CN101726453A - Method for characterizing grain size of magnetic nanometer grains - Google Patents

Abstract

The invention discloses a method for characterizing grain size of magnetic nanometer grains. The method comprises the following steps of: on the basis of the Fourier analysis of a magnetization curve of an analysis model, working out the minimum number of discretization points needed for the characterization of the magnetization curve within a given tolerance error; providing a weighted error based optimal quantization method on the basis of the Lloyd-Max optimal quantization method; and applying the weighted error based optimal quantization method to the problem of solving a grain size distribution function according to a magnetization numerical equation of the magnetic nanometer grains, so that the condition number of a matrix equation is reduced, and the solving precision is improved.

Description

A kind of method for characterizing grain size of magnetic nanometer grains

Technical field

The present invention relates to the nanometer technical field of measurement and test, be specifically related to a kind of diameter characterization method of magnetic nano-particle.

Background technology

Magnetic nanometer (magnetic nanoparticles, be called for short MNP) adopts the magnetic particle colloid of 1～100nm yardstick as molecular biosciences mark of new generation and control technology, for the Long-distance Control of particular organisms incident in vivo provides possibility.Yet the special feature that nanoscale information is obtained is that Heisenberg's indeterminacy effect manifests gradually, and measurement result shows as distribution function but not a fixing amount.Generally speaking, distribution function often obtains by statistic processes, and this just means the balance between high precision and the real-time.Atomic force microscope (AFM), transmission electron microscope (TEM) are the classical mechanics of representative, the explanation of optics statistics nanometer measuring technology, and the diameter characterization technology based on the statistical method does not still possess the online in real time power of test so far.

Based on the experience of mr imaging technique, the researchist thinks that the magnetics measuring method possesses the remote measurement and control ability under the complex dielectrics condition, maybe will provide technical feasibility for the magnetic nanometer online in real time detects.Based on this understanding, the researchist furthers investigate the magnetic characterization technique of magnetic nanometer.The Bo Kefu of Jena, Germany university (D.V.Berkov) professor takes the lead in that the magnetics measuring technology is applied to the MNP size distribution and characterizes, he is with the known variables of particle size distribution function as the magnetization model matrix equation, and then employing singular values of a matrix (Singular Value Decomposition, the thinking of SVD) inverting is found the solution, and directly obtains whole size distribution information.Because people such as Berkov decide characteristic understanding deficiency to the discomfort of magnetization curve matrix equation, comparatively serious false oscillator signal has appearred in solving result in the part.Subsequently, the round good fortune of Kyushu University is respected two (Keiji Enpuku) and is adopted the method for ac magnetic susceptibility the magnetic nanometer label in the solution to be carried out the estimation of particle size distribution function, and the estimated result of this method and the result of dynamic light scattering (optical dynamic light scattering) compared, think that itself and SVD method coincide better, yet this method does not still solve even does not mention false concussion problem.The inventor has found that in research before SVD method consistance on small particle diameter is estimated is better, and false concussion often appears in the big particle diameter estimation, but also never finds effective treating method.False concussion becomes the obstacle that further raising SVD particle diameter estimates to measure the upper limit.The uncertainty of the measuring error of outwardness, limited computational accuracy and time response velocity fails or the like information causes the falseness concussion in the size distribution problems of measurement.Yet, be which kind of factor plays a major role on earth, and whether can improve that this class problem does not obtain clear conclusions as yet by the method for optimizing.

Summary of the invention

The object of the present invention is to provide a kind of method for characterizing grain size of magnetic nanometer grains, overcome the false oscillatory occurences of bringing when prior art adopts singular value decomposition method to find the solution size distribution, and find to contain offspring information in particle size distribution function, thereby improve the sign precision of particle diameter.

