina- tion of p. The particles form chains because of the lo- cal magnetic field even in the absence of any applied ex- ternal magnetic field. According to Weiss’ model when χL ≥ 3/4π this local magnetic field predicts an infi- nite initial magnetic susceptibility. The problem with this model is that it assumes that the orientation of the lo- cal magnetic field is parallel with the external magnetic field. Nevertheless, Weiss’ model is applicable to predict the extent of the local magnetic field by the expression (when H0 = 0): Hl = 4π 3 ρmL ( mHl kBT ) . (9) H∗l as a function of the reduced density ρ∗ at three dif- ferent dipole moments is presented in Fig. 1. The defini- tions of the reduced quantities are H∗ = H √ σ3/kBT ; M∗ = M √ σ3/kBT ; m∗ = m/ √ σ3kBT . All particles are considered to be influenced by this local magnetic field Hl, in the absence of any external magnetic field H0 when calculating the initial magnetic susceptibility. As was mentioned before, the most ac- cepted criterion for chaining is to determine an energy level and if the dipolar energy between two given parti- cles is under this level, the particles are in a bound re- lationship. Generally [15–17], this energy level is 70-75 % of the minimum of the dipolar energy, i.e. U∗lim = −0.7 ∗ 2(m∗)2. Here the well-known dipolar energy is defined as the interaction between point dipoles: Udd ij = −m 2 r3 ij [3 (m̂i · r̂ij) (m̂j · r̂ij)− (m̂i · m̂j)] , (10) where the particles have dipole moments of strength m as well as an orientation given by unit vectors m̂i and Figure 2: The feasible location of a particle between two fixed particles. m̂j . Furthermore, the distance between the centers of the particles is denoted by rij and r̂ij = rij/rij . As is shown in Fig. 2, according to our model parti- cle j can move along the direction of the chain between the two fixed adjacent particles, namely i and the grey one, in the tube with a light blue background. Logically, the minimum distance between two particles in a hard sphere system is σ, on a reduced scale d∗min = 1, while for the maximum distance d∗max = 2 〈r∗〉 − 1, where the reduced average distance between two adjacent particles is denoted by 〈r∗〉. Obviously the maximum distance between two neigh- bouring particles could be greater than d∗max but at higher densities in particular the surrounding particles obstruct the movement of the central particle. Assuming that the distance distribution is isotropic, it is given by 〈r∗〉 = 3 √ 1/ρ∗. Taken all round to calculate p the probability of those states of particle pairs should be totalled when the dipolar interaction energy is less than or equal to the aforemen- tioned energy limit Ulim and the interval of integration in distance is [d∗min, d ∗ max], that is p = ∫ Udd≤Ulim P (θi)dθiP (θj)dθjP (φi)dφiP (φj)dφj , (11) where 0 ≤ θ < π and 0 ≤ φ < 2π are the usual spherical angles of the dipoles and the probabilities when magnetic field H (here H = Hl) is applied in general are P (θ)dθ = exp ( mH kBT cos θ ) sin θdθ ∫ π 0 exp ( mH kBT cos θ ) sin θdθ (12) and P (φ) dφ = dφ/2π. The calculation of p was performed by numerical in- tegration. Particles i and j (Fig. 2) are under the influ- ence of the local magnetic field independently from each other. For both particles, all possible values of θ, φ, and r are swept and taken into account if the dipolar interaction energy between particles i and j is less than or equal to Ulim. This is expressed by Eq. 11. For instance, when ρ∗ = 0.8 and m∗ 2 = 3.0 then H∗l = 5.154 and d∗max = 1.154435. The probability of chaining between particles i and j as a function of dis- tance is shown in Fig. 3. The requested probability p is Hungarian Journal of Industry and Chemistry THE INITIAL MAGNETIC SUSCEPTIBILITY OF DENSE AGGREGATED DIPOLAR FLUIDS 51 Figure 3: An example calculation of p. The parame- ters are rho∗ = 0.8, (m∗)2 = 3.0, H∗ l = 5.154, d∗max = 1.154435, and U∗ lim = 0.7U∗ min. The requested probability p is the average of this curve in the interval [d∗min, d ∗ max]. the average of this curve from 1 to d∗max, i.e. p = 0.3454, if U∗lim = 0.7U∗min. 3. Simulation results and discussion To determine