Nghiên cứu một số tính chất cơ bản của hệ hạt nano từ bằng phương pháp mô phỏng trên máy tính

ĐẠI HỌC QUỐC GIA HÀ NỘI ĐẠI HỌC QUỐC GIA TP HỒ CHÍ MINH TRƯỜNG ĐẠI HỌC CƠNG NGHỆ PTN CÔNG NGHỆ NANO Trần Nguyên Lân NGHIÊN CỨU MỘT SỐ TÍNH CHẤT CƠ BẢN CỦA HỆ HẠT NANO TỪ BẰNG PHƯƠNG PHÁP MƠ PHỎNG TRÊN MÁY TÍNH Chun ngành: Vật liệu Linh kiện Nanơ (Chun ngành đào tạo thí điểm) LUẬN VĂN THẠC SĨ Người hướng dẫn khoa học: PGS TS Trần Hồng Hải Thành phố Hồ Chí Minh - 2010 Lời cảm ơn Đầu tiên cho đợc cảm ơn ĐH Công Nghệ - ĐHQG H Nội v PTN Công Nghệ Nano - ĐHQG Tp Hồ Chí Minh đà tạo điều kiện để đợc học tập v hon thnh luận văn ny Tôi xin đợc chân thnh cảm ơn Thầy Cô giáo đà truyền đạt cho kiến thức quý báu suốt hai năm học qua Đặc biệt cho đợc tỏ lòng biết ơn sâu sắc đến PGS TS Trần Hong Hải Thầy không trực tiếp hớng dẫn m tạo nhiều hội để đợc nghiên cứu khoa học Tôi xin đợc ghi nhớ đến Thầy nh l ngời Thầy đờng nghiên cứu khoa học Cho đợc gởi lời cảm ơn ®Õn TS Ngun M¹nh Tn, ViƯn phã ViƯn VËt lý Tp Hồ Chí Minh kiêm Trởng phòng Vật liệu vμ vËt liƯu cÊu tróc nano, ViƯn VËt lý Tp Hồ Chí Minh Thầy đà quan tâm v tạo điều kiện tốt để hon thnh luận văn ny Cuối cùng, xin chân thnh cảm ơn đồng nghiệp, bạn bè v gia đình đà động viên, giúp đỡ suốt trình học tập nh thực luận văn Tp Hồ Chí Minh, ngy 25 tháng 02 năm 2010 Tác giả luận văn Trần Nguyên Lân Nội dung Trang Bảng chữ viết tắt Mở đầu Chơng 1- Tổng quan 1.1 Mô hình hóa v mô khoa học vật liệu 8 1.1.1 ý tởng việc mô hình hoá v mô 1.1.2 Xây dựng mô hình toán học từ tranh tợng 1.1.3 Một số lu đồ mô tả trình mô hình hóa v mô 1.1.4 Phân loại mô hình 1.2 Phơng pháp Monte Carlo 1.2.1 Giới thiệu 1.2.2 Phơng pháp Monte Carlo lấy mẫu đơn giản 1.2.3 Phơng pháp Monte Carlo lấy mẫu quan trọng - thuật toán Metropolis 1.3 Khái niệm v tính chất hạt nano từ 1.3.1 Sự phân chia domain vật liệu sắt từ 1.3.2 Hạt đơn domain 1.3.3 Sự tõ hãa cđa h¹t nano tõ 1.3.4 TÝnh chÊt cđa hạt nano từ nhiệt độ hữu hạn 1.3.5 Một số phép đo xác định tính chất hệ hạt nano từ 1.3.6 Sự tơng tác hạt nano tõ 11 11 11 12 12 15 15 16 17 19 24 25 1.4 øng dơng cđa h¹t nano tõ y sinh häc 1.4.1 T¸ch tõ 1.4.2 Trun dÉn thuốc 1.4.3 Nâng thân nhiệt cục 1.4.4 Tăng tính tơng phản cho MRI 27 27 28 29 30 Chơng 2- mô hình v mô 2.1 Năng lợng hệ hạt nano từ 2.2 Phơng pháp mô 2.2.1 Hệ tọa độ 2.2.2 Tính toán lợng lỡng cực 2.2.3 Thuật toán mô 2.2.4 Lựa chọn thông số 33 33 35 35 35 36 38 Ch−¬ng 3- kÕt v thảo luận 39 3.1 Nhiệt độ khóa hệ hạt nano từ 3.1.1 Độ từ hóa trình zero-field-cooled 3.1.2 Sự phân bố ro hệ hạt nano từ 3.1.3 Sự tán sắc mẫu 3.1.4 Tơng tác tĩnh từ hạt 3.1.5 Sự phơ thc cđa ®Ønh ZFC vμo tõ tr−êng ngoμi 3.2 Chu trình từ trễ hệ hạt nano từ 39 39 40 41 42 43 47 3.2.1 ¶nh h−ëng cđa nhiệt độ 47 3.2.2 Sự tán sắc mẫu 48 3.2.3 Tơng tác tĩnh từ hạt 3.3 Tính chất tập hợp hệ hạt nano từ 49 51 Kết luận v hớng nghiên cứu tơng lai Ti liệu tham khảo 53 55 Phụ lục: Các bi báo liên quan đến luận văn 59 Bảng chữ viết tắt STT Chữ viết tắt DDI FC MCM MNPs SPM ZFC NghÜa tiÕng Anh Dipolar Interaction Field-Cooled Monte Carlo method MagneticNanoparticles Superparamagnetism Zero-Field-Cooled Nghĩa tiếng Việt Tơng tác lỡng cực Lm lạnh có từ trờng Phơng pháp Monte Carlo Những hạt nano từ Siêu thuận từ Lm lạnh không từ trờng Mở đầu Cïng víi xu h−íng ph¸t triĨn chung cđa khoa häc, ngμnh vËt liƯu ngμy cμng gãp phÇn to lín vμo nhiều mặt đời sống ngời Không chế tạo công cụ đại giúp đỡ ngời m mở khả việc trị bệnh nh bảo vệ môi trờng Vì việc nghiên cứu khoa học vật liệu, lý thut vμ thùc nghiƯm, mang tÝnh chÊt cÊp b¸ch Mét số vật liệu m công nghệ tiên tiến ®em l¹i ®ã lμ vËt liƯu tõ cÊu tróc nano bao gåm h¹t nano tõ vμ mμng máng tõ ThËt ra, vật liệu từ đà đợc ứng dụng từ sím vμ hiƯn vËt liƯu tõ cÊu tróc nano høa hĐn nh÷ng øng dơng réng r·i rÊt nhiỊu lĩnh vực Những hệ hạt nano từ bao gồm hạt nano từ đợc phân bố môi trờng nh chất rắn (granular solids) chất lỏng (magnetic fluid) Các môi trờng ny l cách điện không cách điện, tinh thể vô định hình v có vi pha khác cđa vËt liƯu Theo ®ã, tÝnh chÊt vËt lý cđa hệ hạt nano từ có khả đợc điều chỉnh để tùy vo mục đích ứng dụng nghiên cứu ë n−íc ta hiƯn nay, viƯc nghiªn cøu vËt liƯu từ cấu trúc nano đợc thực số nhóm Hầu hết nghiên cứu trọng vo vấn ®Ị øng dơng cđa h¹t nano tõ, ®ã tính chất cha đợc tìm hiểu cách sâu sắc Do tiến hnh hnh nghiên cứu tính chất hệ hạt nano từ phơng pháp mô máy tính Trên giới, nhiều năm gần đây, nh khoa học đà nỗ lực nghiên cứu nhằm đa lý thuyết tổng quát cho hệ hạt nano từ bao gồm ảnh hởng tơng tác v tán sắc Những mô hình ny dừng lại trờng hợp mẫu loÃng [21,43] tơng tác yếu [1,21,24] Gần đây, dựa phơng pháp mô Monte Carlo, nhiều nghiên cứu đà thăng giáng nhiệt hệ hạt nano từ [10,15], ảnh hởng tơng tác mạnh lên tính chất hệ hạt nano từ [3,10,12,16,22,23,27,40,41,42,45] Tuy nhiên, số vấn đề cha đợc sáng tỏ nh l phụ thc cđa nhiƯt ®é khãa vμo tr−êng thÊp, cịng nh− ảnh hởng liên kết tán sắc v tơng tác tĩnh từ hạt lên tính chất từ trễ hệ Đây l lý tập trung nghiên cứu hai vấn đề ny Bi luận văn gồm bốn chơng, (i) chơng một, giới thiệu sơ lợc ngnh khoa học vật liệu tính toán, non trỴ so víi lý thut vμ thùc nghiƯm nh−ng tÝnh toán số góp phần không nhỏ vo phát triển chung khoa học vật liệu, đồng thời chơng ny đề cập đến phơng pháp Monte Carlo, phơng pháp hữu hiệu v thờng đợc sử dơng viƯc nghiªn cøu hƯ nano tõ Mét sè vấn đề v ứng dụng hệ hạt nano từ y sinh học đợc tóm tắt chơng ny (ii) Chơng hai, đa mô hình chi tiết v trình tính toán, chơng ny quan trọng ảnh hởng trùc tiÕp ®Õn ý nghÜa vËt lý