Research article Topical Sections

Inkjet printed drug-releasing polyelectrolyte multilayers for wound dressings

  • Inkjet printing was used as a novel processing method for the preparation of polyelectrolyte multilayers. Conformal, consistent coatings were formed on a cotton substrate. As a demonstration of a potential application of this processing method, polyelectrolyte multilayers were assembled on cotton for wound dressing. When loaded with gentamicin, these coatings demonstrated burst release of 50% of the loaded gentamicin over the first five hours, followed by consistent release of 0.15 µg/(cm2-h) for at least four days. Significant antimicrobial activity of the gentamicin-releasing polyelectrolyte multilayer-coated cotton was observed, with a zone of inhibition of 1.575 ± 0.03 cm. This result is comparable to the zone of inhibition for cotton soaked in gentamicin (1.75 ± 0.04 cm), indicating that the inkjet printing processing method does not degrade gentamicin. Inkjet printing shows promise as a low cost, versatile option for polyelectrolyte multilayer fabrication. Additionally, as a scalable process, inkjet printed samples exhibited consistent antibacterial function for over three months after preparation.

    Citation: Huilin Yang, Amy M. Peterson. Inkjet printed drug-releasing polyelectrolyte multilayers for wound dressings[J]. AIMS Materials Science, 2017, 4(2): 452-469. doi: 10.3934/matersci.2017.2.452

    Related Papers:

    [1] Minghuan Guo, Hao Wang, Zhifeng Wang, Xiliang Zhang, Feihu Sun, Nan Wang . Model for measuring concentration ratio distribution of a dish concentrator using moonlight as a precursor for solar tower flux mapping. AIMS Energy, 2021, 9(4): 727-754. doi: 10.3934/energy.2021034
    [2] Yasong Sun, Jiazi Zhao, Xinyu Li, Sida Li, Jing Ma, Xin Jing . Prediction of coupled radiative and conductive heat transfer in concentric cylinders with nonlinear anisotropic scattering medium by spectral collocation method. AIMS Energy, 2021, 9(3): 581-602. doi: 10.3934/energy.2021028
    [3] Sunimerjit Kaur, Yadwinder Singh Brar, Jaspreet Singh Dhillon . Multi-objective real-time integrated solar-wind-thermal power dispatch by using meta-heuristic technique. AIMS Energy, 2022, 10(4): 943-971. doi: 10.3934/energy.2022043
    [4] Elio Chiodo, Pasquale De Falco, Luigi Pio Di Noia, Fabio Mottola . Inverse Log-logistic distribution for Extreme Wind Speed modeling: Genesis, identification and Bayes estimation. AIMS Energy, 2018, 6(6): 926-948. doi: 10.3934/energy.2018.6.926
    [5] Jin H. Jo, Zachary Rose, Jamie Cross, Evan Daebel, Andrew Verderber, John C. Kostelnick . Application of Airborne LiDAR Data and Geographic Information Systems (GIS) to Develop a Distributed Generation System for the Town of Normal, IL. AIMS Energy, 2015, 3(2): 173-183. doi: 10.3934/energy.2015.2.173
    [6] Jin H. Jo, Jamie Cross, Zachary Rose, Evan Daebel, Andrew Verderber, David G. Loomis . Financing options and economic impact: distributed generation using solar photovoltaic systems in Normal, Illinois. AIMS Energy, 2016, 4(3): 504-516. doi: 10.3934/energy.2016.3.504
    [7] Yuan Liu, Ruilong Deng, Hao Liang . Game-theoretic control of PHEV charging with power flow analysis. AIMS Energy, 2016, 4(2): 379-396. doi: 10.3934/energy.2016.2.379
    [8] M. Boussaid, A. Belghachi, K. Agroui, N.Djarfour . Mathematical models of photovoltaic modules degradation in desert environment. AIMS Energy, 2019, 7(2): 127-140. doi: 10.3934/energy.2019.2.127
    [9] Constantinos Charalampopoulos, Constantinos S. Psomopoulos, George Ch. Ioannidis, Stavors D. Kaminaris . Implementing the EcoDesign Directive in distribution transformers: First impacts review. AIMS Energy, 2017, 5(1): 113-124. doi: 10.3934/energy.2017.1.113
    [10] Arqum Shahid, Ahsan Ali, Faheem Ashiq, Umer Amir Khan, Muhammad Ilyas Menhas, Sajjad Manzoor . Comparison of resistive and inductive superconductor fault current limiters in AC and DC micro-grids. AIMS Energy, 2020, 8(6): 1199-1211. doi: 10.3934/energy.2020.6.1199
  • Inkjet printing was used as a novel processing method for the preparation of polyelectrolyte multilayers. Conformal, consistent coatings were formed on a cotton substrate. As a demonstration of a potential application of this processing method, polyelectrolyte multilayers were assembled on cotton for wound dressing. When loaded with gentamicin, these coatings demonstrated burst release of 50% of the loaded gentamicin over the first five hours, followed by consistent release of 0.15 µg/(cm2-h) for at least four days. Significant antimicrobial activity of the gentamicin-releasing polyelectrolyte multilayer-coated cotton was observed, with a zone of inhibition of 1.575 ± 0.03 cm. This result is comparable to the zone of inhibition for cotton soaked in gentamicin (1.75 ± 0.04 cm), indicating that the inkjet printing processing method does not degrade gentamicin. Inkjet printing shows promise as a low cost, versatile option for polyelectrolyte multilayer fabrication. Additionally, as a scalable process, inkjet printed samples exhibited consistent antibacterial function for over three months after preparation.


    1. Introduction

    The increasing presence of wind farms (WF) in electrical power networks has made it important to simulate correlated wind speeds. The use of wind speed series in combination with the wind turbine (WT) power curves is common for the resolution of more than one typical problem in electrical power network analysis.

    In order to attain a solution for some of these problems it is necessary to deal with wind speed series satisfying features regarding the frequency distribution of wind speeds and the correlation between series at different sites. There is wide agreement on considering the Weibull distribution as the best continuous approximation for the frequency distribution of wind speed in a site [1] and Kavasseri presents a study of the phenomena associated to the correlation [2]. Spatial correlation is explicity mentioned by Damousis [3] and autocorrelation has been studied by Brown et al. [4], and also by Song and Hsiao [5].

    Other solutions to the problem stated above have been proposed in different papers, and a review of them has been presented by Feij\'oo et al. [6], especially for the case when no additional chronological constraints are imposed.

    Correia and Ferreira de Jesus [7] use the process of obtaining a Weibull distribution as a combination of two Normal distributions while Segura et al. [8] obtain correlated Weibull distributions from Uniform distributions.