A kind of diameter characterization method of magnetic nano-particle is specially: the sampling spot H that at first determines discrete magnetization curve _i, i=1 ... Z, number of sampling Z 〉=21 obtain the discrete magnetization curve M (H of magnetic nanometer colloidal solution again _i), last finding the solution according to discrete magnetization curve obtained grading curve; Wherein, the sampling spot H of discrete magnetization curve _i, determine in the following manner:

(a1) initialization H _i, satisfy H _i＜H _I+1, x ₁=-∞, x _Z+1=+∞;

(a2) calculate $x_j=\frac{{2H}_{j-1}{H}_j}{H_{j-1}+H_j},j=2,...,Z;$

(a3) upgrade $H_i=\frac{{\∫}_{x_i}^{x_{i+1}}{x}^{2}p\left(x\right)\mathrm{dx}}{{\∫}_{x_i}^{x_{i+1}}\mathrm{xp}\left(x\right)\mathrm{dx}},$ P (x) is the probability density function of quantizer input signal x;

(a4) calculate ${\mathrm{\σ}}^2=\underset{i=1}{\overset{Z}{\mathrm{\Σ}}}{\∫}_{x_i}^{x_{i+1}}{\left(\frac{x-H_i}{H_i}\right)}^{2}p\left(x\right)\mathrm{dx};$

(a5) if σ ²＜allowable error threshold epsilon then finishes; Otherwise return step (a2).

Technique effect of the present invention is embodied in: the present invention is by carrying out Fourier analysis to the magnetic nanometer magnetization curve, obtaining characterizing under the given allowable error condition the required smallest discreteization of magnetization curve counts, then according to the theoretical resulting discretize strategy of improved optimum quantization, reduce the conditional number of the super paramagnetic magnetization of Langevin numerical value equation, thereby further suppress the uncomfortable false oscillator signal that characteristic causes of deciding, make and found the particle diameter information of offspring and the distribution variable concentrations thereof from the angle of granularmetric analysis first.

Description of drawings

Fig. 1 is a schematic flow sheet of the present invention.

Fig. 2 is the frequency domain characteristic synoptic diagram of magnetization curve, the amplitude versus frequency characte synoptic diagram of magnetization curve when wherein Fig. 2 (a) is 0.01Gauss for excitation field discretize step-length, Fig. 2 (b) be among Fig. 2 (a) curve to peaked normalized curve result schematic diagram, the phase-frequency characteristic result schematic diagram of magnetization curve when Fig. 2 (c) is 0.01Gauss for excitation field discretize step-length.

Fig. 3 is the optimum quantization solving result synoptic diagram of 30 discretizes of magnetization curve, wherein Fig. 3 (a) is 30 grades and successively decreases and quantize in the progression the 30th, 29,28,27,26 grade and the 21st, 16,11,1 grade, and Fig. 3 (b) is the 5th, 4,3,2 and 1 grade in 30 grades of quantizing processs that successively decrease.

Fig. 4 is the variation synoptic diagram of the matrix A conditional number of discretize optimizing process.

Fig. 5 contrasts synoptic diagram for embodiment size distribution result.

Fig. 6 contrasts synoptic diagram for embodiment size distribution result.

Embodiment

The present invention carries out on the basis of Fourier analysis at the magnetization curve to analytical model, draws under given allowable error to characterize the required smallest discreteization of magnetization curve and count.A kind of optimum quantization method based on the error weighting has been proposed on the basis of Laue moral-Marx (Lloyd-Max) optimum quantization method then, it is applied in according to the magnetization numerical value equation of magnetic nanometer finds the solution on the problem of particle size distribution function, can reduce the conditional number of matrix equation, thereby improve solving precision.With reference to Fig. 1, concrete steps are:

1) chooses the sampling spot H of discrete magnetization curve _i, i=1 ... Z.

1.1) the determining of number of sampling

1.1.1) set up analytical model

Analytical model is the MNP particle size distribution function with lognormal distribution, specifically can be divided into two groups: the first is on 0.3 the basis in fixing particle size distribution function distribution variance, according to the particle size of the existing magnetic nanometer of being produced, choosing the particle diameter average is 7.6nm, 12nm and 18nm; It two is that variance is set at 0.3,0.5 and 0.8 respectively on the basis of fixing average 7.6nm.

1.1.2) magnetization curve of normal research is obeyed expression formula (1) in the frequency domain Physics of Magnetism of analytical model magnetization curve

$M\left(H\right)=\∫L(D,H)\frac{\mathrm{\π}{D}^3}{6}f\left(D\right)\mathrm{dD}---\left(1\right)$

Wherein L () represents the Langevin equation, and D represents diameter of nano particles, and f () represents particle size distribution function, and H is an externally-applied magnetic field.