cịng nh− kÕt mô (iii) Trong chơng ba, thảo luận kết đà thu đợc, kết mô đợc so sánh với kết thực nghiệm nh tiên đoán tính chất hệ hạt nano từ Tất kết ny đợc giải thích rõ rng (iv) Ci cïng, mét sè vÊn ®Ị chÝnh u cđa ln văn nh dự định nghiên cứu tơng lai đợc tóm tắt phần kết luận Chơng - TổNG QUAN 1.1 Mô Hình hóa V Mô Pháng Trong Khoa Häc VËt LiƯu Trong phÇn chóng ta sơ lợc số vấn đề khoa häc vËt liƯu tÝnh to¸n Nh− chóng ta thÊy hình 1.1, mô máy tính l mắc xÝch kh«ng thĨ thiÕu khoa häc vËt liƯu hiƯn đại Thực nghiệm Mô máy tính Lý thuyết Hình 1.1 Sự liên hệ thực nghiệm, lý thuyết v mô máy tính khoa học vật liệu đại 1.1.1 ý tng c bn ca việc mô hình hoá v mô Mc ích chung ca khoa hc để t×m hiểu điều khiển giới vật cht Tuy nhiên, có rt nhiu không th quan sát mt cách y hoc không th hiu thấu điều khiển kh«ng cã trừu tượng hãa khoa học Trừu tượng hãa khoa học cã nghÜa lμ thay phần giới thực xem xét bng mt mô hình Quá trình thit k nhng mô hình c xem nh l nguyên lý tng quát v c bn nht ca vic mô phng Nó mô t phng pháp khoa hc ca vic a “bắt chước” đơn giản với hệ thực mμ bảo tồn đặc tÝnh quan trọng hệ thc ó Nói cách khác, mt mô hình mô t hệ thực c¸ch sử dụng cấu tróc tng t nhng n gin hn Nhng mô hình tru tượng cã thể xem điểm bắt u c bn ca lý thuyt Tuy nhiên, cần phi nhấn mạnh kh«ng cã tồn thống hon ton gia nhng mô hình v h thc Hay nói cách khác mi mô hình không th bao gm c¸ch đầy đủ tÝnh chất hệ thực Và điều đóng khoa học vật liệu, v× nã bao gồm nhiều kích thc v c ch khác 1.1.2 Xây dựng mô hình toán hc từ tranh tợng Trớc đa phơng pháp giải số, nh khoa học tính toán phải đa mô hình toán học vừa phù hợp với tính chất đà quan sát từ thực nghiệm, vừa có khả giải đợc Một mô hình toán học sau đợc xây dựng áp dụng cho trờng hợp với thông số v điều kiện khác V để xây dựng mô hình toán học từ tranh tợng cần phải thực bớc xác định (hoặc lựa chọn) vấn đề sau: Biến độc lập l biến đợc lựa chọn tự do: thời gian v không gian Biến trạng thái l hm biến độc lập: nhiệt độ, nồng độ, độ dịch chuyển, Phơng trình động học l phơng trình mô tả thay đổi tọa độ chất điểm m không xem xét đến ảnh hởng lực tác dụng vo nó: phơng trình tính sức căng, phơng trình mô tả quay, phơng trình chuyển động, Phơng trình trạng thái l phơng trình độc lập đờng v mô tả trạng thái thật vật liệu thông qua biến trạng thái Những phơng trình trạng thái vi cấu trúc thờng xác định tính chất vật liệu tơng ứng với thay đổi bên bên ngoi giá trị biến trạng thái Tức l phơng trình trạng thái đặc trng cho vật liệu Phơng trình tiến triển cấu trúc l phơng trình phụ thuộc đờng v mô tả thay đổi vi cấu trúc thông qua thay đổi biến trạng thái (ngợc với phơng trình trạng thái) Những thông số vật lý l thông số cho phép xác đinh biến trạng thái Những thông số ny phải có ý nghĩa vật lý v tuân theo thực nghiệm lý thuyết Điều kiện biên v điều kiện đầu l giá trị để giới hạn bi toán Thuật giải số phơng pháp phân tích số dùng để giải phơng trình Phải l thuật toán tối u vμ cho kÕt qu¶ xÊp xØ tèt nhÊt 1.1.3 Mét số lu đồ mô tả trình mô hình hóa v mô Nh đà nói, mục đích khoa học vật liệu tính toán l để tìm hiểu giới vật chất nhờ vo tính toán Theo đó, cã rÊt nhiỊu l−u ®å tht tãan ®−a nh»m mô tả cách tổng quát ý nghĩa việc mô [7]: mô hình Asby (1992), mô hình Preziosi (1995), mô hình Bunge (1997), mô hình biến trạng thát tổng quát (1998) Trong số đó, ®Ị cËp ®Õn hai l−u ®å th«ng dơng nhÊt nh− bên dới a, Lu đồ Bellomo and Preziosi (1995) A: Quan sát v đo đạc tợng hệ vật lý B1: Phạm vi biến độc lập B2: Lựa chọn biến trạng thái B3: Định nghĩa thông số C: Xây dựng mô hình toán học D: Phân tích mô hình E: Xem xét tính hợp lệ mô 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and Rivas J (2008), “Interplay between the magnetic field and the dipolar interaction on a magnetic nanoparticle system: a Monte Carlo study”, Journal of Non-Crystalline Solids 354, pp 5224-5226 42 Sharma R., Pratima C., Lamba S and Annapoorni S (2005), “Interaction effects on magnetic oxide nanoparticle systems”, Pramana: Journal of Physics 65, pp 739-743 43 Suzuki S., Fullem S I., and Suzuki I S (2009), “Scaling form of zerofield-cooled and field-cooled susceptibility in superparamagnet”, arXiv: Condensed Matter/ 0903.0537v1(9) 44 Walton D (2006), “A theory for spin glass phenomena in interacting nanoparticle systems”, arXiv: Condensed Matter, pp 0608696v1(9) 45 Yang Y., Shen S., Ye Q., Lin L., Huang Z (2006), “The roles of the exchange and dipolar coupling on the magnetoresistance for the nanoparticle arrays”, Journal of Magnetism and Magnetic Materials 303, pe312 - e314 46 Zheng R K., Gu H., Xu B., Zhang X X (2006), “The origin of the nonmonotonic field dependence of the blocking temperature in magnetic nanoparticles”, Journal of Physics: Condensed Matter 18, 5905 (2006) 58 Phơ lơc: C¸c bμi b¸o liên quan đến luận văn A Tran Nguyen Lan, Tran Hoang Hai (2010), “Monte Carlo Simulation of Magnetic Nanoparticle Systems”, accepted to be published on Computational Material Science B Tran Nguyen Lan, Tran Hoang Hai (2010), “Monte Carlo Study of Collective Behavior of Magnetic Nanoparticle System”, Asia - Europe Physics Summit (AEPS), ID 60 (accepted) C Tran Nguyen Lan, Tran Hoang Hai (2010), “Magnetic Properties of Interacting Nanoparticle Systems”, submitted to Journal of Magnetism and Magnetic Materials 59 MONTE CARLO SIMULATION OF MAGNETIC NANOPARTICLE SYSTEMS Tran Nguyen Lan, Tran Hoang Hai Ho Chi Minh City Institute of Physics, VAST 01 Mac Dinh Chi Street, District 01, Ho Chi Minh City, Vietnam E-mail address: trannguyenlan@gmail.com Abstract: We use the Monte Carlo method to study the influence of the low field on the blocking temperature of magnetic nanoparticle systems at different sample conditions, namely the poly-dispersity and the concentration We find the increase of the blocking temperature at the low field in the dilute sample, and the slow decrease in the concentrated sample We extract the barrier distribution to explain and compare the magnetic properties of two above samples We also provide short discussions about the origin of the increase of the peak temperature at low field and the influence of the interaction on the blocking state of particles in the high field Key words: magnetic nanoparticles, Monte Carlo simulation, peak temperature, poly-dispersity, concentration INTRODUCTION Magnetic nanoparticles have been applied more and more in technologies, then the experimental and theoretical investigation is very important and essential The number of papers performing the simulation of magnetic nanoparticle systems has increased fast during this decade These papers are considered as the thermal effects on magnetic properties [1,2], or studied interactions between nanoparticles [3-8] However, the influence of the low field on the maximum temperature of the zero-field-cooled (ZFC) magnetization has been not well understood and this is our motivation for doing this work Properties of magnetic nanoparticles are described by the supper-paramagnetic theory [9] In this theory, the uniaxial anisotropy leads to two equilibrium states of the magnetic moment of particles The relaxation time τ is essentially the average time to reverse a particle’s magnetization from one of equilibrium states to the other through the relation ⎛ ΔE ⎞ τ = τ exp ⎜ a ⎟ (1) ⎝ k BT ⎠ In here, ∆Ea is the barrier energy of each particle, and τ0 is the characteristic constant time (of the order of 10-10 second) In presence of the external field, the barrier energy is changed and has form [9] ∆Ea = KuV (1 – h)α (2) Where V as the particle volume, Ku as the uniaxial anisotropy constant, α as the parameter determines the dependence of the barrier energy on the field, in every case α ≤ [9,18], and h = H/Ha as the reduced field, Ha = 2KuV/Ms as the anisotropy field, with Ms as saturated magnetization The temperature determined from Eq.1 with τ set equal to the experimental observation time τm defines the blocking temperature TB, which separates the two regimes However, the time τm is defined by the requirements, so the definition of TB is not unique, but depends on the type of experiment In the present paper, we consider a system of single-domain particles having a lognormal distribution of sizes f (σ , D) = exp ⎡⎣ − ln ( D / Dm ) / ( 2σ ) ⎤⎦ / 2πσ D , with σ is the width of distribution and Dm is the median size ( ) MODEL AND SIMULATION In such assemblage of N particles, each particle has a constant absolute value of its total magnetic moment |μi|=Ms Vi In the presence of the dipolar interaction, the energy of each particle is [9] N ⎛ μ μ ⎛ μ i ni ⎞ ( μi rij )( μ j rij ) ⎞⎟ i j (i ) − μi H + g ∑ ⎜ − E = − K uVi ⎜ (3) ⎟ ⎜ μ ⎟ ⎟ rij5 j ≠ i ⎜ rij ⎝ i ⎠ ⎝ ⎠ The first term in Eq.3 is the anisotropy energy, ni is the direction of the anisotropy axis, |ni| = The second term is the Zeeman energy, H is the external field The last time is the dipolar energy between two particles i and j separated by rij, and constant g = μ0/4π The effect of the dipolar interaction can be more easily seen by defining the dipolar local field acting on each particle i [12] N ∂U ij N ⎛μ rij ( μ j rij ) ⎞ dipol j ⎟ = −g∑ ⎜ − H idipol = −∑ (4) ⎟ ∂μ i rij5 j ≠i j ≠ i ⎜ rij ⎝ ⎠ i Then, the dipolar energy of the particle i can rewrite in the simple form U dipol = μ i H idipol And the energy of the particle i as 60 ⎛ μ n ⎞ E (i ) = − K uVi ⎜ i i ⎟ − μ i H ieff (5) ⎜ μ ⎟ ⎝ i ⎠ Now, the system can be thought as an ensemble of the non-interacting particles feeling an effective field that is sum of an external and a local field H ieff = H + H idipol We assume that the external field H is applied along the z-axis of the particle system, and easy axis of particles aligned at an angle ψ with the field and a magnetization orientated (in spherical coordinates) along (θ,φ) At the beginning of each simulation, an assemblage of 64 particles was generated and random values of φ, and θ drawn from a uniform distribution θ ∈ [ 0, 2π ] , and ϕ ∈ [ 0, 2π ] The ψ value of each particle is constant throughout the simulations and the variation of φ and θ is of interest The Metropolis procedure [10] is utilized to find the free energy minima of particles in given field conditions Therefore, each Monte Carlo step consists of the following steps: (i) using Eq (5) the energy of each particle is determined base on the applied field and the current values of θ,φ, this value is E (ii) A new orientation of the magnetization is selected at random within even angles (dφ and dθ, these values are determined randomly from [-ηmax, ηmax]) (iii) The energy Etrial is calculated for the particle along with the new values of the magnetization (iv) The difference ∆E is calculated for the two possible orientations of the magnetization, ∆E = Etrial – E (v) The magnetization of the particle is moved to the new orientation with the probability min[1, exp(-∆E/KBT)] In this paper, we use the parameters reported by Josson et al.