    A method based on the conditional probability theorem was presented by Karaki et al. [9] and an approach based on the inverse discrete Fourier transform was presented by Young and Beaulieu [10].

    Huang and Chalabi [11] present a method based on chronological series and Shamshad et al. [12] a different one based on Markovian models.

    Villanueva et al. [13] propose a solution for an application to the economic dispatch problem in networks with penetration of wind farms. It consists of the simulation of wind speed series satisfying statistical constraints such as those mentioned above together with an additional one regarding autocorrelation, which added a chronological nuance to the proposed method. As a result, series of correlated wind speeds are obtained with a very high degree of accuracy regarding correlations, Weibull parameters and autocorrelation of the series, although it is not so clear why these features are retained through the proposed transformation, which is a transformation from Normal to Weibull distributions.

    The main objective of this paper is to return to this subject and offer an approach which helps understand and discuss why those features are retained through such a transformation. Polynomial approximations of first and second degree will be used to achieve this.

    In the rest of the paper, a Normal distribution with mean μ

    and standard deviation σ
    is denoted as N(μ,σ)
    , and a Weibull distribution with scale parameter c
    and shape parameter k
    is denoted as W(c,k)
    . Uniform distributions are also mentioned, and U(0,1)
    denotes one of these distributions in the interval [0,1]. Subindices such as in μx
    or in μu
    are used for distinguishing between different data series, for example, for denoting the mean of the series {xi}i{1,2,...,M}
    , or of the series {uj}j{1,2,...,M}
    respectively.

    2. Normal to Weibull transformation

    A N(0,1)

    distribution can be converted into a W(c,k)
    one by means of the following transformation [13]:

    u=c(log(1erf(x2)2))1k
    (1)

    where x

    is a value belonging to a normally distributed series, u
    to a Weibull one, log
    represents the natural logarithm and erf
    the error function defined as follows:

    erf(x)=2πx0et2dt
    (2)

    For such a transformation the fact must be taken into account that the cumulative distribution function (CDF) of normally distributed data with mean value μx

    and standard deviation σx
    can be expressed as Fx(x)=12(1+erf(xμxσx2))
    . For the N(0,1)
    distribution the transformation is Fx(x)=12(1+erf(x2))
    . In the case of data belonging to a Weibull distribution, with scale parameter c
    and shape parameter k
    , the transformation is Fu(u)=1e(uc)k
    . According to this notation, (1) is obtained by matching both distribution functions and clearing up the variate u
    , as u=F1u(Fx(x))
    , which is distributed as a Weibull one. Both functions can be equalled due to the fact that any distribution function provides a Uniform distributed variate, i.e., Fx(x)U(0,1)
    and Fu(u)U(0,1)
    .

    When such a transformation is performed, a representation of the normally distributed values against the Weibull distributed ones gives Figure 1 as a result, where the Weibull values cover an interval including [3,25], i.e., the interval of wind speed values in which most WTs can run, which is the reason for being considered an important interval in this paper.

    The graph represented in Figure 1 has been obtained for a Weibull distribution with parameters c=7

    and k=2
    . A value of k=2
    corresponds to a particular case of the Weibull distribution known as Rayleigh distribution. Similar curves are obtained for other values of the pair (c,k)
    , although they have not been represented here.

    So far, in Figure 1 attention must be paid to the line with the legend Exact, obtained from (1).

    Figure 1. Normal against Weibull data.


    3. Polynomial approximations

    A visual inspection of Figure 1 and the experience of having carried out many different simulations lead the authors to think that polynomial approximations to (1) could give accurate results.

    If a least squares approximation is applied to the set of values of such a transformation, then the other curves of Figure 1 are obtained, i.e., those corresponding to the legends 1st order and 2nd order.

    Both curves have been obtained by means of polynomial approximations of first and second order, respectively, such as:

    f(x)=Dn=0anxn
    (3)

    where D{1,2}

    .

    This means that u

    can be expressed as u=f(x)+ϵ(x)
    , where ϵ(x)
    is a small error that, when neglected, involves accepting that uf(x)
    .

    The values of the constants obtained for these approximations are a0=5.7660

    , a1=4.1384
    in the case of the first degree polynomial (i.e., D=1
    ) and a0=5.9301
    , a1=3.5227
    and a2=0.1539
    for the second degree one (i.e., D=2
    ).

    Summarizing, for the transformation proposed in (1) there are the following possible approximations.

    f1(x)=5.7660+4.1384x
    (4)
    \begin{eqnarray}\label{f_12}
    f_2(x)&=&5.9301+3.5227\cdot x+0.1539\cdot x^2
    \end{eqnarray}
    (5)

    For each approximation, a measure of error can be defined as:

    efk(x)=Mi=1(uifk(xi))2M
    (6)

    where k{1,2}

    , M
    is the number of samples, ui
    are calculated from (1) and fk(xi)
    are calculated from (4).

    The error made in a set of 100, 000 samples, according to (6), when obtaining the approximations of Figure 1 came to 0.0043 for the first order polynomial and 0.0011 for the second order one. In different simulations the values of these errors can be slightly different, but very close to the one given here.

    Although the differences in error according to (6) are not so important, they are bigger when dealing with absolute errors. If an absolute error is measured according to:

    eabsfk(x)=max{|uifk(xi)|,i{1,2,...,M}}
    (7)

    then in different simulations, values around 1.27 have been found for the first degree approximation and around 0.34 for the second degree one.

    In Figure 2, the histogram obtained by means of a transformation based on (1) has been included with the notation Exact. In the same Figure, 1st deg. pol. and 2nd deg. pol. denote the histograms corresponding to both proposed approximations. This histogram has been obtained with 100, 000 samples.

    Figure 2. Weibull distribution and approximations.

    An observation must be made here. A first order polynomial does not convert a Normal distribution into a Weibull one. It is a linear conversion and the result has to be a Normal distribution, and this can be appreciated in Figure 2. However, a second order polynomial has the contribution of the second order term, with a strong trend to make the function asymmetric, and its similarity with a Weibull distribution is stronger.

    Finally, in Table 1 some of the moments of u

    , f1(x)
    and f2(x)
    are given. As can be seen, both approximations have a lower mean and a higher standard deviation. It is intuitive that the three series are highly correlated. And indeed, correlations between them are ρuf1(x)=0.9862
    , ρuf2(x)=0.9943
    . If Spearman rank correlations, ρufi(x)Sp
    , i{1,2}
    , are calculated, then the result is ρuf1(x)Sp=1
    and ρuf2(x)Sp=1
    . As f1(x)
    is a linear relationship, the calculation of ρxf1(x)
    should give 1 as result, and this is exactly what happens. And also, ρxu=0.9862
    , just like ρuf1(x)
    .