the initial magnetic susceptibility, Monte Carlo simulations of DHS fluids were performed using a canonical NVT ensemble. Boltzmann sampling [18], pe- riodic boundary conditions and the minimum-image con- vention were applied. In order to take into account the long-range character of the dipolar interaction, the re- action field method under boundary conditions of con- duction was used. After 100,000 equilibration cycles, be- tween 1 and 10 million production cycles were conducted involving N = 512 particles. In the absence of an exter- nal magnetic field, the initial magnetic susceptibility was obtained from the following fluctuation formula: χ0 = 1 3kBTV (〈 M2 〉 0 − 〈M〉20 ) , (13) where M = ∑N i=1 mi. The exact results of the probability of chain formation from Eq. 11 applied to the local magnetic field, given by Eq. 9, are shown in Table 2. The data associated with the aforementioned dipole moments were rounded to three non-zero decimals. According to Fig. 1 at low densities the values of the local magnetic field are zero, neverthe- less, the rates of the probability of chaining are not equal to zero. The value of the energy limit was fitted to the best agreement between the simulation data and our the- ory lines. Our theoretical findings (green lines, Eq. 7 ) in terms of the initial magnetic susceptibility according to our Monte Carlo simulation data (blue dots) and the values of Ivanov’s theory (grey lines, Table 1: 4.4) are compared in Figs. 4-6. The variability is not indicated where its mag- nitude is comparable to the size of the dot. The values of dipole moments in the order m∗ = 1, m∗ = √ 2 and m∗ = √ 3 are shown in Figs. 4-6. Table 2: The probability of chaining at three different dipole moments. The applied energy limit at m∗ = 1 and m∗ = √ 2 is Ulim = 0.77Umin and Ulim = 0.71Umin at m∗ = √ 3. p ρ∗ m∗ = 1 m∗ = √ 2 m∗ = √ 3 0.1 0.000260 0.000263 0.000580 0.2 0.000426 0.000429 0.000943 0.3 0.000615 0.000617 0.00794 0.4 0.000852 0.00292 0.0250 0.5 0.00117 0.0110 0.0521 0.6 0.00164 0.0253 0.0953 0.7 0.00241 0.0515 0.170 0.8 0.0135 0.107 0.320 0.85 0.0269 0.164 0.471 0.9 0.0539 0.273 0.649 0.95 0.101 0.416 0.775 In the case of m∗ = 1 (Fig. 4), the energy limit is Ulim = 0.77Umin. A significant difference was only observed between Ivanov’s theory and the simulation data beyond a reduced density of 0.8 and the simu- lation dots were tracked by our present theory. The maximum relative deviation from the simulation data is |χ0 sim − χ0 th| /χ0 sim = 4.175%. When m∗ = √ 2, the appropriate energy criterion is Ulim = 0.77Umin as well. Up to a reduced density of 0.6 the former theory and simulation are in good agreement (Fig. 5), but in the present theory more than double the surplus is shown in the region of high density compared to Table 1: 4.4. The maximum relative deviation from the simulation data is 8.024 %. Here it is quite conceivable that the simple series ex- pansion of initial magnetic susceptibility as the summa- tion of positive integer powers of Langevin susceptibil- ity is unsatisfactory. By increasing the third coefficient in Figure 4: Initial magnetic susceptibility of DHS fluid as a function of reduced density with dipole moment m∗ = 1. Monte Carlo simulation results are denoted by symbols and the solid lines correspond to the present (Eq. 7) and Ivanov’s (Table 1: 4.4) theories. 46(2) pp. 47–54 (2018) 52 NAGY Figure 5: Initial magnetic susceptibility of DHS fluid as a function of reduced density with dipole moment m∗ =√ 2. Monte Carlo simulation results are denoted by sym- bols and the solid lines correspond to the present (Eq. 7) and Ivanov’s (Table 1: 4.4) theories. Figure 6: Initial magnetic susceptibility of DHS fluid as a function of reduced density with dipole moment m∗ =√ 3. Monte Carlo simulation results are denoted by sym- bols and the solid lines correspond to the present (Eq. 7) and Ivanov’s (Table 1: 4.4) theories. Table 1: 4.4, the initial magnetic susceptibility at lower densities is also increased. When m∗ = √ 3 the difference is even more spec- tacular between the theories (Fig. 6). At high densities the uncertainty of initial magnetic susceptibility is