[11] for the γ-Fe2O3 nanoparticle, Ku = 1.9x105 erg/cm3, and the mean diameter Dm = 7.5 nm RESULTS AND DISCUSSION Fig.1a shows ZFC curves of the dilute sample at the different values of the width distribution, arrows signal the peak of each ZFC curve Garcia-Otero et al [3] asserted that the blocking temperature is not dependent on the poly-dispersity of the sample even the interactions can be negligible, however, our results show that the ZFC-peaks, Tp, shift extremely rapidly towards higher temperatures when the distribution width increases This means that in the dilute sample the magnetic properties of the particle system strongly depend on the poly-dispersity, or the anisotropy distribution In Fig.2b, we show the influence of the sample concentration on the ZFC-peak With increasing the interacting strength, the blocking temperature increases [3-6] The Monte Carlo simulation does not include the relaxation time, so the shape of ZFC curves is not completely realistic However, the present variations of the ZFC-peaks with different conditions are good in conformity with the reported results [15,16] a) b) Fig 1: ZFC curves at different values of (a) the distribution width, and (b) the sample concentration It is very difficult to build a numerical model that finds the barrier distribution However, we recognize that the energy difference ∆E in the translation probability is always equal to one of the actual energy barriers of the system [12] Therefore, we can extract the barrier distribution by using the Monte Carlo method to sample the individual energy barriers of all the particles Fig.2 performs the barrier distribution of the magnetic nanoparticle system in the dilute sample (Fig 2a) and the dense sample (Fig 2b) Therefore, we can easily see that in the dilute sample the energy barrier distribution is similar with the size distribution, namely the lognormal distribution This means that the peak temperature in the dilute sample is simply the average blocking temperature of all the particles [16,17] However, in the concentrated sample the barrier distribution changes very strongly as the interacting strength increases [12] At the same time, the barrier distribution shifts to the larger energy barrier responding to the increase of the blocking temperature This can be explained that at the high concentration, the particles associate closely together, so the rotation of a particle moment can excite the rotation of others, in other words, the assembly behaves a collective manner [9] These results provide the way, which cannot infer from numerical analyses [12], to explain the behavior of the Tp vs h curvature at the low field [15,16] 61 The non-monotonic field dependence of the peak temperature in the dilute sample was found in the many various magnetic nanoparticle systems, such as Fe3O4 [13,16], Ferritin [14,15], γ-Fe2O3 [15],and Co, FePt [16], and some authors provide numerical analyses to explain this problem They are based on different ways, such as the Kramer theory of the escape of particles [17], or the non-linearity of the Langevin function [18] These results ensure that the size distribution and the anisotropy are main causes of the non-monotonousness, however, as we discussed, they was still explained unclearly the influence of the low field on the barrier distribution finding by experiment [15,16] a) b) Fig 2: The barrier distributions along with the variation of (a) the field in the dilute sample, and (b) the concentration in the dense sample In the present paper, we have used the Monte Carlo method to consider this non-monotonousness, and similar results with the experiment are found in Fig 3a When the reduced field value is smaller than 0.3, the peak temperature increases along with the increase of the reduced field, and then it continuously decreases to increase the reduced field The increase of the peak temperature in the low field expresses strongly at the large-σ As saw in Fig 2a, the distribution of the energy barrier is sensitive to the applied field, and they is broadened as the field increases at the low values Sappey et al [15] suggested that the effect of low fields on the energy barrier distribution could be due to disorder of orientations, or the defects of each particle Zheng et al [16] explained that this non-monotonous was due to combining the size distribution and the slow decrease of the magnetization (or non-Curie’s law dependence of magnetization) above the blocking temperature in the field Although these explanations satisfy with the present model (the unixial anisotropy model), they are simple and not sufficient Recently, Perez et al [20] showed the effect of the anisotropy ratio (core-shell anisotropy) on the energy barrier distribution of the dilute systems and with increasing the surface anisotropy the energy barrier distribution is enlarged even when the size distribution is quite narrow Basing on this, we can imagine that under the influence of the low field, the contribution of the surface anisotropy may become dominant in the comparison with the core anisotropy, so the energy barrier distribution of the system is enlarged At the high field, the Zeeman energy exceeds the anisotropy energy of each particle Therefore, atomic moments of each particle containing the atomic surface and the atomic core orient along with the field direction, this means that the disorder at the particle surface disappears In other words, the contribution of the surface anisotropy is not dominant, so the barrier distribution becomes narrow, and the peak temperature