    Table 1. Moments of u, f1(x) and f2(x).
    mean μ
    std. dev. σ
    u
    6.19753.2355
    f1(x)
    5.77074.1289
    f2(x)
    6.07873.5213
     | Show Table
    DownLoad: CSV

    4. Influence of the constants c
    and k

    The next question to be answered is about the influence of constants c

    and k
    of the Weibull distribution on the transformation.

    In this section the variation of constants ai

    of the polynomial representation are studied as a function of c
    and k
    .

    4.1 The effect of the variation of c

    In (1) it can be observed that the constant c

    is a factor of the equation, i.e., it multiplies the rest of the transformation. This is equivalent to saying that (1) can be written as an equation such as u=cg(x)
    where g(x)
    is a nonlinear function, or that u
    is proportional to g(x) because c
    is a constant.

    It is not difficult to deduce that variations of c

    should lead to proportional values of u
    for the same values of x
    . Consequently, this fact should be reflected in the polynomial approximation. And it is exactly what happens as can be seen in Figure 3, where these variations can be observed for a fixed value of k=2
    , although similar results are obtained for different values of k
    .

    Figure 3. Variation of a0, a1 and a2 of the polynomial approximation with the constant c of the Weibull distribution.

    For this value of k=2

    , the values of a0
    , a1
    and a2
    , of the polynomial approximation, which will be denoted as a0c
    , a1c
    and a2c
    , can be calculated directly as a function of c
    , according to the following equations:

    a0c(c)=0.8464ca1c(c)=0.4972ca2c(c)=0.0243c
    (8)

    As has been reflected in Figure 3 the simulation was made for values of c

    in the interval [0, 14].

    The following comment should be added. As mentioned before, sometimes the Rayleigh distribution has been recommended as a good continuous approximation of the frequency distribution of wind speeds at a given site. In a Weibull distribution the mean and standard deviation are calculated by means of the Gamma function first described by Euler, Γ(p)=0exxp1dx

    , according to μWeibull=cΓ(1+1k)
    for the mean, and σ2Weibull=c2(Γ(1+2k)Γ2(1+1k))
    for the standard deviation. However, one of the advantages of the Rayleigh distribution is the fact that the calculation of these statistical values is simpler, and they can be expressed such as μRayleigh=cπ2
    for the mean and σ2Rayleigh=c2(1π4)
    for the variance.

    Taking the previous paragraph into account, the use of (8) can be of interest when using the Rayleigh distribution. In this case, k=2

    and the value c
    is calculated from the mean value as c=2μRayleighπ
    . And after that, the transformation of the Normal values to the Rayleigh ones can be performed with a high degree of accuracy by means of (8), which can be alternatively expressed as follows:

    a0c(μ)=0.84642πμ=0.9551μa1c(μ)=0.49722πμ=0.5610μa2c(μ)=0.02432πμ=0.0274μ
    (9)
    where μ
    denotes the mean value of the Rayleigh distribution, that in the previous paragraph was denoted as μRayleigh
    .

    4.2 The effect of the variation of k

    The presence of k

    in (1) is quite different from the case of the presence of c
    .

    As can be expected, changes in the value of k

    produce nonlinear effects. This can be seen in Figure 4, where the value of c
    has been kept constant, allowing changes in k
    in the interval [1,4].

    Figure 4. Variation of a0, a1 and a2 of the polynomial approximation with the constant k of the Weibull distribution.

    A first order approximation does not seem to be able to fit these curves, for which a second order one is here recommended, and it reveals that a0

    , a1
    and a2
    , now denoted as a0k
    , a1k
    and a2k
    can be expressed as follows:

    a0k(k)=0.0715k2+0.4862k+0.9886a1k(k)=0.0967k20.8153k+2.2533a2k(k)=0.1503k20.9132k+1.3177
    (10)

    Both effects of the variations of c

    and k
    can be studied in combination, and this is explained in next section.

    4.3 Combined effect of the variation of c
    and k

    The results of the variation of both c

    and k
    values have been combined for the constants a0
    , a1
    and a2
    , and can be seen in Figures4, 5 and 6.

    Figure 5. Variation of a0 in the polynomial approximation (degree 2) with the constants c and k of the Weibull distribution.
    Figure 6. Variation of a1 in the polynomial approximation (degree 2) with the constants c and k of the Weibull distribution.

    This combined dependency can also be approximated by means of a polynomial transformation such as:

    ai(c,k)=1m=02n=0pmnicmkni{0,1,2}
    (11)

    where the constants pmni

    can be calculated for each ai
    . Summarizing, (1) can be substituted by an equation like (3), where the constants ai
    can be calculated as functions of the constants of the Weibull distribution, c
    and k
    , by means of (11).

    Under the assumption of an interval for c

    equal to [0,14] and an interval for k
    equal to [1,4], the use of the above mentioned tool gives as a result the values for the constants given in Table 2 (the first subscript is for c
    and the second one for k
    ).

    Figure 7. Variation of a2 in the polynomial approximation (degree 2) with the constants c and k of the Weibull distribution.
    Table 2.Constants pmn in the expresions of ai, i∈{1, 2, 3}.
    p00
    p10
    p01
    p11
    p02
    a0
    -0.97430.68910.89380.0644-0.1788
    a1
    1.31700.8632-1.209-0.16600.2417
    a2
    2.04800.2492-1.8790-0.08080.3758
     | Show Table
    DownLoad: CSV

    The coefficients p12

    have not been included because they are very close to 0 in all cases (in fact, their values are lower than 1015).

    5. Consequences of the proposed approximations

    Some consequences of the proposed approximations are given in this section. The conservation of autocorrelation when going from Normal distributions to Weibull distributions is not clearly easy to deduce directly from (1). But some operations with the statistical values based on the approximations can be of help for giving some explanation about why these values are retained. At the end of the section there are also some considerations about negative values in the simulations.

    5.1 Conservation of the autocorrelation

    It has been mentioned that (1) was presented by Villanueva et al. in [13] as an option for the obtaining of Weibull distributions, with given mean, standard deviation and also lag 1 autocorrelation. To achieve such an objective, an autoregresive process known as AR(1) was used for randomly simulating Normal distributions with mean 0 and standard deviation 1 and then a transformation based on the Choleski decomposition of the covariance matrix was used to obtain the Weibull distributions.

    Something that was observed when using this transformation process was the fact that lag 1 autocorrelation was apparently retained when converting data from the initial Normal distribution to the final Weibull one, i.e., the autocorrelation of each of the simulated Weibull series had a value very close to the value of the autocorrelation of the initial Normal series. In view of (1) it is not so evident why such statistical values are retained through the transformation.

    The results obtained with the polynomial approximations presented in this paper can be used as an interesting approach to explain the fact given above.

