工程地质学报  2018, Vol. 26 Issue (5): 1237-1242   (#KB#)    
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  • 收稿日期:2018-05-14
  • 接受日期:2018-07-18
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    徐永福

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    徐永福. 2018. 基于膨润土凝胶分形结构的屈服强度理论[J]. 工程地质学报, 26(5): 1237-1242. doi: 10.13544/j.cnki.jeg.2018071.
    XU Yongfu. 2018. Fractal model for yield stress of bentonite colloids[J]. Journal of Engineering Geology, 26(5): 1237-1242. doi: 10.13544/j.cnki.jeg.2018071.

    基于膨润土凝胶分形结构的屈服强度理论
    徐永福①②    
    ① 上海交通大学土木工程系 上海 200240;
    ② 皖江工学院 马鞍山 243000
    摘要:在核废料处置库安全使用的设计年限(数万年至数十万年)内,膨润土遇水侵蚀,导致缓冲/回填层致密性降低、渗透性增加,危及核废料处置库安全。膨润土侵蚀是以膨润土凝胶形式迁移,膨润土凝胶的屈服强度就是侵蚀的临界剪切应力。膨润土凝胶颗粒之间的联结靠颗粒间的长程作用,即van der Waals力,颗粒间的联结作用取决于凝胶的结构。本文基于膨润土凝胶结构的分形模型,假设凝胶的屈服强度等于冲刷面上单位面积的van der Waals力的总和,导出了膨润土凝胶的屈服强度(σy)的表达式,表示为凝胶的固体体积率(φs)的幂函数,即σy=σy0φsm,幂函数的指数是凝胶结构分维的函数。膨润土凝胶的屈服强度理论得到了试验数据的验证。
    关键词膨润土    侵蚀    凝胶    屈服强度    分维    
    FRACTAL MODEL FOR YIELD STRESS OF BENTONITE COLLOIDS
    XU Yongfu①②    
    ① Department of Civil Engineering, Shanghai Jiaotong University, Shanghai 200240;
    ② Wanjiang Institute of Technology, Maanshan 243000
    Abstract: The performance of the bentonite buffer in nuclear waste repository concept relies to a great extent on the buffer surrounding the canister having sufficient dry density. Loss of buffer material caused by erosion remains as the most significant process reducing the density of the buffer. Yield stress of bentonite colloids is a key parameter to express the erosion process of bentonite aqueous solution. A model for the yield stress of aggregates is presented. It incorporates fractal dimension taking into account the solid volume fraction and the aggregate diameter. The model shows that the yield stress(σy) of aggregates increases with the solid volume fraction (φs) as a power law, and is given by σy=σy0φsm, where the exponent(m) is related to fractal dimension(D), and σy0 is a referenced parameter. The relationship between exponent(m) and fractal dimension is validated by published data of aggregates and represents the measured data very well, over a wide range of the solid volume fractions. The referenced parameter(σy0) is calibrated from experiments of yield stress using power law fittings. The agreement between theory and experiments supports the idea that yielding is ultimately caused by the rupture of a few interparticle bonds within aggregates.
    Key words: Bentonite    Erosion    Colloid    Yield strength    Fractal dimension    

    0 引言

    因其具有渗透性低、膨胀性大和吸附性强的特点,压实膨润土被广泛用于高放废物处置库的理想缓冲/回填材料(秦冰等,2008叶为民等,2009孙德安等,2013姜昊等,2014项国圣等,2015)。膨润土遇水产生侵蚀,分为化学侵蚀和物理侵蚀,膨润土的化学侵蚀是凝胶的形成过程,物理侵蚀是凝胶随水的迁移铛(Xu et al., 2014a, 2014b, 2016, 2018; 徐永福,2017)。高放废物处置库缓冲/回填层在膨润土侵蚀过程中,密实度变小,水渗流速度增加,可能导致放射性核元素随流水迁移泄漏,导致核素泄漏的灾害(Birgersson et al., 2009)。