With particle size distribution function f (D) and excitation field H difference discretize, then the magnetization curve response equation becomes equation (2) in equation (1)

$M\left(H_i\right)=\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{\μ}}_0{M}_d\frac{\mathrm{\π}}{6}{D_j}^{3}L({\mathrm{\μ}}_0{M}_d\frac{\mathrm{\π}}{6}{D_j}^3{H}_i/\mathrm{kT})f\left(D_j\right)\mathrm{\Δ}{D}_j,i=1...Z---\left(2\right)$

Wherein N represents the number of sampling of particle size distribution function, and Z is the number of sampling of excitation field Hi, μ ₀Represent the magnetic permeability of vacuum, M _dRepresent the saturation magnetic moment of material, π is a circular constant, D _jBe j sampling value of particle diameter, k is a Boltzmann constant, and T is an absolute temperature, Δ D _jDiscretize step-length for particle diameter.

For the described magnetic history of equation (2), the discretize strategy of excitation field H has determined the real-time that the length of test duration is just tested, the more direct quality that has determined that M (H) calibration curve information obtains.Information science research points out that the sampling problem of discretize process can be summed up in the point that celestial agricultural sampling thheorem usually.In order to realize this purpose, with step 1.1.1) two groups of particle size distribution functions of the analytical model set up substitution equations (2) respectively, equation (2) is carried out Fourier transform, obtain the spectrum information of magnetization curve, as shown in Figure 2.A1 wherein, A2, A3 represents that respectively grain size of magnetic nanometer grains distribution function variance is 0.3, and average is respectively 7.6nm, 12nm, the frequency spectrum of the magnetization curve during 18nm, B1, B2 represents that respectively grain size of magnetic nanometer grains distribution function average is 7.6nm, and variance is respectively 0.5 and the frequency spectrum of 0.8 o'clock magnetization curve.

Think that from the spectrum analysis of Fig. 2 if magnetic history is carried out 21 discretize, its spectrum signal amplitude has decayed to peaked 5%; Signal amplitude has decayed to peaked 3.5% in the time of 30; Signal amplitude has decayed to peaked about 1% in the time of 100.Certainly, it can also be seen that the effect of statistics, obeying linear change basically greater than X-axis after 50 and logarithmic coordinate Y-axis, just in the coincidence loss statistical theory about the square root of variance and sampling number this theory expectation that is inversely proportional to.Allowable error is generally below 5%, so sampling number should be no less than 21 points, considers the length that how much has determined the test duration of sampling number, recommends to get 21～50 points.

It seems from normalized curve, the final frequency spectrum basically identical of magnetization curve that different-grain diameter distributes, this illustrates that the distribution of this class magnetization curve has identical feature.In addition, Fig. 2 (c) shows that the phase frequency feature of magnetization curve is also basic identical.The feature of frequency domain is identical to mean that " time domain " also possesses certain identical feature.Therefore, this is hinting that the MNP magnetization curve for different size distribution can adopt a kind of unified discretize strategy.

1.2) the sampling spot position determines.

The discretize policing issue of magnetization curve is belonged to the research category of optimum quantization problem in fact.Obviously, uniform discrete cloth point mode is not optimum scheme, and the matrix conditional number of its generation is very big, and ill characteristic performance is especially obvious under the very little situation of counting especially.Thereby under the condition that limited discretize is counted, magnetic history is carried out optimum quantization (optimum discretize) research seem very necessary.

Optimum quantization method of the present invention is the improvement of doing on Lloyd-Max optimum quantization method basis.In Lloyd-Max optimum quantization method, if whole quantification (discretize) interval division is Z interval, H _i(i=1,2 ..., Z) be output signal after the discretize, promptly quantizer input signal x drops on x _iWith x _I+1Between the time quantizer output level be H _i, wherein any one value is H less than its adjacent back value _i＜H _I+1, H _Z+1=+∞, x _i＜x _I+1, x ₁=-∞, x _Z+1=+∞.Its square error σ then ²Be defined as expression formula (3)

${\mathrm{\σ}}^2=\underset{i=1}{\overset{Z}{\mathrm{\Σ}}}E{[x-H_i]}^2=\underset{i=1}{\overset{Z}{\mathrm{\Σ}}}{\∫}_{x_i}^{x_{i+1}}{(x-H_i)}^{2}p\left(x\right)\mathrm{dx}---\left(3\right)$

Wherein, p (x) is the probability density function of quantizer input signal x.