    In previous sections it has been shown that good approximations with polynomials of degrees 1 and 2 can be obtained for (1). The proposal is to look for relationships between the statistical values by means of these approximations.

    5.1.1 First degree

    From this section on, some new notation will be used, and so x=[x1,x2,...,xM],\[xi=[x1,x2,...,xM1],and\[xi+1=[x2,x3,...,xM].Fortheapproximation,thenotationwillbe\[f1(x)=[f1(x1),f1(x2),...,f1(xM)],\[f1(x)i=[f1(x1),f1(x2),...,f1(xM1)]andinthesamewayto\[xi+1

    , f1(x)i+1=[f1(x2),...,f1(xM)].Similarnotationwillbeemployedfor\[f2(x)
    .

    It is a well known fact that if the mean value of a Normal distribution is μx=E[x]=Mk=1xkM

    , where E represents the expected value of x
    , then the mean value of the approximated Weibull distribution, if the approximation is by means of a first degree polynomial, i.e., f1(x)=a0+a1x
    , can be estimated as μf1(x)=E[f1(x)]=E[a0+a1x]=a0+a1μx=a0
    , taking into account that μx=0
    , because xN(0,1)
    . It must not be forgotten that the transformed distribution is a Normal one because the transformation is linear, as the polynomial is of first degree.

    If σ2x=E[(xμx)2]=Mk=1(xkμx)2M

    is the variance of the Normal distribution, then the variance of the approximated Weibull one is σ2f1(x)=E[(f1(x)μu)2]=E[(a0+a1xa0)2]=a21\[,because\[E[x2]=E[(xμx)2]=1
    , as xN(0,1)
    .

    Lag 1 autocorrelation is obtained as the correlation between a given series and the same series shifted by one position. For calculating covariances between two series, for example, x=[x1,x2,...,xM]and\[y=[y1,y2,...,yM],aformulationhastobeusedwheretermslikethefollowingappear,\[(xiμx)(yiμy)

    . However, when calculating lag 1 autocorrelation, only one series is involved, x=[x1,x2,...,xM]andthetermsaresuchas\[(xiμx)(xi+1μx)
    , i.e., the values are grouped as follows, (x1,x2)
    , (x2,x3)
    , ..., (xM1,xM)
    .

    The covariance between a series of a Normal distribution and the shifted one will be denoted as σxixi+1

    and can be calculated as:

    σxixi+1=M1i=1(xiμx)(xi+1μx)M=M1i=1xixi+1M

    by taking into account that μx=0

    because xN(0,1)
    . One fact to bear in mind is that there are only M1
    addends in the previous sum. The value of M
    in the denominator can remain, instead of being changed to M1
    , because this does not affect the rest of the reasoning, and also because it is generally accepted in the calculation of lag s
    autocorrelation, for a given value of s
    .

    The correlation between both series can be defined as

    ρxixi+1=σxixi+1σxiσxi+1=σxixi+1σ2=σxixi+1

    assuming that σ=σxi=σxi+1=1

    .

    This correlation between the two series is the lag 1 autocorrelation of the series.

    What happens with the values of the approximated Weibull distribution is the following:

    σf1(x)if1(x)i+1=1MM1i=1(f1(xi)μf1(x))(f1(xi+1)μf1(x))=a21σxixi+1

    taking into account that f1(xi)μf1(x)=a0+a1xia0

    , and f1(xi+1)μf1(x)=a0+a1xi+1a0
    , the autocorrelation is, then, calculated, as a correlation between both series, i.e.

    ρf1(x)if1(x)i+1=σf1(x)if1(x)i+1σf1(x)iσf1(x)i+1

    Previously it has been shown that σ2f1(x)=a21

    , which leads to σ2f1(x)i=σ2f1(x)i+1=a21
    , and σf1(x)i=σf1(x)i+1=a1
    . Finally:

    ρf1(x)if1(x)i+1=a21σxixi+1a1σxia1σxi+1=σxixi+1σxiσxi+1=ρxixi+1

    The conclusion is that substituting (1) by a first degree polynomial involves a transformation where lag 1 autocorrelation is retained.

    In fact, if subindices i+1

    are changed to subindices i+s, then s
    lag autocorrelation must be kept in such a transformation. Thus, if an AR(s
    ) instead of an AR(1) process is used for generating the Normal distribution, the transformation to approximated Weibull series should keep s
    lag autocorrelations for all s
    .

    A summary of all these moments can be read in Table 3.

    Table 3. Moments of x
    and f(x)
    .
    Variableμ
    σ
    x
    01
    f1(x)=a0+a1x
    a0
    a21
     | Show Table
    DownLoad: CSV

    5.1.2 Second degree

    The conclusions of the previous section are not surprising because the proposed transformation by means of f1

    is linear.

    A better approximation to (1) consists of a second order polynomial, such as f2(x)=a0+a1x+a2x2

    . The term of second order contributes to confer a certain degree of asymmetry to the distribution, which is a feature that makes it more similar to a Weibull one.

    The mean value of the given distribution is μf2(x)=E[a0+a1x+a2x2]=a0+a2

    , again because E[x]=μx=0
    and in addition because E[x2]=σ2x=1
    .

    In the case of the variance the calculation is as follows, σ2f2(x)=E[(a0+a1x+a2x2μf2(x))2]=E[(a1x+a2(x21))2],bytakingintoaccountthefactthat\[μf2(x)=a0+a2

    , shown in the previous paragraph. If some operations are performed, the following transformation can be obtained, \[\sigma_{{\bf f_2(x)}}^2=E[a_2^2{\bf x}^4+2a_1a_2{\bf x}^3+(a_1^2-2a_2^2){\bf x}^2-2a_1a_2{\bf x}+a_2^2].

    The values of the moments for a Normal distribution can be seen in appendix A, and substituting them in this transformation, the result is that σ2f2(x)=a21+2a22

    .

    Autocorrelation can be defined with the help of σxixi+1=M1i=1xixi+1M

    , for the case of the original Normal series.

    For the transformed one: σf2(x)if2(x)i+1==1MM1i=1(f2(xi)μf2(x))(f2(xi+1)μf2(x))==1MM1i=1(a1xi+a2(x2i1))(a1xi+1+a2(x2i+11))

    For simplicity, M

    will not be taken into account now. By operating the previous equation:

    M1i=1(a21xixi+1+a1a2xix2i+1a1a2xi+a1a2x2ixi+1a1a2xi+1+a22x2ix2i+1a22x2ia22x2i+1+a22)

    This transformation can be simplified by taking into account that, due to properties of the N(0,1) distribution, with the help of expressions given in appendix A, the final result is that σf(x)if(xi+1)=a21σxixi+1+2a22σ2xixi+1

    .