quite large with regard to the simulation data. The best fit curve belongs to an energy limit of 71 %, which is very close to the value from references [15–17] of 70 %. The maxi- mum relative deviation from the simulation data is 15.850 %. When m∗ = 1 and ρ∗ = 0.95, with an energy limit of 70 %, the theoretical value of initial magnetic suscep- tibility is χ0 = 1.042, and if the energy limit is 75 %, χ0 = 0.937. Both values are higher than the correspond- ing simulation data, thus, the energy limit has to be raised to 77 %. The situation is similar when m∗ = √ 2 and ρ∗ = 0.95, namely χ0 = 8.701 when the energy limit is 70 % and χ0 = 6.363 when it is 75 %. Probably at lower dipole moments and higher densi- ties the two adjacent particles cannot be considered as a chain even though the interaction energy exceeds an en- ergy limit of 70 % or 75 % for example because the av- erage kinetic energy is closer to this interaction energy than is the case when m∗ = √ 3. Therefore, the duration of chain formation is short to draw the particles together. 4. Conclusion The initial magnetic susceptibility of dipolar hard sphere fluids was described by the help of the probability vari- able p supplemented by Pshenichnikov’s well-known the- ory. The validity of the present theory is up to ρ∗ = 0.95 and at leastm∗ = √ 3. In addition to the theoretical work, Monte Carlo simulations were run to confirm our investi- gation. At higher densities, especially with higher dipole moments, the former related theories significantly under- estimate the simulation data but good results are also pro- vided by the presented theory within this range. By con- sidering the green curves and blue dots in Figs. 4-6, it is obvious that the simple quadratic or tertiary polynomial approach is outdated, therefore, the Taylor series expan- sion of (1 + p)/(1− p) contains powers even as high as infinity. 5. Appendix 5.1 Appendix A The particle distribution can be obtained by di- viding the number of particles in chains of length k by the total number of particles: hk = kgk∑ kgk = kqpk−1∑ kqpk−1 = kpk−1∑ kpk−1 = kpk−1 1 + 2p+ 3p2 + · · · = = kpk−1 (1+p+ p2 + · · · ) + (p+ p2 + p3 + · · · ) + (p2 + p3 + p4 + . . .) + · · · = = kpk−1 (1 + p+ p2 + · · · ) + (p+ p2 + p3 + · · · ) + (p2 + p3 + p4 + · · · ) + · · · = = kpk−1 1 1− p + p 1− p + p2 1− p + · · · = qkpk−1 1 + p+ p2 + · · · = qkpk−1 1 1− p = q2kpk−1. (14) Hungarian Journal of Industry and Chemistry THE INITIAL MAGNETIC SUSCEPTIBILITY OF DENSE AGGREGATED DIPOLAR FLUIDS 53 The sum of hk terms must be equal to one: ∞∑ k=1 hk = ∞∑ k=1 q2kpk−1 = q2 ( 1 + 2p+ 3p2 + . . . ) = q2 [( 1 + p+ p2 + · · · ) + ( p+ p2 + p3 · · · ) + · · · ] = = q2 [ 1 1− p + p 1− p + p2 1− p + · · · ] = q [ 1 + p+ p2 + . . . ] = q 1 1− p = 1. (15) 5.2 Appendix B χ0 = ∂ ∂H0 ∣∣∣∣ H0=0 ρm ∞∑ k=1 hkL ( kmHe kBT ) = ∂ ∂H0 ∣∣∣∣ H0=0 ρm ∞∑ k=1 q2kpk−1L km ( H0 + 4π 3 ρmL ( mH0 kBT )) kBT  = = q2ρm ∂ ∂H0 ∣∣∣∣ H0=0 ∞∑ k=1 kpk−1L kmH0 kBT + 4π 3 kρm2L ( mH0 kBT ) kBT  = = q2ρm ∞∑ k=1 kpk−1 1 3  km kBT + 4π 3 kρm2 1 3 m kBT kBT  = q2 ∞∑ k=1 k2pk−1 ( 1 3 ρm2 kBT + 1 9 4π 3 ρ2m4 k2 BT 2 ) = q2 ( χL + 4π 3 χ2 L ) ∞∑ k=1 k2pk−1 = 1 + p 1− p χL ( 1 + 4π 3 χL ) . (16) REFERENCES [1] Langevin, P.: Sur la théorie du magnétisme, Jour- nal de Physique Théorique et Appliquée, 1905 4(1), 678–693 DOI: 10.1051/jphystap:019050040067800 [2] Weiss, P.: L’hypothése du champ moléculaire et la propriété ferromagnétique, Journal de Physique Théorique et Appliquée, 1907 6(1), 661–690 DOI: 10.1051/jphystap:019070060066100 [3] Pshenichnikov, A. F.; Mekhonoshin, V. 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J.: Computer Simulation of Liquids (Clarendon Press, Oxford) 1987 ISBN: 978-0198556459 Hungarian Journal of Industry and Chemistry https://doi.org/10.1103/PhysRevLett.71.2729 https://doi.org/10.1103/PhysRevE.49.5131 https://doi.org/10.1103/PhysRevE.49.5131 https://doi.org/10.1103/PhysRevE.51.5962 Introduction Theory The appearance of probability analysis in the initial magnetic susceptibility The numerical calculation of the probability of chain formation ``p'' Simulation results and discussion Conclusion Appendix Appendix A Appendix B