decreases In our opinion, the non-monotonic behavior of the Tp vs h curvature expresses strongly in the independent particle systems possessing the strong surface anisotropy We can use the atomistic simulation methods, such as the ab initio calculation, to predict the influence of the applied field on the particle surface structure However, this is beyond the scope of this paper, and we plan to consider it in the future work a) b) Fig.3: The applied field dependence of peak temperature at difference of (a) the distribution width in the dilute sample, and (b) the concentration in the dense sample, the dot line indicates the value of the anisotropy field Ha 62 To give a sufficient comparison, we considered the influence of the dipolar interaction on the behavior of the ZFC-peak vs applied field curve A recent numerical study was based on the Gittlmen-Abeles-Bozowski model [19] to perform the change of the shape of this curve for the weak interaction, but it was complex and at the high concentrations, the numerical analyses is impossible [9] We need to remember that the dipolar energy i) of the particle i will make a dipolar field H (dipol acting on it As we saw in Fig 3b, the curvature change from the non-monotonousness (bell-like shape [17,19]) to monotonousness, and at the very strong interaction, the curvature becomes flatter (plateau-like shape [6]) As in Fig.2b, when the interacting strength increases, the relation between the size distribution and the barrier distribution disappears, therefore, the non-monotonousness will change to the monotonousness However, if the sample is very dense, the dipolar field seems to equal the small values of the applied field, then the energy barrier of each particle slowly decreases along with the applied field, this means that the curvature is less sloping Because the high dipolar field will remain the barrier, the blocking temperature exists even the applied field exceeds the anisotropy field Ha This scenario also found by Serantes et al [6], however, the behaviors at the low field are rather different from our results We find that the curvatures are separated clearly at the low field, because at this region the local field plays dominant role, then the blocking temperature increases along with the increase of the concentration CONCLUSION In summary, we present some important properties of the single-domain system by simulating computer In this paper, a comparison between the non-interacting and the interacting samples are considered The variations of the blocking temperature along with the distribution width and the concentration are good in conformity with the experiments To can see clearly the difference, we extracted the energy barrier distribution, and we found that, in the dilute samples, the main effect on the magnetic properties is the polydispersity, or the anisotropy distribution On the contrary, the concentration plays dominant role in the dense samples Finally, we think that to understand clearly the increase of the blocking temperature at low field, we may base the recent results performing the anisotropy surface of magnetic nanoparticle system [21], and we also agree with the argument of Serantes et al [6] about improving the expression that describes the relation between the blocking temperature and the field [9] Recently, Denardin et al [22] provided the random anisotropy model (RAM) to consider the effect of interactions on the blocking temperature However, this model seems to fit experiments, and it has not yet presented a general view about this problem Acknowledgments: This work done by the financial sponsor of the fund support of NAFOSTED base research program, Vietnam Academy of Science and Technology (VAST) and the Department of Science and Technology of Dong Nai province References 10 11 12 13 14 15 16 17 18 19 D A Dimitrov, and G M Wysin, Phys Rev B 54, 9237 (1996) D Heslop, Stud Geophys Geod 49, 163(2005) J Garcia-Otero, M Porto, J Rivas, and A Bunde, Phys Rev Lett 84, 167 (2000) R Sharma, C Pratima, S Lamba and S Annapoorni, Pramana : J Phys 65, 739 (2005) D Baldomir, J Rivas, D Serantes, M Pereiro, J.E Arias, 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very long, and it is promised potential in spintronic applications In addition, some recent results found that the collective beahvior at the high temperature can open a new applied way in the biomedicine However, our understanding about the origin as well as the difference of above behaviors is quite limitted In this paper, we use the Monte Carlo simulation to show collective behavior at the low and high temperature of colloidal magnetic nanoparticle systems possessing the size distribution and the random anisotropy The barrier distribution is extracted to consider effect of interparticle interaction and polydispersity on magnetic properties of the sample Therefore, the role of the interaction and the polydispersity are showed clearly At the low temperature, the dipolar interaction makes a decrease of the corecive field While the blocking temperature increases along with the increase of the concentration Beside, the dipolar interaction remains the blocking temperature even the applied field exceeds the anisotropy field A comparison is also disscussed to show the applied potential of magnetic nanoparticle systems 65 MAGNETIC PROPERTIES OF INTERACTING NANOPARTICLE SYSTEMS Tran Nguyen Lan, Tran Hoang Hai Ho Chi Minh City Institute of Physics, VAST 01 Mac Dinh Chi