    According to this:

    ρf2(x)if2(x)i+1=σf2(x)if2(x)i+1σf2(x)iσf2(x)i+1=ρxixi+1a21+2a22ρxixi+1a21+2a22
    (12)

    where ρxixi+1=σxixi+1

    .

    It is interesting to point out the values of ρf2(x)if2(x)i+1

    in some particular cases, which can be considered extreme cases:

    ρf2(x)if2(x)i+1={1if \[\rho_{\bf x_ix_{i+1}}=1\]0if \[\rho_{\bf x_ix_{i+1}}=0\]a21+2a22a21+2a22if \[\rho_{\bf x_ix_{i+1}}=-1\]

    Anyway, as generally a1a2

    , it can be concluded that ρf2(x)if2(x)i+1ρxixi+1
    . For example, by using the values obtained for a second degree approximation, i.e., a1=3.5227 and a2=0.1539, the result is that if ρxixi+1=1
    , then ρf2(x)if2(x)i+1=0.9924
    . The meaning of this is that the autocorrelation of the distribution obtained through the second degree polynomial is greater than 99\% of the autocorrelation of the initial Normal distribution.

    All that has been explained in this section can be summarized as follows: assuming a Normal distribution and the Weibull distribution obtained from this Normal one by means of (1), different approximations to the Weibull distribution can be run on a polynomial basis.

    A degree one polynomial is a linear approximation which retains variance and autocorrelation without dependency of the lag.

    A degree two polynomial includes a certain degree of asymmetry, which makes the transformed distribution be further from the Normal one and closer to the Weibull one. In this case, the autocorrelation is not retained and its degree of approximation to the values of the original distribution depends on the value itself.

    As a second degree approximation is closer to the first degree approximation this allows the conclusion to be made that the autocorrelation is not retained through the exact transformation given by (1).

    Table 4. Moments of x
    and f(x)
    .
    Variableμ
    σ
    x
    01
    f2(x)=a0+a1x+a2x2
    a0+a2
    a21+2a22
     | Show Table
    DownLoad: CSV

    5.2 Conservation of the correlation

    Correlations between series are retained through linear transformations, and this can be argued in a similar way to the assertion made for the lag 1 autocorrelations.

    If xN(0,1)

    and yN(0,1)
    , and f1(x)=a0+a1x
    and g1(y)=b0+b1y
    , where f1(x)
    and g1(y)
    are now two new series transformed from x
    and y
    , then the correlation between x
    and y
    is given by ρxy=σxyσxσy
    . As σx=σy=1
    , it can be expressed as ρxy=σxy
    . On the other side, σxy=E[(xμx)(yμy)],where\[μx=μy=0
    , by which \[\sigma_{{\bf xy}}=E[{\bf xy}].

    This means that σf1(x)f1(y)=E[(f1(x)μf1(x))(f1(y)μf1(y))]=E[(a0+a1xa0)(b0+b1yb0)]=a1b1σxy

    .

    As σf1(x)=a1

    and σg1(y)=b1
    , then ρf1(x)g1(y)=σf1(x)g1(y)σf1(x)σg1(y)=a1b1σf1(x)g1(y)a1b1=ρf1(x)g1(y)
    , and this proves that the correlation is also retained.

    However, in nonlinear transformations, i.e., in the case of second degree approximations, things operate in a different manner. If new approximations are taken, such as f2(x)=a0+a1x+a2x2

    and g2(y)=b0+b1y+b2y2
    , then both means are μx=a0+a2
    and μy=b0+b2
    and the standard deviations are σx=a1+2a22
    and σy=b1+2b22
    .

    The variance between f2(x)

    and g2(x)
    is calculated as \[\sigma_{{\bf f_2(x)g_2(y)}}=E[({\bf f_2(x)}-\mu_{\bf f_2(x)})({\bf g_2(y)}-\mu_{\bf g_2(y)})], which is $E[(a_1{\bf x}+a_2({\bf x}^2-1))(b_1{\bf y}+b_2({\bf y}^2-1))].

    By rearranging the previous expression and by taking into account appendix A, it can be written as:

    σf2(x)g2(y)=σxy(a1b1+a2b2σxy)

    Now, as σf2(x)=(a21+2a22)σx

    and σf2(y)=(a21+2a22)σy
    , and σxy=ρxy
    finally:

    ρf2(x)g2(y)=ρxya1b1+2a2b2ρxya21+2a22b21+2b22
    (13)

    As can be deduced, things operate in a similar way to the case of lag 1 autocorrelation if both distributions coincide, i.e., if they have identical c

    and k
    parameters, because in this case a1=b1
    , a2=b2
    and (13) is an equation similar to (12). In this case the conclusions achieved in (12) can be applied here.

    However, when both distributions differ, then the four parameters a1

    , a2
    , b1
    and b2
    in (13) cannot be substituted by only two of them, and the dependency on them is more complex. But the conclusion is that correlation is not completely retained when using (1).

    5.3 Negative values

    Another consequence of the approximation is the appearance of negative values in the conversion, a problem that was detected by Feij\'oo et al. [14], in a work where correlated Weibull and Rayleigh distributed series of wind speeds were simulated, and then avoided with new methods by Feij\'oo and Sobolewski [15], with the use of nonparametric correlations, i.e., Spearman rank correlations.

    For representing wind speed values, the Weibull distribution can be treated just as it has so far, i.e., assuming that its minimum value is 0.

    But more generally, the Weibull distribution CDF with an origin γ0

    can be defined as $W: [\gamma, +\infty) \rightarrow [0, 1]. As mentioned, in the case of wind speed distributions, \[\gamma=0\] because wind speeds are supposed to be positive, so there are no negative values in the distribution. The application of the explained approximations gives a certain number of negative values as a result. This can be observed in Figure 2. By multiple simulations, it has been estimated that no more than 0.5\% of the data are negative.

    According to this, it is natural that when data represent wind speeds, there is a tendency to reject negative values. However, other errors are accepted in all simulations and an interesting question that begs to be answered is how much of a problem it would be to accept negative values as wind speed values.

    And the answer is that it does not necessary involve making a significant error in the calculations.

    A typical situation consists of combining wind speed data with WT power curves with the aim of calculating either the power generated or the total energy produced during a certain period of time.

    In order to check the error made when substituting the exact formulation given by (1) by a polynomial approximation of degree 2, a power curve for a WT has been combined with data corresponding to a site with a Weibull distribution of parameters c=7 and k=2

    .

    The power curve has been proposed by Carta et al. [16] and described in appendix B. For the following values, vCI=4$ $ms1

    as cut-in wind speed, vR=14$ $ms1
    as rated wind speed, and vCO=25
    ~ms1
    as cut-out wind speed, the error made in the calculation of energy generated by the WT rises to a 3\%. The value of this error does not seem to depend on the maximum power. For WTs of rated powers 2, 3, ..., 7~MW
    the error is also around 3\%.