Generally speaking, the Lloyd-max method is a kind of optimum quantization method commonly used.If the σ that formula (3) is provided ²Get minimum value, then necessary condition is (4a) and (4b):

$\frac{{\∂\mathrm{\σ}}^2}{\∂x_i}=0,i=\mathrm{2,3},...,Z---\left(4a\right)$

$\frac{{\∂\mathrm{\σ}}^2}{\∂H_i}=0,i=\mathrm{2,3},...,Z---\left(4b\right)$

Obviously, for the error at unlike signal place, the weight that aforementioned Lloyd-max method adds up is identical.Yet the error that the particle size distribution function of magnetization curve is found the solution concentrates under the little magnetic field excitation situation basically, and promptly little magnetic moment error is not after the match allowed to ignore for the particle diameter estimation effect.Thereby the present invention is at Lloyd-max error metrics (x-H _i) ²Under a kind of relative error is proposed

Evaluation method, with of the contribution of outstanding little field excitation situation magnetic moment, be about to formula (3) and be rewritten as formula (5) for error

${\mathrm{\σ}}^2=\underset{i=1}{\overset{Z}{\mathrm{\Σ}}}E{\left[\frac{x-H_i}{H_i}\right]}^2=\underset{i=1}{\overset{Z}{\mathrm{\Σ}}}{\∫}_{x_i}^{x_{i+1}}{\left(\frac{x-H_i}{H_i}\right)}^{2}p\left(x\right)\mathrm{dx}---\left(5\right)$

Formula (5) substitution formula (4a) is got

$\frac{\∂}{\∂x_i}[{\∫}_{x_{i-1}}^{x_i}{\left(\frac{x-H_{i-1}}{H_{i-1}}\right)}^{2}p\left(x\right)\mathrm{dx}+{\∫}_{x_i}^{x_{i+1}}{\left(\frac{x-H_i}{H_i}\right)}^{2}p\left(x\right)\mathrm{dx}]=0$

Promptly ${\left(\frac{x_i-H_{i-1}}{H_{i-1}}\right)}^2-{\left(\frac{x_i-H_i}{H_i}\right)}^2=0$

$(\frac{x_i}{H_{i-1}}-\frac{x_i}{H_i})(\frac{x_i}{H_{i-1}}+\frac{x_i}{H_i}-2)=0$

$\frac{x_i}{H_{i-1}}+\frac{x_i}{H_i}=2$

$x_i=\frac{{2H}_{i-1}{H}_i}{H_{i-1}+H_i},i=\mathrm{2,3},...,Z---\left(6a\right)$

Formula (5) substitution formula (4b) is got

$\frac{\∂}{\∂H_i}\left[{\∫}_{x_i}^{x_{i+1}}{\left(\frac{x-H_i}{H_i}\right)}^{2}p\left(x\right)\mathrm{dx}\right]=0$

Promptly ${\∫}_{x_i}^{x_{i+1}}2\left(\frac{x-H_i}{H_i}\right)\·\frac{-x}{H_i^2}p\left(x\right)\mathrm{dx}=0$

${\∫}_{x_i}^{x_{i+1}}x(x-H_i)p\left(x\right)\mathrm{dx}=0$

${\∫}_{x_i}^{x_{i+1}}{x}^{2}p\left(x\right)\mathrm{dx}=H_i{\∫}_{x_i}^{x_{i+1}}\mathrm{xp}\left(x\right)\mathrm{dx}$

$H_i=\frac{{\∫}_{x_i}^{x_{i+1}}{x}^{2}p\left(x\right)\mathrm{dx}}{{\∫}_{x_i}^{x_{i+1}}\mathrm{xp}\left(x\right)\mathrm{dx}},i=\mathrm{1,2},...,Z---\left(6b\right)$