    膨润土侵蚀在静水和渗流环境下都可能发生(Missana et al., 2011Xu et al., 2016)。在静水环境下,膨润土表现为自由膨胀,形成膨润土絮状凝胶,出现化学侵蚀现象。化学侵蚀与膨润土类型和裂隙水的离子类型和浓度有关。遇到渗流流速大的地下水环境,膨润土凝胶剥离缓冲/回填层表面,产生物理侵蚀(Missana et al., 2011)。凝胶的物理侵蚀速率表示为:E=M(σ-σy),这里E是侵蚀速率,M是侵蚀速率常数,σσy分别是流水产生的剪切应力和屈服强度(又称为临界剪切应力)(Pusch,1999Sanford,2008)。发生物理侵蚀的前提是凝胶表面的剪切应力超过凝胶屈服强度(Xu et al., 2014a, 2014b)。

    膨润土凝胶的屈服强度取决于凝胶的密实程度。凝胶的密实程度用固体体积率表示(Xu et al., 2014a, 2014b徐永福,2017)。膨润土凝胶的结构可以采用分形模型描述(Xu et al., 2001, 2003, 2004a, 2014b, 2006),凝胶的屈服强度与固体体积率的关系用幂函数表示(Shih et al., 1990de Rooij et al., 1993, 1994Kranenburg,1994Uriev et al., 1996Piau et al., 1999Son et al., 2009Zhou et al., 2009Studart et al., 2011Nasser et al., 2012Kataoka et al., 2013Xu et al., 2014a, 2014bXi et al., 2015):

    $ {\sigma _y} = {\sigma _{y0}}\varphi _s^m $ (1)

    式中,σy为屈服强度;φs为固体体积率;σy0为参考参数,为φs=1的屈服强度;m为常数,列于表 1。表中D是凝胶结构的分维;d为欧拉维数,0<ε<1。从凝胶结构的分形模型可以导出m的表达式,但表达式差别很大(Xu et al., 2014a, 2014bXi et al., 2015徐永福,2017);另外,与X-射线小角度衍射实测的凝胶分维不一致(Franks et al., 2004Zhou et al., 2009)。针对凝胶屈服强度的研究现状,基于膨润土凝胶结构和颗粒间联结数的分形模型,假设屈服强度等于冲刷面上单位面积的联结力总和,导出膨润土凝胶的屈服强度,采用凝胶屈服强度和凝胶结构分维的试验数据验证凝胶的屈服强度理论。本文提出的凝胶的屈服强度理论是基于凝胶结构的分形模型,是当前各种屈服强度的统一理论,具有普适性,已被试验数据验证。

    表 1 参数m的表达式 Table 1 Expression of parameter m

    1 凝胶的分形模型

    膨润土遇到水溶液,由于黏土表面带电荷,颗粒间存在电作用,形成絮状凝胶,具有自相似的分形结构(Meakin,1987)。膨润土凝胶的嵌套结构如图 1所示(Krone,1963)。膨润土颗粒在电的吸附力和黏结力作用下,联结形成一级团粒;一级团粒在黏结力和吸附力作用下,联结形成二级团粒。以此类推,联结形成三级团粒、四级团粒等更大的团粒。假设各级团粒孔隙比不变,凝胶的颗粒数和粒径的关系为(Xu et al., 2003, 2014a, 2014bXu et al., 2018):

    $ N = \frac{{cd_c^D}}{{{f_c}{\rho _p}d_p^3}}d_P^{ - D} $ (2)

    图 1 黏性团粒的自相似嵌套结构(Krone,1963) Fig. 1 Fractal structure of cohesive aggregate(Krone, 1963)

    式中,N为颗粒数;c为凝胶的浓度(kg·m-3);fc为形状因子,对于球状凝胶,fc=π/6;ρp为颗粒的密度;dp为颗粒的粒径;dc为凝胶的粒径;D为凝胶的分维。球状凝胶的固体体积率定义为φs=dc3/(Ndp3),膨润土凝胶的固体体积率表示为:

    $ {\varphi _s} = \frac{c}{{{f_c}{\rho _p}}}{\left( {\frac{{{d_c}}}{{{d_p}}}} \right)^{D - 3}} $ (3)

    式中,φs为凝胶的固体体积率。凝胶的密度是固体颗粒的密度与水的密度的平均值,ρc=φsρp+(1-φs)ρw;膨润土凝胶的有效密度(ρc-ρw)为:

    $ \frac{{{\rho _c} - {\rho _w}}}{{{\rho _p} - {\rho _w}}} = \frac{c}{{{f_c}{\rho _p}}}{\left( {\frac{{{d_c}}}{{{d_p}}}} \right)^{D - 3}} $ (4)

    式中,ρc为膨润土凝胶的密度;ρw为水的密度。

    Kunigel V1膨润土在膨胀变形试验中,形成膨润土凝胶,凝胶的固体体积率和有效密度与膨胀后试样尺寸的关系如图 2所示。图中L为试样膨胀后的试样尺寸,与凝胶颗粒粒径之比为常数,即L=ndc图 2中分别表示了膨润土凝胶的固体体积率和有效密度与试样尺寸的相关关系,相关直线的斜率为D-3,图中直线斜率为- 0.85,因此膨润土凝胶的分维为2.15。图 2中,φs-L和(ρc-ρp)-L的相关关系验证了膨润土凝胶结构的分形模型。

    图 2 Kunigel V1凝胶的分维 Fig. 2 Fractal dimension of Kunigel V1 colloids

    图 3 凝胶结构的散射试验结果(Zhou et al., 2006) Fig. 3 SAXS test results of colloids(Zhou et al., 2006) a.掺入10%带电聚合物;b.掺入40%带电聚合物

    X-射线是测量凝胶结构分维的最常用的方法,根据Rayleigh-Gans-Debye准则,X-射线的散射强度(I)与散射矢量的幅值(Q)满足(Schaefer et al., 1984Sinkó et al., 2008):

    $ I\left( Q \right) \propto {Q^{ - D}} $ (5)

    式中,Q为散射矢量,Q=4πsinθ/λθ为散射角;λ为波长。式(5)适用条件是:2/dcQ<2/dp

    Zhou et al.(2006)采用He-Ne激光作为光源,利用Mastersizer散射仪测量了氧化硅凝胶的分维。散射角介于0°~46°,氧化硅颗粒粒径为90 nm。图 3a图 3b分别掺入了10%和40%带电聚合物。对于分别掺入了10%和40%带电聚合物的氧化硅凝胶的分维分别为2.4和2.6。

    2 凝胶的屈服强度

    基于van der Waals长程吸引力机理,Boller et al.(1998)给出凝胶颗粒间的联结力的表达式:

    $ {F_H} = \frac{{{A_H}C}}{{24{K^2}}}\frac{1}{{{d_p}}} $ (6)

    式中,AH为Hamaker常数,≈10-20J;C为压实因子,=π/6;K为压实方程;dp为颗粒的粒径。

    在冲刷面上,凝胶颗粒间的联结键的数目为(Son et al., 2009):

    $ n = \frac{{\rm{ \mathsf{ π} }}}{4}{\left( {\frac{{\rm{ \mathsf{ π} }}}{6}} \right)^{ - \frac{2}{3}}}{\left( {\frac{{{d_c}}}{{{d_p}}}} \right)^{\frac{{2X}}{3}}} $ (7)

    式中,n为联结键的数目;X为凝胶骨架的分维,1<XD。因此,在冲刷面上,凝胶颗粒间的联结力总和由式(6)和式(7)的乘积得到,即:

    $ {F_c} = n{F_H} = {\left( {\frac{{9{\rm{ \mathsf{ π} }}}}{{16}}} \right)^{\frac{1}{3}}}\frac{{{A_H}C}}{{24K}}\frac{1}{{{d_p}}}{\left( {\frac{{{d_c}}}{{{d_p}}}} \right)^{\frac{{2X}}{3}}} $ (8)

    基于凝胶的分形结构特性,冲刷面积表示为(Xu et al., 2014a, 2014bXi et al., 2015徐永福,2017):

    $ {A_c} = {C_c}{\left( {\frac{{{d_c}}}{{{d_p}}}} \right)^D} $ (9)

    式中,Ac为冲刷面积;Cc为凝胶截面形状系数。由式(8)和式(9),凝胶的屈服强度为:

    $ {\sigma _y} = \frac{{{F_c}}}{{{A_c}}} = {\sigma _{y0}}{\left( {\frac{{{d_c}}}{{{d_p}}}} \right)^{ - \frac{{2X - 3D}}{3}}} $ (10)

    式中,σy0=[AHC(9π/16)1/3]/(24KCcdp),为φs=1时的屈服强度。由式(10)和式(3),凝胶的屈服强度用固体体积率表示为式(1)的形式,其中参数m为:

    $ m = \frac{{2X - 3D}}{{3\left( {D - 3} \right)}} $ (11)

    X=D,式(11)与Xu et al.(2014a, 2014b)的结果相同;当X=5D/2-3,m=2/3,式(11)与Son et al. (2009)Nasser et al. (2012)的结果相同。

    参数m随分维D的变化规律如图 4所示,参数m随分维D增加而增大;分维D大于2.5以后,参数m随分维增加而快速增大。随着多孔絮状凝胶的粒径增加,分维减小,固体体积率减小,屈服强度减小。因此,参数m随分维D增加而增大。

    图 4 m值随分维D的变化 Fig. 4 Variation of m with D

    Woignier et al. (1998)采用X-射线小角度衍射法测量了氧化硅凝胶结构的分维,并测量了凝胶的屈服强度(图 5)。图 5a是氧化硅凝胶结构的X-射线小角度衍射试验结果,凝胶结构的分维为2.4。

    图 5 氧化硅凝胶的屈服强度的预测结果与试验结果比较 Fig. 5 Comparison of prediction with experiments of yield strength of silica aerogels a.氧化硅凝胶的分维;b.氧化硅凝胶的屈服强度

    一般情况下,凝胶的骨架分维小于凝胶结构的分维。为了简单起见,假设凝胶的骨架分维(X)与凝胶结构的分维(D)的关系为:

    $ X = D - 1 $ (12)

    参数σy0按下列公式计算:

    $ {\sigma _{y0}} = \sum\limits_{i = 1}^n {\frac{{{\sigma _{yi}}}}{{\varphi _{si}^m}}/n} $ (13)

    式中,n为试验个数。根据氧化硅凝胶结构的分维,按照式(11)~式(13),计算参数mσy0: m=2.44和σy0=17 MPa。按照式(1)和式(11)计算氧化硅凝胶的屈服强度,屈服强度的计算值与试验值比较于图 5b中。按照式(1),氧化硅凝胶屈服强度的计算值与试验值符合得很好。

    Zhou et al. (2009)采用X-射线小角度衍射法测量了膨润土凝胶结构的分维,给出了凝胶屈服强度与固体体积率的关系参数m值。基于式(12)的假设,根据式(11)由参数m值算出凝胶结构的分维。膨润土凝胶结构的分维的计算值与由X-射线小角度衍射法测量的分维比较于图 6中。由图 6中看出,de Rooij et al. (1993)Uriev et al. (1996)Zhou et al. (2009)Piau et al. (1999)给出的分维计算值与分维的测量值差别很大;相对于de Rooij et al. (1993)Uriev et al. (1996)Zhou et al. (2009)Piau et al. (1999)Xu et al. (2014)Kataoka et al. (2013)计算的分维值与测量值比较接近。由式(11)反算的凝胶分维与测量值基本一致,分维计算结果最理想。

    图 6 分维的计算值与试验值的比较 Fig. 6 Comparison of prediction with experiments of fractal dimension

    3 结论

    本文得到以下几个结论:

    (1) 基于凝胶结构的分形模型,建立了膨润土凝胶的屈服强度的计算方法,表示为凝胶粒径和固体体积率的幂函数,给出屈服强度公式中参数的计算方法。

    (2) 引进了凝胶的骨架分维(X),假设凝胶的骨架分维与凝胶结构分维(D)的关系为X=D-1。

    (3) 在膨润土凝胶结构的屈服强度的分形模型中,凝胶结构的分维是唯一的计算参数,膨润土凝胶结构的分维采用X-射线小角度衍射法测量,凝胶结构的分维介于0~3之间。

    (4) 膨润土凝胶的屈服强度公式得到了试验数据的广泛验证。

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