    As a conclusion, the acceptance of negative values as wind speeds for the calculation of power or energy values in a simulation is not so important, as they are filtered by the WT power curve, i.e., if the value of wind speed is negative, then the power generated by the WT will be 0. In many cases it will be just like when the value is positive but under 3 or 4~ms1

    or above 25~ms1
    .

    An approximation with a polynomial of degree one is much less satisfactory, as the errors made when calculating energy rise to values close to 13\% for all the rated powers given.

    6. Conclusions

    In this paper two different polynomial approximations have been proposed for the transformation of sets of Normally distributed data to sets of Weibull distributed series, satisfying not only the parameters of Weibull distributions, but also their correlations and even autocorrelations.

    The approximations have been used to provide an approach to a better understanding of why these features are retained when an exact transformation is carried out, with the following consequences:

    1. The use of (1) for the Normal to Weibull transformation is very adequate and gives very good results. It has no disadvantages from a computational point of view. However, it is not easy to explain why certain statistical values, such as correlation and autocorrelation are highly retained, which has been previously obtained by simulations.

    2. In order to explain such phenomena, polynomial approximations based on the least square method were used for the CDFWeibull=f(\[CDF

    _{Normal})\] curve.

    3. A second degree approximation shows a high degree of accuracy, and also shows that correlations and autocorrelations are not exactly retained, but seem to explain why they are highly retained.

    4. The appearance of negative values of the wind speed in simulations does not involve an important error. Although there are no negative values of wind speed, they can appear in the simulation, but even so, they are filtered by the power curves of the WTs when used for estimating power or energy captured from the wind.

    Appendi A Moments of the Normal distribution

    The following are the expressions of moments of the normal distribution that have been necessary in the paper: E[x]=μx=0E[x2]=σ2x=1E[x3]=μ3x+3μxσ2x=0E[x4]=μ4x+6μ2xσ2x+3σ4x=3E[xi]=E[xi+1]=0E[x2i]=E[x2i+1]=1E[xixi+1]=σxixi+1E[xix2i+1]=E[x2ixi+1]=0E[x2ix2i+1]=σxiσxi+1+2σ2xixi+1=1+2σ2xixi+1

    Appendi B Power curves description

    A WT power curve can be described by means of a function such as the following [17]:

    P={00vwvCIh(vw)PRvCIvw<VRPRvRvw<vCO0vwvCO

    where vw

    is the input variable, i.e., the wind speed, vCI
    stands for cut-in wind speed, 4~ms1
    in the example proposed in the paper, vCO
    for cut-out wind speed, 25~ms1
    , vR
    is the rated wind speed, 14~ms1
    , and PR
    the rated power, 1~MW
    .

    The function h(vw)