The same with the Lloyd-max method, formula (6a) with (6b) can only find the solution by alternative manner in the ordinary course of things.Putative signal is for being symmetrically distributed, so only need to calculate the value of x＞0.If iterations be q (q=1,2,3 ...), σ _q ²The σ that calculates by (5) formula when being the q time iteration ²Value, then formula (6a) and (6b) basic solution procedure can be with reference to following Lloyd-max process of iteration:

(a1) all H of initialization _i(i=1,2 ..., Z), q=1;

(a2) calculate all x by formula (6a) _i(i=2,3 ..., Z);

(a3) according to the x that obtains _i(i=2,3 ..., Z), by the result of calculation of formula (6b) upgrade all Hi (i=1,2 ..., Z);

(a4) calculate σ by formula (5) _q ²

(a5) if ${\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="H2009102731852E00031.png"\; state="\{\{state\}\}">q</figure-callout>}}^2<\mathrm{\&epsiv;}$ (ε is given allowable error threshold value, gets below 5%) then stops; Otherwise q=q+1 forwards (a2) to and continues to calculate.

Can obtain { H by above iterative computation ₁H ₂H _Z.Thereby obtain the quantification progression of optimum quantization, i.e. excitation field H _i(i=1 ... Z) value.When Fig. 3 has provided excitation field number of sampling Z=30 point, H _iOptimum quantization progression result.

2) determine the sampling spot position after, utilize vibration magnetometer (VSM) to obtain the discrete magnetization curve M (H of magnetic nanometer colloidal solution _i).

3) according to discrete magnetization curve M (H _i), adopt the SVD method to carry out the matrix equation approximate solution to obtain MNP grading curve f (j).

Equation (2) is rewritten into matrix form equation (7)

M(i)＝A(i，j)f(j)

And $A(i,j)={\mathrm{\&mu;}}_0{M}_d\frac{\mathrm{\&pi;}}{6}{D_j}^{3}L({\mathrm{\&mu;}}_0{M}_d\frac{\mathrm{\&pi;}}{6}{D_j}^3{H}_i/\mathrm{kT})\mathrm{\&Delta;}{D}_j,i=1...Z,j=1...N---\left(7\right)$

Wherein A (i, j) depend on fully the magneto dynamics process physics principle (A in the follow-up expression formula (i j) is abbreviated as matrix A), M (i) is a magnetization curve, the particle size distribution function of f (j) expression nano particle all has non-negative parameter characteristic.Particle size distribution function f (j) is unique unknown term, thereby nano particle particle size distribution measurement problem transforms nonnegative matrix equation solution problem.For the magnetization curve that VSM test (Lake Shore7410) obtains, available svd (Singular value decomposition, SVD) find the solution by method.Be specially: at first svd (SVD) method of matrix A is found the solution, promptly matrix A according to

A＝USV′ (8)

Here U and V all are orthogonal matrixes, and S then is a diagonal matrix; Be exactly on the basis that SVD decomposes, to carry out equation solution then, promptly

x＝A ⁺b＝VS ^-1U′b。

Above-mentioned solution procedure can adopt the SVD instrument of MATLAB to find the solution.

4) to the performance test of the improved discretize strategy of the present invention: the matrix computations theoretical research thinks, ill characteristic is the important parameter of the matrix equation solving precision that obtains after the decision discretize.Thereby, the discretize strategy of excitation field is optimized, can reduce the conditional number of matrix, thereby improve resolution characteristic signal detail.Simultaneously, conditional number also can be used for estimating the performance of discretize strategy.Concrete test mode is:

The scope of diameter of nano particles D is determined at 0～100nm particle diameter discretize step delta D _jBe 0.1nm, probe temperature is 25 degrees centigrade, and saturation magnetic moment is 65emu/g.The excitation field scope is 0～20000Gauss.By above-mentioned given parameter and excitation field H _iDiscretize numerical value, obtain matrix A by formula (7).And singular value quantity is 7～10 in the setting SVD calculating, and the SVD instrument among the employing MATLAB is found the solution the ratio between maximum singular value and the minimum singular value, thereby obtains conditional number.