Street, District 01, Ho Chi Minh City, Vietnam Tel: +84 22 101094, E-mail: trannguyenlan@gmail.com Abstract: In this paper, the role of the dipolar interaction on the magnetic nanoparticle systems is study by Monte Carlo method A poly-dispersity sample containing the uniaxial anisotropy and the random orientation is used to investigate The dipolar interaction makes a decrease of the coercive field at the low temperature and an increase of blocking temperature With increasing the dipolar interaction, the role of the size distribution becomes dim The coercive field can be destroyed by the high concentration even the sample has the strong polydispersity Beside, the dipolar interaction remains the energy barrier, so the field dependence of the blocking temperature changes from the non-monotonous to the plateau-like Key words: magnetic nanoparticle, dipolar interaction, coercive field, peak temperature, Monte Carlo simulation Introduction The collective behavior at the low temperature has been investigated very long (see [1] and references in there), however, some recent results showed that this behavior at the high temperature open a new applied way in the biomedicine [2,3,4] In this paper, the effects as well as the difference of the dipolar interaction at the low temperature and the high temperature are investigated Properties of magnetic nanoparticles are described by the Néel-Brown theory [5] In this theory, the uniaxial anisotropy leads to two equilibrium states of the magnetic moment of particles The relaxation time τ is essentially the average time to reverse a particle’s magnetization from one of equilibrium states to the other through the relation ⎛ ΔE ⎞ τ = τ exp ⎜ a ⎟ (1) ⎝ k BT ⎠ In here, ∆Ea is the barrier energy of each particle, and τ0 is the characteristic constant time (of the order of 10-10 second) In presence of the external field, the barrier energy is changed and has form [6] ∆Ea = KuV (1 – h) α (2) Where V as the particle volume, Ku as the uniaxial anisotropy constant, α as the parameter determines the dependence of the barrier energy on the field, in every case α ≤ [6], and h = H/Ha as the reduced field, Ha = 2KuV/Ms as the anisotropy field, with Ms as saturated magnetization The temperature determined from Eq.1 with τ set equal to the experimental observation time τm defines the blocking temperature TB, which separates the two regimes However, the time τm is defined by the requirements, so the definition of TB is not unique, but depends on the type of experiment An ensemble of nanoparticles is denoted as SPM, when the magnetic interactions between the particles are sufficiently small Then the magnetic behavior of the ensemble is essentially given by the configurational average over a set of independent particles More generally, one can denote the magnetic behavior as SPM in the sense of a thermodynamic phase No collective inter-particle order exists, while the intra-particle spin structure is FM ordered In the case of small concentrations of particles, only SPM behavior is observed However, for increasing concentrations the role of magnetic interactions becomes non-negligible Then the system behaves the collective states as we will investigate below Model and simulation In such assemblage of N particles, each particle has a constant absolute value of its total magnetic moment |μi| = Ms Vi In the presence of the dipolar interaction, the energy of each particle is [6] N ⎛ μ μ ⎛ μ i ni ⎞ ( μi rij )( μ j rij ) ⎞⎟ i j (i ) E = − K uVi ⎜ (3) − μi H + g ∑ ⎜ − ⎟ ⎜ μ ⎟ ⎟ rij5 j ≠ i ⎜ rij ⎝ i ⎠ ⎝ ⎠ The first term in Eq.3 is the anisotropy energy, ni is the direction of the anisotropy axis, |ni| = The second term is the Zeeman energy, H is the external field The last time is the dipolar energy between two particles i and j separated by rij, and constant g = μ0/4π The effect of the dipolar interaction can be more easily seen by defining the dipolar local field acting on each particle i 66 N ⎛μ rij ( μ j rij ) ⎞ j ⎟ = −g∑ ⎜ − ⎟ rij5 ∂μ i j ≠i j ≠ i ⎜ rij ⎝ ⎠ i Then, the dipolar energy of the particle i can rewrite in the simple form U dipol = μ i H idipol N H idipol = −∑ ij ∂U dipol (4) And the energy of the particle i as ⎛ μ n ⎞ E = − K uVi ⎜ i i ⎟ − μ i H ieff (5) ⎜ μ ⎟ ⎝ i ⎠ Now, the system can be thought as an ensemble of the non-interacting particles feeling an effective field that is sum of an external and a local field H ieff = H + H idipol (i ) We assume that the external field H is applied along the z-axis of the particle system, and easy axis of particles aligned at an angle ψ with the field and a magnetization orientated (in spherical coordinates) along (θ,φ) At the beginning of each simulation, an assemblage of 64 particles was generated and random values of φ, and θ drawn from a uniform distribution θ ∈ [ 0, 2π ] , and ϕ ∈ [ 0, 2π ] The ψ value of each particle is constant throughout the simulations and the variation of φ and θ is of interest The Metropolis procedure [7] is utilized to find the free energy minima of particles in given field conditions Therefore, each Monte Carlo step consists of the following steps: (i) using Eq (5) the energy of each particle is determined base on the applied field and the current values of θ,φ, this value is E (ii) A new orientation of the magnetization is selected at random within even angles (dφ and dθ, these values are determined randomly from [-ηmax, ηmax]) (iii) The energy Etrial is calculated for the particle along with the new values of the magnetization (iv) The difference ∆E is calculated for the two possible orientations of the magnetization, ∆E = Etrial – E (v) The magnetization of the particle is moved to the new orientation with the probability [1, exp (-∆E/KBT)] In the present paper, we consider a system of single-domain particles having a lognormal distribution of sizes f (σ , D) = exp ⎡⎣ − ln ( D / Dm ) / ( 2σ ) ⎤⎦ / 2πσ D , with σ is the width of distribution and Dm is the median size We use the parameters similarly to the previous work [8] ( ) Results and discussion Firstly, we will consider the influence of the dipolar interaction and the size distribution on the barrier distribution of the sample Fig.1a shows the variation of the barrier distribution at the different concentrations In the dilute sample, the barrier distribution responds to the size distribution, namely log-normal distribution, however, with increasing the concentration, the barrier distribution shifts to the large-energy and the relation disappears A similar result also found by Iglesias et al [9], with increasing the interacting strength, the distribution peaks appears at the lower energy and the high energy tails become longer This result has an importance significant to show the role of the size distribution and the dipolar interaction The size distribution plays dominant role in the non-interacting sample, and with increasing the concentration it is replaced by the dipolar interaction Fig.1 The barrier distribution of magnetic nanoparticle Fig performs the effect of the dipolar interaction on magnetic properties of sample, the hysteresis and the zero - field - cooled (ZFC) At the low temperature, the hysteresis becomes narrow as the concentration raises This phenomenon, which is found in many experiments, is due to the collective behavior arising from the dipolar interactions among individual particles Knobel et al [10] suggested that this can be replaced by the framework of independent “particle clusters” after a renormalization that takes into account the correlation length in the random anisotropy model (RAM) However, this assumption only suits as the correlation length is enough short, namely the weak interaction case If the dipolar interaction is so strong that all particles in the system can be associated together, the correlation length is very long, the above assumption is broken We 67 believe that the way considering the strong interacting system as a “super-spin glass” system is complete The role of the dipolar interaction is demagnetizing, so the dipolar energy is minimal as the moments parallel together Therefore, the rotation of a moment can excite the rotation of another In other words, the system is easy to magnetize, so the coercive field is close to zero At the high temperature, under the influence of the thermo fluctuation the magnetic moment of each particle orients randomly, so, the total moment of the sample seem to equal zero However, when the dipolar interaction is enough strong to make associations between some nearest particles Therefore, the blocking temperature, at where the particle moment can completely move from an equilibrium state to another, increases At this time, the sample contains individual particle clusters, however, not that the behavior was mentioned by Knobel et al [10] In this case, the sample is similar to the ferromagnetic material including domain walls a) b) Fig.2 a) The hysteresis at the low temperature, T = 0.1 K; and b) the ZFC curve with the value different of the concentration We can image that under the influence of the dipolar interaction, the magnetic nanoparticle system has properties responding to spin glass material at the low temperature and to the multi-domain wall material at the high temperature However, the magnetic nanoparticle system differs from these materials in several behaviors [11] (i) Firstly, the magnetic moment of each particle (102 – 104 μB) is much larger than the atomic magnetic moment (a few μB) (ii) The nature and the range of interactions are different (anisotropic and long range dipolar interaction for magnetic nanoparticles vs short range exchange or longer range RKKY interaction for atomic spins) (iii) The time to flip the atomic spin is much shorter than the time to reserve the particle moment because thermal activation of the particle moment depends on the ratio between the magnetic anisotropy energy Ea and the thermal energy kBT Recently, Schaller et al [12,13], by the Monte Carlo method, presented the effective magnetization of multi-core magnetic nanoparticles containing many separate particles These results opened a new way in the biomedical application of magnetic nanoparticles a) b) Fig.3 a) The variation of coercive field along with temperature, the dot line indicates the classical case (c = 0); and b) the field dependence of the peak temperature, the dot line indicates the anisotropy field Ha = 2Ku/Ms The sample concentration is varied to consider the effect the dipolar interaction The variation of the coercive field along with the temperature and the field dependence of the blocking temperature are presented in Fig.3 In Fig.3a, at the low temperature, the coercive fields completely separate, and as saw in Fig.2a, the coercive field decreases along with the increase of the concentration, however, at the certain temperature, about 10 K in our case, the coercive field starts to increase In the other words, below this temperature the system behaves the collective state and it is so called the glass translation, Tg, or superferromagnetic (SFM) translation temperature, Tc (see Fig.1 in Ref.1) It is worth commenting that the temperature dependence of coercive field does not follow the classical theoretical prediction, Hc ~ – (T/TB) 1/2, as represented by dot line in Fig.3a This was found by many experiments, such as Bae et al [13] It is equally 68 interesting to note that the dipolar interaction makes the coercive field decreases slowly along with the temperature Therefore, the temperature Tg (Tc) raises as the dipolar interaction is strong, and it can exceed the blocking temperature if the sample is very dense [1] Now, we consider the change of the curvature performing the field dependence of the blocking temperature as in Fig.3b This problem was reported in the previous work [8], and in here we only remark some main arguments to give the sufficient understanding about the influence as well as the role of the dipolar i) interaction We need to remember that the dipolar energy of the particle i will make a dipolar field H (dipol acting on it As we saw in Fig 3b, the curvature change from the non-monotonousness (bell-like shape) to monotonousness, and at the very strong interaction, the curvature becomes flatter (plateau-like shape) By experiment, many papers showed the cause of non-monotonous in the dilute sample is due to the respond of the energy barrier distribution to the size distribution, (for details see [8] and references in there) As in Fig.1, when the interacting strength increases, the relation between the size distribution and the barrier distribution disappears, therefore, the non-monotonousness will change to the monotonousness However, if the sample is very dense, the dipolar field seems to equal the small values of the applied field, then the energy barrier of each particle slowly decreases along with the applied field, this means that the curvature is less sloping Because the high dipolar field will remain the barrier, the blocking temperature exists even the applied field exceeds the anisotropy field Ha Notice that the curvatures are separated clearly at the low field, because at this region the local field plays dominant role, then the blocking temperature increases along with the increase of the concentration Conclussion In summary, we presented the influence as well as the role of the dipolar interaction on the magnetic nanoparticle system It deduces the decrease of the coercive field below the translation temperature Tg (or Tc) and the increase of blocking temperature The translation temperature increases with increasing the interacting strength As mentioned in the previous work [8], in the dense sample, the role of the size distribution is dim, and the dipolar interaction plays dominant role In addition, the dipolar interaction makes the blocking temperature varies slowly along with the field, therefore, the peak temperature vs the applied field curvature changes from the bell-like to plateau-like Finally, the expressions represent the relation between the field and the blocking temperature not satisfy in the very dense sample A most recent paper [15] basing the mean field theory to study spin glass phase transition, and may be it will promote in the construction a general theory describing the strong interacting system In our opinion, an effective method to investigate this problem is the spin wave between the magnetic nanoparticles under the influence of the dipolar interaction constructed by the quantum field theory Acknowledgments This work done by the financial sponsor of the fund support of NAFOSTED base research program, Vietnam Academy of Science and Technology (VAST) and the Department of Science and Technology of Dong Nai province References [1] O Petracic, X Chen, S Bedanta, W Kleemann, S Sahoo, S Cardoso, P P Freitas, J Mag Mag Mat 300 (2005), 192 [2] C L Dennis et al., J Appl Phys 103 (2008), 07A319 [3] C L Dennis et al., Nanotechnlogy 20 (2009), 395103 [4] P Tartaj et al., J Phys D: Appl Phys 36, R182 (2003) [5] L Néel, Ann Geophys (1949) 99; W F Brown, Phys Rev 130 (1963) 1677 [6] D Kechrakos, preprint 2009, arXiv: Cond-mat/ 0907.4417 [7] H Gould, J Tobochnik, An introduction to computer simulation methods, Addison-Wesley Publishing Company, 1996 [8] T N Lan, T H Hai, Comput Mater Sci (2010), doi: 10.1016/commatsci.2010.01.025 [9] O Iglesias and A Labarta, Phys Rev B 70 (2004), 144401 [10] M Knobel, W C Nunes, L M Socolovsky, E De Biasi, J M Vargas, J C Denardin, J Nanosci Nanotechno (2008), 2836 [11] D Parker et al., Phys Rev B 77 (2008), 104428 [12] V Schaller et al., J Mag Mag Mat 321 (2009), 1400 [13] V Schaller et al., Phys Rev B 80 (2009), 092406 [14] C J Bae, Y Hwang, J Park, K An, Y Lee, J Lee, T Hyeon, J.-G Park, J Mag Mag Mat 310 (2007), e806 [15] V Dotsenko, preprint 2010, arXiv: Cond-mat/1001.3023 69 ... dụng hạt nano từ, tính chất cha đợc tìm hiểu cách sâu sắc Do tiến hnh hnh nghiên cứu tính chất hệ hạt nano từ phơng pháp mô máy tính Trên giới, nhiều năm gần đây, nh khoa học đà nỗ lực nghiên cứu. .. s¾t từ thang nano, trờng hợp l hạt nano từ Các giá trị trờng khử từ Hc v độ từ d Mr (hình 3.8) đợc tính toán để xác định tính chất trễ hệ hạt nano từ Để mô chu trình từ trễ, sử dụng thông số tơng... ro hệ hạt nano từ Trong hệ hạt nano từ, để hạt chuyển từ trạng thái ny sang trạng thái khác lợng nhiệt tác dụng lên hạt phải đủ lớn để hạt vợt qua ro (hình 2.3) Do phân bố ro hệ hạt nano từ có