    is calculated as h(vw)=A+Bvw+Cv2w
    , and

    A=a(vCIb4vCIvRC)B=a(4bd3vCIvR)C=a(24d)a=1(vCIvR)2b=vCI+vRd=(vCI+vR2vR)3

    [1] Sen CK, Gordillo GM, Roy S, et al. (2009) Human skin wounds: A major and snowballing threat to public health and the economy. Wound Repair Regen 17: 763–771. doi: 10.1111/j.1524-475X.2009.00543.x
    [2] Mangoni ML, Mcdermott AM, Zasloff M (2016) Antimicrobial peptides and wound healing: Biological and therapeutic considerations. Exp Dermatol 25: 167–173. doi: 10.1111/exd.12929
    [3] Fahs F, Bi X, Yu FS, et al. (2015) New insights into microRNAs in skin wound healing. IUBMB Life 67: 889–896. doi: 10.1002/iub.1449
    [4] You HJ, Han SK (2014) Cell therapy for wound healing. J Korean Med Sci 29: 311–319. doi: 10.3346/jkms.2014.29.3.311
    [5] Boateng JS, Matthews KH, Stevens HNE, et al. (2008) Wound healing dressings and drug delivery systems: A review. J Pharm Sci 97: 2892–2923. doi: 10.1002/jps.21210
    [6] Zelikin AN (2010) Drug Releasing Polymer Thin Films: New Era of Surface-Mediated Drug Delivery. ACS Nano 4: 2494–2509. doi: 10.1021/nn100634r
    [7] Delcea M, Möhwald H, Skirtach AG (2011) Stimuli-responsive LbL capsules and nanoshells for drug delivery. Adv Drug Deliver Rev 63: 730–747. doi: 10.1016/j.addr.2011.03.010
    [8] Kataoka K, Harada A, Nagasaki Y (2001) Block copolymer micelles for drug delivery: design, characterization and biological significance. Adv Drug Deliver Rev 47: 113–131. doi: 10.1016/S0169-409X(00)00124-1
    [9] Qiu Y, Park K (2001) Environment-sensitive hydrogels for drug delivery. Adv Drug Deliver Rev 53: 321–339.
    [10] Yamada Y, Harashima H (2008) Mitochondrial drug delivery systems for macromolecule and their therapeutic application to mitochondrial diseases. Adv Drug Deliver Rev 60: 1439–1462. doi: 10.1016/j.addr.2008.04.016
    [11] Huang X, Brazel CS (2001) On the importance and mechanisms of burst release in matrix-controlled drug delivery systems. J Control Release 73: 121–136. doi: 10.1016/S0168-3659(01)00248-6
    [12] Kost J, Langer R (2001) Responsive polymeric delivery systems. Adv Drug Deliver Rev 46: 125–148.
    [13] Wohl BM, Engbersen JFJ (2012) Responsive layer-by-layer materials for drug delivery. J Control Release 158: 2–14.
    [14] Meyer F, Dimitrova M, Jedrzejenska J, et al. (2008) Relevance of bi-functionalized polyelectrolyte multilayers for cell transfection. Biomaterials 29: 618–624. doi: 10.1016/j.biomaterials.2007.10.027
    [15] Lee IC, Wu YC (2014) Assembly of Polyelectrolyte Multilayer Films on Supported Lipid Bilayers To Induce Neural Stem/Progenitor Cell Differentiation into Functional Neurons. ACS Appl Mater Inter 6: 14439–14450. doi: 10.1021/am503750w
    [16] Fakhrullin RF, Zamaleeva AI, Minullina RT, et al. (2012) Cyborg cells: functionalisation of living cells with polymers and nanomaterials. Chem Soc Rev 41: 4189–4206. doi: 10.1039/c2cs15264a
    [17] Fakhrullin RF, Lvov YM (2012) "Face-Lifting" and "Make-Up" for Microorganisms: Layer-by-Layer Polyelectrolyte Nanocoating. ACS Nano 6: 4557–4564. doi: 10.1021/nn301776y
    [18] He W, Frueh J, Shao J, et al. (2016) Guidable GNR-Fe3O4-PEM@SiO2 composite particles containing near infrared active nanocalorifiers for laser assisted tissue welding. Colloid Surface A 511: 73–81. doi: 10.1016/j.colsurfa.2016.09.052
    [19] He W, Frueh J, Hu N, et al. (2016) Guidable Thermophoretic Janus Micromotors Containing Gold Nanocolorifiers for Infrared Laser Assisted Tissue Welding. Adv Sci 3: 1600206. doi: 10.1002/advs.201600206
    [20] Moskowitz JS, Blaisse MR, Samuel RE, et al. (2010) The effectiveness of the controlled release of gentamicin from polyelectrolyte multilayers in the treatment of Staphylococcus aureus infection in a rabbit bone model. Biomaterials 31: 6019–6030. doi: 10.1016/j.biomaterials.2010.04.011
    [21] Shukla A, Fang JC, Puranam S, et al. (2012) Release of vancomycin from multilayer coated absorbent gelatin sponges. J Control Release 157: 64–71. doi: 10.1016/j.jconrel.2011.09.062
    [22] Shukla A, Fuller RC, Hammond PT (2011) Design of multi-drug release coatings targeting infection and inflammation. J Control Release 155: 159–166. doi: 10.1016/j.jconrel.2011.06.011
    [23] Teng XR, Shchukin DG, Möhwald H (2008) A Novel Drug Carrier: Lipophilic Drug-Loaded Polyglutamate/Polyelectrolyte Nanocontainers. Langmuir 24: 383–389. doi: 10.1021/la702370k
    [24] Shchukina EM, Shchukin DG (2011) LbL coated microcapsules for delivering lipid-based drugs. Adv Drug Deliver Rev 63: 837–846. doi: 10.1016/j.addr.2011.03.009
    [25] González-Toro DC, Ryu JH, Chacko RT, et al. (2012) Concurrent binding and delivery of proteins and lipophilic small molecules using polymeric nanogels. J Am Chem Soc 134: 6964–6967. doi: 10.1021/ja3019143
    [26] Shukla A, Fleming KE, Chuang HF, et al. (2010) Controlling the release of peptide antimicrobial agents from surfaces. Biomaterials 31: 2348–2357. doi: 10.1016/j.biomaterials.2009.11.082
    [27] Huang Y, Luo Q, Li X, et al. (2012) Fabrication and in vitro evaluation of the collagen/hyaluronic acid PEM coating crosslinked with functionalized RGD peptide on titanium. Acta Biomater 8: 866–877. doi: 10.1016/j.actbio.2011.10.020
    [28] Palankar R, Skirtach AG, Kreft O, et al. (2009) Controlled intracellular release of peptides from microcapsules enhances antigen presentation on MHC class I molecules. Small 5: 2168–2176.
    [29] Zhang J, Chua LS, Lynn DM (2004) Multilayered thin films that sustain the release of functional DNA under physiological conditions. Langmuir 20: 8015–8021. doi: 10.1021/la048888i
    [30] Kim SH, Jeong JH, Lee SH, et al. (2008) Local and systemic delivery of VEGF siRNA using polyelectrolyte complex micelles for effective treatment of cancer. J Control Release 129: 107–116. doi: 10.1016/j.jconrel.2008.03.008
    [31] Cho HJ, Chong S, Chung SJ, et al. (2012) Poly-L-arginine and dextran sulfate-based nanocomplex for epidermal growth factor receptor (EGFR) siRNA delivery: its application for head and neck cancer treatment. Pharm Res 29: 1007–1019. doi: 10.1007/s11095-011-0642-z
    [32] Magboo R, Peterson AM (2016) Polyelectrolyte multilayers for controlled release of biologically relevant molecules. In: Somasundaran P, Encyclopedia of Surface and Colloid Science, Taylor and Francis, 1519–1533.
    [33] Shah NJ, Macdonald ML, Beben YM, et al. (2011) Tunable dual growth factor delivery from polyelectrolyte multilayer films. Biomaterials 32: 6183–6193. doi: 10.1016/j.biomaterials.2011.04.036
    [34] Crouzier T, Ren K, Nicolas C, et al. (2009) Layer-by-layer films as a biomimetic reservoir for rhBMP-2 delivery: Controlled differentiation of myoblasts to osteoblasts. Small 5: 598–608. doi: 10.1002/smll.200800804
    [35] Peterson AM, Pilz-Allen C, Kolesnikova T, et al. (2014) Growth factor release from polyelectrolyte-coated titanium for implant applications. ACS Appl Mater Inter 6: 1866–1871.
    [36] Gand A, Hindié M, Chacon D, et al. (2014) Nanotemplated polyelectrolyte films as porous biomolecular delivery systems: Application to the growth factor BMP-2. Biomatter 4: e28823. doi: 10.4161/biom.28823