Fig. 4 has provided C1～C4 and has represented that respectively singular value quantity is the variation tendency of the conditional number of 7～10 o'clock matrix A.As can be seen from Figure 4, conditional number places an order at improved optimum quantization strategy and downgrades lowly, illustrates that improving algorithm has realized re-set target.The reduction of conditional number helps to improve the solving precision of matrix equation.C4 just also exists singular value not have the situation of (discontinuous) under 10 maximum singular value algorithms among Fig. 4, means that just singular value calculation process parameter has exceeded the computational accuracy of MATLAB and can't calculate under this computational accuracy.Yet after improving the discretize strategy, this phenomenon has disappeared.C4 has reduced about 5 times with respect to the conditional number initial value among Fig. 4.

Example: the EMG1111 magnetic nanometer colloidal solution that utilizes VSM (Lake Shore 7410) that certain company is produced is tested the magnetization curve of acquisition, and the particle diameter estimated result of 30 optimum quantization progression of this test employing as shown in Figure 5 and Figure 6.The result of Fig. 5 and Fig. 6 shows to have big preferably particle diameter resolution characteristic.Being embodied in 2 points, at first is that solving result has been eliminated negative substantially.What is more important because the further reduction of finding the solution error at the bulky grain particle diameter, false oscillator signal obtained inhibition.Thereby oarse-grained size distribution signal is more outstanding, and particle size distribution function shows the peak value feature among Fig. 5 near 6.3nm and 14nm.This result is former can't Direct observation.If there is not the particle of adhesion to be called primary particle, two particles are sticked together and can be described as offspring so.

S1 and S2 represent 100 points among Fig. 6, and S3 represents 30 particle diameter estimated results under the discretize strategy, and its colloid concentration is respectively 100 times of original solution dilutions, 250 times with 40 times.Particle size distribution function is with respect to peaked normalized curve among the figure.The presentation of results of Fig. 6 shows flex point between 13nm to 14nm, can think the performance of offspring.Generally speaking, near the peak value the 6.3nm can be thought the peak value of primary particle.Yet near the peak value the 14nm then is worth thinking, is after the spurious signal if can get rid of, and perhaps can think the result of magnetic nanometer adhesion.Just 13-14nm is slightly greater than the twice of 6.3nm, and whether this point means the formation of offspring along the direct adhesion of direction of magnetization, and bigger particle more the probability of adhesion is bigger, be worth carrying out more meticulous research from the statistical physics angle.The result shows, adopts the present invention further to suppress false vibration, has found to contain the information of the offspring in particle size distribution function, thereby has obtained more accurate estimated result.

Claims (2)

1. the diameter characterization method of a magnetic nano-particle is specially: the sampling spot H that at first determines discrete magnetization curve _i, i=1 ... Z, number of sampling Z 〉=21 obtain the discrete magnetization curve M (H of magnetic nanometer colloidal solution again _i), last finding the solution according to discrete magnetization curve obtained grading curve; Wherein, the sampling spot H of discrete magnetization curve _i, determine in the following manner:

(a1) initialization H _i, satisfy H _i＜H _I+1, H _Z+1=+∞, x ₁=-∞, x _Z+1=+∞;

(a2) calculate $x_j=\frac{{2H}_{j-1}{H}_j}{H_{j-1}+H_j},j=2,...,Z;$

(a3) upgrade $H_i=\frac{{\&Integral;}_{x_i}^{x_{i+1}}{x}^{2}p\left(x\right)\mathrm{dx}}{{\&Integral;}_{x_i}^{x_{i+1}}\mathrm{xp}\left(x\right)\mathrm{dx}},$ P (x) is the probability density function of quantizer input signal x;

(a4) calculate ${\mathrm{\&sigma;}}^2=\underset{i=1}{\overset{Z}{\mathrm{\&Sigma;}}}{\&Integral;}_{x_i}^{x_{i+1}}{\left(\frac{x-H_i}{H_i}\right)}^{2}p\left(x\right)\mathrm{dx};$

(a5) if σ ²＜allowable error threshold epsilon then finishes; Otherwise return step (a2).

2. the diameter characterization method of kind of magnetic nano-particle according to claim 1 is characterized in that, 21≤Z≤50.

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