    [37] Seon L, Lavalle P, Schaaf P, et al. (2015) Polyelectrolyte Multilayers: A Versatile Tool for Preparing Antimicrobial Coatings. Langmuir 31: 12856–12872. doi: 10.1021/acs.langmuir.5b02768
    [38] Grunlan JC, Choi JK, Lin A (2005) Antimicrobial behavior of polyelectrolyte multilayer films containing cetrimide and silver. Biomacromolecules 6: 1149–1153.
    [39] Lichter JA, Van Vliet KJ, Rubner MF (2009) Design of Antibacterial Surfaces and Interfaces: Polyelectrolyte Multilayers as a Multifunctional Platform. Macromolecules 42: 8573–8586. doi: 10.1021/ma901356s
    [40] Gomes AP, Mano JF, Queiroz JA, et al. (2012) Layer-by-Layer Deposition of Antibacterial Polyelectrolytes on Cotton Fibres. J Polym Environ 20: 1084–1094. doi: 10.1007/s10924-012-0507-5
    [41] Agarwal A, Nelson TB, Kierski PR, et al. (2012) Polymeric multilayers that localize the release of chlorhexidine from biologic wound dressings. Biomaterials 33: 6783–6792. doi: 10.1016/j.biomaterials.2012.05.068
    [42] Yang JM, Yang JH, Huang HT (2014) Chitosan/polyanion surface modification of styrene-butadiene-styrene block copolymer membrane for wound dressing. Mater Sci Eng C 34: 140–148. doi: 10.1016/j.msec.2013.09.001
    [43] Dai J, Bruening ML (2002) Catalytic Nanoparticles Formed by Reduction of Metal Ions in Multilayered Polyelectrolyte Films. Nano Lett 2: 497–501. doi: 10.1021/nl025547l
    [44] Rieger KA, Birch NP, Schiffman JD (2013) Designing electrospun nanofiber mats to promote wound healing-a review. J Mater Chem B 1: 4531.
    [45] Wood KC, Chuang HF, Batten RD, et al. (2006) Controlling interlayer diffusion to achieve sustained, multiagent delivery from layer-by-layer thin films. Proceedings of the National Academy of Sciences, 103: 10207–10212. doi: 10.1073/pnas.0602884103
    [46] Kiryukhin MV, Man SM, Tonoyan A, et al. (2012) Adhesion of polyelectrolyte multilayers: Sealing and transfer of microchamber arrays. Langmuir 28: 5678–5686. doi: 10.1021/la3003004
    [47] Kiryukhin MV, Gorelik SR, Man SM, et al. (2013) Individually addressable patterned multilayer microchambers for site-specific release-on-demand. Macromol Rapid Comm 34: 87–93. doi: 10.1002/marc.201200564
    [48] Schlenoff JB, Dubas ST, Farhat T (2000) Sprayed Polyelectrolyte Multilayers. Langmuir 16: 9968–9969. doi: 10.1021/la001312i
    [49] Patel PA, Dobrynin AV, Mather PT (2007) Combined effect of spin speed and ionic strength on polyelectrolyte spin assembly. Langmuir 23: 12589–12597. doi: 10.1021/la7020676
    [50] Kiel M, Mitzscherling S, Leitenberger W, et al. (2010) Structural characterization of a spin-assisted colloid-polyelectrolyte assembly: stratified multilayer thin films. Langmuir 26: 18499–18502. doi: 10.1021/la103609f
    [51] Andres CM, Kotov NA (2010) Inkjet deposition of layer-by-layer assembled films. J Am Chem Soc 132: 14496–14502. doi: 10.1021/ja104735a
    [52] Schmidt DJ, Moskowitz JS, Hammond PT (2011) Electrically Triggered Release of a Small Molecule Drug from a Polyelectrolyte Multilayer Coating. Chem Mater 22: 6416–6425.
    [53] Owens DK, Wendt RC (1969) Estimation of the surface free energy of polymers. J Appl Polym Sci 13: 1741–1747. doi: 10.1002/app.1969.070130815
    [54] Frutos Cabanillas P, Diez Pena E, Barrales-Rienda JM, et al. (2000) Validation and in vitro characterization of antibiotic-loaded bone cement release. Int J Pharmaceut 209: 15–26. doi: 10.1016/S0378-5173(00)00520-2
    [55] Tan H, Peng Z, Li Q, et al. (2012) The use of quaternised chitosan-loaded PMMA to inhibit biofilm formation and downregulate the virulence-associated gene expression of antibiotic-resistant staphylococcus. Biomaterials 33: 365–377. doi: 10.1016/j.biomaterials.2011.09.084
    [56] Peterson AM, Pilz-Allen C, Möhwald H, et al. (2014) Evaluation of the role of polyelectrolyte deposition conditions on growth factor release. J Mater Chem B 2: 2680–2687. doi: 10.1039/c3tb21757d
    [57] Liu TY, Lin YL (2010) Novel pH-sensitive chitosan-based hydrogel for encapsulating poorly water-soluble drugs. Acta Biomater 6: 1423–1429. doi: 10.1016/j.actbio.2009.10.010
    [58] Siepmann J, Peppas NA (2001) Modeling of drug release from delivery systems based on hydroxypropyl methylcellulose (HPMC). Adv Drug Deliver Rev 48: 139–157. doi: 10.1016/S0169-409X(01)00112-0
    [59] von Klitzing R (2006) Internal structure of polyelectrolyte multilayer assemblies. Phys Chem Chem Phys 8: 5012–5033. doi: 10.1039/b607760a
    [60] Ghostine RA, Markarian MZ, Schlenoff JB (2013) Asymmetric growth in polyelectrolyte multilayers. J Am Chem Soc 135: 7636–7646. doi: 10.1021/ja401318m
    [61] Schoeler B, Sharpe S, Hatton TA, et al. (2004) Polyelectrolyte multilayer films of different charge density copolymers with synergistic nonelectrostatic interactions prepared by the layer-by-layer technique. Langmuir 20: 2730–2738. doi: 10.1021/la035909k
    [62] Armstrong G, Buggy M (2005) Hydrogen-bonded supramolecular polymers: A literature review. J Mater Sci 40: 547–559. doi: 10.1007/s10853-005-6288-7
    [63] Abidi N, Hequet E, Cabrales L (2011) Applications of Fourier transform infrared spectroscopy to study cotton fibers, In: Nikolic G, Fourier Transforms-New Analytical Approaches and FTIR Strategies, InTech, 90–114.
    [64] Moulds RFW, Jeyasingham MS (2010) Gentamicin: A great way to start. Aust Prescr 33: 35–37.
    [65] Schönhoff M, Ball V, Bausch AR, et al. (2007) Hydration and internal properties of polyelectrolyte multilayers. Colloid Surface A 303: 14–29.
    [66] Hu N, Frueh J, Zheng C, et al. (2015) Photo-crosslinked natural polyelectrolyte multilayer capsules for drug delivery. Colloid Surface A 482: 315–323.
    [67] Salvi C, Lyu X, Peterson AM (2016) Effect of assembly pH on polyelectrolyte multilayer surface properties and BMP-2 release. Biomacromolecules 17: 1949–1958. doi: 10.1021/acs.biomac.5b01730
    [68] Schafer T, Pascale A, Shimonaski G, et al. (1972) Evaluation of gentamicin for use in virology and tissue culture. Appl Microbiol 23: 565–570.
    [69] Berg MC, Zhai L, Cohen RE, et al. (2006) Controlled drug release from porous polyelectrolyte multilayers. Biomacromolecules 7: 357–364. doi: 10.1021/bm050174e
    [70] Siepmann J, Peppas NA (2011) Higuchi equation: derivation, applications, use and misuse. Int J Pharmaceut 418: 6–12. doi: 10.1016/j.ijpharm.2011.03.051
  • This article has been cited by:

    1. Elio Chiodo, Pasquale De Falco, Luigi Pio Di Noia, Fabio Mottola, Inverse Log-logistic distribution for Extreme Wind Speed modeling: Genesis, identification and Bayes estimation, 2018, 6, 2333-8334, 926, 10.3934/energy.2018.6.926
    2. Andrés Feijóo, Daniel Villanueva, Assessing wind speed simulation methods, 2016, 56, 13640321, 473, 10.1016/j.rser.2015.11.094
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5985) PDF downloads(1085) Cited by(7)

Article outline

Figures and Tables

Figures(8)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog