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Size exclusion chromatography with superficially porous particles
超微孔颗粒尺寸排除色谱法

Mark R. Schure a , a , ^(a,**){ }^{\mathrm{a}, *}, Robert E. Moran b ^("b "){ }^{\text {b }}a Theoretical Separation Science Laboratory, Kroungold Analytical, Inc., 1299 Butler Pike, Blue Bell, PA, 19422 USA b b ^(b){ }^{\mathrm{b}} Advanced Materials Technology, Inc., 3521 Silverside Road, Suite 1-K, Quillen Building, Wilmington, DE, 19810, USA
b b ^(b){ }^{\mathrm{b}} Advanced Materials Technology, Inc.,3521 Silverside Road, Suite 1-K, Quillen Building, Wilmington, DE, 19810, USA

A R T IC LE IN F O
在 F O

Article history: 文章历史:

Received 24 September 2016
2016 年 9 月 24 日收到

Received in revised form 7 November 2016
2016 年 11 月 7 日收到修订稿

Accepted 9 December 2016 2016 年 12 月 9 日接受
Available online 9 December 2016
2016 年 12 月 9 日在线提供

Keywords: 关键词:

Size-exclusion chromatography
尺寸排阻色谱法

Core-shell particles 核壳粒子
Superficially porous particles
表面多孔的颗粒

Efficiency 效率
Pore diffusion 孔隙扩散
Peak capacity 峰值容量

Abstract 摘要

A comparison is made using size-exclusion chromatography (SEC) of synthetic polymers between fully porous particles (FPPs) and superficially porous particles (SPPs) with similar particle diameters, pore sizes and equal flow rates. Polystyrene molecular weight standards with a mobile phase of tetrahydrofuran are utilized for all measurements conducted with standard HPLC equipment.
本研究使用尺寸排阻色谱法(SEC)对具有相似颗粒直径、孔径和相同流速的全多孔颗粒(FPPs)和超多孔颗粒(SPPs)的合成聚合物进行了比较。所有测量均采用标准 HPLC 设备,以四氢呋喃为流动相的聚苯乙烯分子量标准。

Although it is traditionally thought that larger pore volume is thermodynamically advantageous in SEC for better separations, SPPs have kinetic advantages and these will be shown to compensate for the loss in pore volume compared to FPPs. The comparison metrics include the elution range (smaller with SPPs), the plate count (larger for SPPs), the rate production of theoretical plates (larger for SPPs) and the specific resolution (larger with FPPs). Advantages to using SPPs for SEC are discussed such that similar separations can be conducted faster using SPPs.
尽管传统观点认为,在 SEC 中,较大的孔隙率在热力学上有利于实现更好的分离,但 SPPs 具有动力学优势,与 FPPs 相比,这些优势将被证明可以弥补孔隙率的损失。比较指标包括洗脱范围(SPPs 较小)、板数(SPPs 较多)、理论板速率(SPPs 较多)和特定分辨率(FPPs 较多)。讨论了使用 SPPs 进行 SEC 的优势,即使用 SPPs 可以更快地进行类似的分离。

SEC using SPPs offers similar peak capacities to that using FPPs but with faster operation. This also suggests that SEC conducted in the second dimension of a two-dimensional liquid chromatograph may benefit with reduced run time and with equivalently reduced peak width making SPPs advantageous for sampling the first dimension by the second dimension separator. Additional advantages are discussed for biomolecules along with a discussion of optimization criteria for size-based separations.
使用 SPP 的 SEC 与使用 FPP 的 SEC 具有相似的峰容量,但运行速度更快。这也表明,在二维液相色谱仪的二维中进行 SEC 可缩短运行时间并相应减少峰宽,从而使 SPPs 在通过二维分离器对一维进行采样时更具优势。此外,还讨论了生物大分子的其他优势,以及基于粒度分离的优化标准。

© 2016 Elsevier B.V. All rights reserved.
© 2016 Elsevier B.V. 版权所有。保留所有权利。

1. Introduction 1.导言

Size exclusion chromatography (SEC) is a form of chromatography that separates molecules by size. It is also known as gel-filtration chromatography (GFC) and gel permeation chromatography (GPC) which often refers to SEC with organic solvents. The primary applications of this technique are polymer and medium to large biomolecule separations. The performance characteristics of SEC are not on the scale of the efficiencies typical of columns used for reversed-phase or normal-phase separations. However, SEC continues to be highly utilized for separations where fractionation can be accomplished based on molecular size.
尺寸排阻色谱法(SEC)是一种按尺寸分离分子的色谱法。它也被称为凝胶过滤色谱法(GFC)和凝胶渗透色谱法(GPC),通常指使用有机溶剂的 SEC。这种技术的主要应用领域是聚合物和大中型生物大分子分离。SEC 的性能特点与反相或正相分离柱的典型效率不同。不过,在可以根据分子大小进行分馏的分离过程中,SEC 仍有很高的利用率。
The first SEC columns were made of starch [1,2], however, dextran gels were soon developed for biochemical applications [3]. After this initial introduction, cross-linked polystyrene was introduced [4] as a material useful for fractionating industrial polymers. Silica is also often utilized as a porous medium and complements other media, especially when small particles are used to minimize zone broadening and provide a stable mechanical particle for use
最早的 SEC 色谱柱是由淀粉制成的[1,2],但很快就开发出了用于生化应用的葡聚糖凝胶[3]。之后,交联聚苯乙烯作为一种可用于分馏工业聚合物的材料问世[4]。二氧化硅也经常被用作多孔介质,与其他介质相辅相成,特别是当使用小颗粒时,可最大限度地减少区域扩展,并提供稳定的机械颗粒,以便使用
at higher pressures [5,6]. In most cases, particles are used for SEC although polymer monoliths have been reported that can function in the size-exclusion mode [7].
5,6] 。大多数情况下,SEC 使用颗粒,但也有报道称聚合物单片可在尺寸排除模式下发挥作用 [7]。
In all cases where particles are used for SEC, the particle is made of organic or inorganic materials and is a fully porous particle (FPP) morphology. Superficially porous particles (SPPs) [8-13], also known as core-shell particles, offer the performance of a smaller diameter particle, for example a sub 2 μ m 2 μ m 2-mum2-\mu \mathrm{m} diameter particle, with the smaller pressure drop of a standard-sized particle, for example a 2.7 μ m a 2.7 μ m a >= 2.7 mum\mathrm{a} \geq 2.7 \mu \mathrm{~m} particle. The SPP morphology has become a serious choice for high performance chromatographic media. In the early development of SPP materials, specifically from 1978, it was written by one of the developers of the SPP technology [14]: “However, it is anticipated that, while wide-linear calibrations should result, resolution of polymers with these particular particles would be relatively poor because of their low specific porosity.”
在所有将颗粒用于 SEC 的情况中,颗粒都是由有机或无机材料制成,并具有全多孔颗粒 (FPP) 形态。表面多孔颗粒 (SPP)[8-13],也称为核壳颗粒,具有较小直径颗粒(例如直径小于 2 μ m 2 μ m 2-mum2-\mu \mathrm{m} 的颗粒)的性能,同时具有标准尺寸颗粒(例如 a 2.7 μ m a 2.7 μ m a >= 2.7 mum\mathrm{a} \geq 2.7 \mu \mathrm{~m} 的颗粒)的较小压降。SPP 形态已成为高性能色谱介质的重要选择。在 SPP 材料的早期开发中,特别是从 1978 年开始,SPP 技术的开发者之一写道[14]:"不过,预计虽然会产生宽线性定标,但由于这些特定颗粒的比孔隙率较低,因此聚合物的分辨率会相对较差"。
The assumption inherent in this quote is that it is necessary to have as much pore volume as possible to get maximum performance from the SEC technique. This assumption is partly driven by the low peak capacity inherent in the SEC technique where getting a 10 peak separation would be considered a large number of peaks [15-17]. This is in contrast to getting peak counts greater than 60 which is not uncommon [18] using the partition
这句话的内在假设是,有必要拥有尽可能多的孔容积,以获得 SEC 技术的最大性能。这一假设的部分原因是 SEC 技术的固有峰容量较低,能分离出 10 个峰值就已经算是大量峰值了 [15-17]。与此形成鲜明对比的是,使用分区技术获得大于 60 个峰值的情况并不少见[18]。

and adsorptive retention mechanisms inherent in reversed-phase liquid chromatography [19].
和反相液相色谱固有的吸附保留机制 [19]。
Recent theoretical work [20,21] has concluded that reducing the pore volume by using a SPP for SEC would not be especially deleterious for large-sized solutes with slow diffusional characteristics. This work highlighted that the loss of pore volume can be compensated by shortening the diffusion length of the solutes inside the pore and theoretical elution curves which showed this effect were given in detail [20,21]. These results suggest that SPPs could be used successfully for SEC.
最近的理论研究[20,21]得出结论,在 SEC 中使用 SPP 减少孔隙体积对具有缓慢扩散特性的大体积溶质不会造成特别严重的影响。这项工作强调,孔体积的损失可以通过缩短孔内溶质的扩散长度来补偿,并详细给出了显示这种效果的理论洗脱曲线[20,21]。这些结果表明,SPPs 可成功用于 SEC。
From the thermodynamic point of view, one will spread the SEC chromatogram across a greater range of retention volume V R V R V_(R)V_{R} when the pore volume is increased. This can easily be shown by Eq (1)
从热力学角度来看,当孔隙体积增大时,SEC 色谱图将在更大的保留体积 V R V R V_(R)V_{R} 范围内展开。公式 (1) 很容易说明这一点

V R = V 0 + K V i V R = V 0 + K V i V_(R)=V_(0)+KV_(i)V_{R}=V_{0}+K V_{i}
where V 0 V 0 V_(0)V_{0} is the interstitial pore volume, V i V i V_(i)V_{i} is the particle pore volume and K K KK is the distribution constant such that K K KK varies from an excluded zone ( = 0 ) ( = 0 ) (=0)(=0) when the solute is larger than the pore to a fully included zone ( = 1 = 1 =1=1 ) when the solute is very much smaller than the pore. As the volume of internal pores V i V i V_(i)V_{i} increases, the retention volume, V R V R V_(R)V_{R}, will also increase. This shows that the range of separation will be increased by a larger pore volume. SPPs have less pore volume than FPPs so SPPs are at a thermodynamic disadvantage for SEC. This was thought to be a limitation of SPPs for use as SEC materials, as mentioned previously.
其中 V 0 V 0 V_(0)V_{0} 是间隙孔隙体积, V i V i V_(i)V_{i} 是颗粒孔隙体积, K K KK 是分布常数,因此 K K KK 会从溶质大于孔隙时的排除区 ( = 0 ) ( = 0 ) (=0)(=0) 变化到溶质远小于孔隙时的完全包含区( = 1 = 1 =1=1 )。随着内部孔隙 V i V i V_(i)V_{i} 体积的增大,截留体积 V R V R V_(R)V_{R} 也会增大。这表明,孔隙体积越大,分离范围就越大。SPP 的孔体积小于 FPP,因此 SPP 在 SEC 中处于热力学劣势。如前所述,这被认为是 SPP 用作 SEC 材料的一个限制因素。
The reduced pore length, inherent in the SPP shell, has specific advantages in reducing the resistance to mass transport in the particle and faster transport kinetics provide for higher efficiency. This is also important because slow diffusional properties of large molecules that are typically separated with SEC, are compounded by the additional slowing when the solute molecular size is on the order of the pore size [ 22 , 23 ] [ 22 , 23 ] [22,23][22,23]. Hence, the experimental determination of whether faster diffusional kinetics can overcome thermodynamic limitations is long overdue. Some of these issues were recently discussed [24] in the experimental demonstration of SEC with SPPs. In one case, SEC has been reported [25] to occur without pores.
SPP 外壳固有的孔隙长度缩短,在减少颗粒内的质量传输阻力方面具有独特的优势,而且更快的传输动力学提供了更高的效率。这一点也很重要,因为通常使用 SEC 分离的大分子的扩散特性较慢,而当溶质分子大小与孔径 [ 22 , 23 ] [ 22 , 23 ] [22,23][22,23] 大小相当时,扩散特性会变得更加缓慢。因此,早就应该通过实验确定更快的扩散动力学是否能克服热力学限制。最近[24]在用 SPPs 演示 SEC 的实验中讨论了其中的一些问题。据报道[25],在一种情况下,SEC 是在没有孔隙的情况下发生的。
In this paper we explore the use of SPP technology for SEC in the context of the interplay between particle pore volume and the efficiency of separation. It will be shown that for small and wide-pore materials that SEC with SPPs can deliver a faster separation while retaining most of the resolution of a FPP. This compromise will be suggested to be extremely important in two-dimensional separations where speed constraints often apply in the second dimension separation system. We also discuss some aspects of bioseparations pertinent to SEC using SPPs where adsorptive and partitioning mechanisms of retention exist alongside with the size-exclusion mechanism.
在本文中,我们将从粒子孔体积与分离效率之间的相互作用角度,探讨如何将 SPP 技术用于 SEC。结果表明,对于小孔和宽孔材料,使用 SPP 的 SEC 分离速度更快,同时保留了 FPP 的大部分分辨率。这种折衷方法对于二维分离极为重要,因为在二维分离系统中,速度往往受到限制。我们还讨论了与使用 SPPs 的 SEC 有关的生物分离的一些方面,其中除了尺寸排阻机制外,还存在吸附和分区保留机制。

2. Experimental and data processing conditions
2.实验和数据处理条件

2.1. Particles and columns
2.1.颗粒和柱

FPPs were obtained from the Osaka Soda Co, (Osaka, Japan). The two silica-based FPPs used in this study are SP-200-3-P and SP-1000-3 which are both of diameter 3.2 μ m 3.2 μ m 3.2 mum3.2 \mu \mathrm{~m} and had pore sizes nominally of 200 200 200"Å"200 \AA and 1000 1000 1000"Å"1000 \AA respectively with pore volumes of 1.06 cc / g 1.06 cc / g 1.06cc//g1.06 \mathrm{cc} / \mathrm{g} and 0.80 cc / g 0.80 cc / g 0.80cc//g0.80 \mathrm{cc} / \mathrm{g} respectively, as provided by the manufacturer. SPPs, also made of silica, are from Advanced Materials Technology (Wilmington, Delaware, USA) and have nominal pore sizes of 160 160 160"Å"160 \AA and 1000 1000 1000"Å"1000 \AA. These particles have outer diameters of 2.7 μ m 2.7 μ m 2.7 mum2.7 \mu \mathrm{~m} and 4.18 μ m 4.18 μ m 4.18 mum4.18 \mu \mathrm{~m} respectively and have solid core diameters of 1.7 μ m 1.7 μ m 1.7 mum1.7 \mu \mathrm{~m} and 3.3 μ m 3.3 μ m 3.3 mum3.3 \mu \mathrm{~m} respectively. These particles had pore volumes of 0.29 cc / g 0.29 cc / g 0.29cc//g0.29 \mathrm{cc} / \mathrm{g} and 0.20 cc / g 0.20 cc / g 0.20cc//g0.20 \mathrm{cc} / \mathrm{g} for the 160 160 160"Å"160 \AA and 1000 1000 1000"Å"1000 \AA particles, as measured in-house using nitrogen adsorption methods. The pore size
FPP 取自大阪纯碱公司(日本大阪)。根据制造商提供的资料,本研究中使用的两种硅基 FPP 为 SP-200-3-P 和 SP-1000-3,直径均为 3.2 μ m 3.2 μ m 3.2 mum3.2 \mu \mathrm{~m} ,孔径分别为 200 200 200"Å"200 \AA 1000 1000 1000"Å"1000 \AA ,孔体积分别为 1.06 cc / g 1.06 cc / g 1.06cc//g1.06 \mathrm{cc} / \mathrm{g} 0.80 cc / g 0.80 cc / g 0.80cc//g0.80 \mathrm{cc} / \mathrm{g} 。SPP 也由二氧化硅制成,来自先进材料技术公司(美国特拉华州威尔明顿),标称孔径为 160 160 160"Å"160 \AA 1000 1000 1000"Å"1000 \AA 。这些颗粒的外径分别为 2.7 μ m 2.7 μ m 2.7 mum2.7 \mu \mathrm{~m} 4.18 μ m 4.18 μ m 4.18 mum4.18 \mu \mathrm{~m} ,实心直径分别为 1.7 μ m 1.7 μ m 1.7 mum1.7 \mu \mathrm{~m} 3.3 μ m 3.3 μ m 3.3 mum3.3 \mu \mathrm{~m} 。根据内部使用氮吸附方法测量的结果,这些颗粒的孔隙体积分别为 0.29 cc / g 0.29 cc / g 0.29cc//g0.29 \mathrm{cc} / \mathrm{g} 0.20 cc / g 0.20 cc / g 0.20cc//g0.20 \mathrm{cc} / \mathrm{g} ,其中 160 160 160"Å"160 \AA 1000 1000 1000"Å"1000 \AA 颗粒的孔隙体积为 0.29 cc / g 0.29 cc / g 0.29cc//g0.29 \mathrm{cc} / \mathrm{g} 0.20 cc / g 0.20 cc / g 0.20cc//g0.20 \mathrm{cc} / \mathrm{g} 。孔径

distribution of the 160 160 160"Å"160 \AA SPPs was narrow and measured in-house. The 1000 1000 1000"Å"1000 \AA SPPs have a have wider distribution and this is shown in a recent paper [26]. The pore size distributions of the 200 200 200"Å"200 \AA and 1000 1000 1000"Å"1000 \AA FPPs were not supplied by the manufacturer nor determined in-house. All particles of both morphologies and both pore sizes were packed into columns of dimension 4.6 mm i.d. and length 50 mm using an in-house developed proprietary packing process.
160 160 160"Å"160 \AA SPP 的分布很窄,是在内部测量的。而 1000 1000 1000"Å"1000 \AA SPP 的分布范围更广,这在最近的一篇论文[26]中有所体现。 200 200 200"Å"200 \AA 1000 1000 1000"Å"1000 \AA FPP 的孔径分布既不是由制造商提供的,也不是内部测定的。两种形态和两种孔径的所有颗粒均采用内部开发的专有填料工艺填入内径 4.6 毫米、长 50 毫米的柱中。

2.2. HPLC conditions 2.2.高效液相色谱条件

Individual solutes were run at 0.25 mL / min 0.25 mL / min 0.25mL//min0.25 \mathrm{~mL} / \mathrm{min} and 0.50 mL / min 0.50 mL / min 0.50mL//min0.50 \mathrm{~mL} / \mathrm{min} flow rates at a temperature of 25 C 25 C 25^(@)C25^{\circ} \mathrm{C} using a Shimadzu Nexera TM X 2 TM X 2 ^(TM)X2{ }^{\mathrm{TM}} \mathrm{X} 2 liquid chromatograph (Shimadzu, Columbus, Maryland). The UV detector wavelength is 254 nm and the solutes were polystyrene standards of molecular weight 2.5 kDa , 5.0 kDa , 9.0 kDa , 17.5 kDa 2.5 kDa , 5.0 kDa , 9.0 kDa , 17.5 kDa 2.5kDa,5.0kDa,9.0kDa,17.5kDa2.5 \mathrm{kDa}, 5.0 \mathrm{kDa}, 9.0 \mathrm{kDa}, 17.5 \mathrm{kDa}, 30 kDa , 50 kDa , 110 kDa , 220 kDa , 400 kDa , 600 kDa , 900 kDa 30 kDa , 50 kDa , 110 kDa , 220 kDa , 400 kDa , 600 kDa , 900 kDa 30kDa,50kDa,110kDa,220kDa,400kDa,600kDa,900kDa30 \mathrm{kDa}, 50 \mathrm{kDa}, 110 \mathrm{kDa}, 220 \mathrm{kDa}, 400 \mathrm{kDa}, 600 \mathrm{kDa}, 900 \mathrm{kDa} and 1.8 mDa . The standards are from a low and high molecular weight polystyrene standards kit (Supelco, Bellefonte, PA, part numbers 4-8937 and 4-8938). The polydispersity index of these standards was not supplied by the manufacturer. The molecular weights and log log log\log molecular weights are given in Table 1 along with the radius of gyration and the diameter of gyration calculated from the formula R g = 0.137 M 0.589 R g = 0.137 M 0.589 R_(g)=0.137M^(0.589)R_{g}=0.137 M^{0.589} [14] for polystyrene in tetrahydrofuran (THF). THF was used for all mobile phase solvents in the unstabilized, HPLC grade form, and was from J. T. Baker (Center Valley, PA). Typically, polystyrene standards were made up as solutions in THF at 1 mg / mL 1 mg / mL 1mg//mL1 \mathrm{mg} / \mathrm{mL} concentration with 1 μ L 1 μ L 1muL1 \mu \mathrm{~L} injections.
使用 Shimadzu Nexera TM X 2 TM X 2 ^(TM)X2{ }^{\mathrm{TM}} \mathrm{X} 2 液相色谱仪(Shimadzu,Columbus,Maryland),在 25 C 25 C 25^(@)C25^{\circ} \mathrm{C} 温度下,以 0.25 mL / min 0.25 mL / min 0.25mL//min0.25 \mathrm{~mL} / \mathrm{min} 0.50 mL / min 0.50 mL / min 0.50mL//min0.50 \mathrm{~mL} / \mathrm{min} 流速运行单个溶质。紫外检测器波长为 254 nm,溶质为分子量为 2.5 kDa , 5.0 kDa , 9.0 kDa , 17.5 kDa 2.5 kDa , 5.0 kDa , 9.0 kDa , 17.5 kDa 2.5kDa,5.0kDa,9.0kDa,17.5kDa2.5 \mathrm{kDa}, 5.0 \mathrm{kDa}, 9.0 \mathrm{kDa}, 17.5 \mathrm{kDa} 30 kDa , 50 kDa , 110 kDa , 220 kDa , 400 kDa , 600 kDa , 900 kDa 30 kDa , 50 kDa , 110 kDa , 220 kDa , 400 kDa , 600 kDa , 900 kDa 30kDa,50kDa,110kDa,220kDa,400kDa,600kDa,900kDa30 \mathrm{kDa}, 50 \mathrm{kDa}, 110 \mathrm{kDa}, 220 \mathrm{kDa}, 400 \mathrm{kDa}, 600 \mathrm{kDa}, 900 \mathrm{kDa} 和 1.8 mDa 的聚苯乙烯标准物质。这些标准物质来自低分子量和高分子量聚苯乙烯标准试剂盒(Supelco, Bellefonte, PA,零件编号 4-8937 和 4-8938)。制造商没有提供这些标准物质的多分散指数。表 1 列出了分子量和 log log log\log 分子量,以及根据聚苯乙烯在四氢呋喃(THF)中的 R g = 0.137 M 0.589 R g = 0.137 M 0.589 R_(g)=0.137M^(0.589)R_{g}=0.137 M^{0.589} [14]公式计算出的回旋半径和回旋直径。所有流动相溶剂均使用未经稳定的 HPLC 级四氢呋喃,该溶剂来自 J. T. Baker 公司(宾夕法尼亚州,Center Valley)。通常情况下,聚苯乙烯标准物质以 1 mg / mL 1 mg / mL 1mg//mL1 \mathrm{mg} / \mathrm{mL} 浓度在 THF 中制成溶液, 1 μ L 1 μ L 1muL1 \mu \mathrm{~L} 进样。
Table 1 表 1
Radius of gyration of Polystyrene in THF.
聚苯乙烯在四氢呋喃中的回转半径。
 分子量 (Da)
Molecular weight
(Da)
Molecular weight (Da)| Molecular weight | | :--- | | (Da) |
 对数分子量
Log molecular
weight
Log molecular weight| Log molecular | | :--- | | weight |

半径 回旋 ( Å ) ( Å )("Å")(\AA)Å
Radius of
gyration ( ) ( ) ("Å")(\AA)
Radius of gyration ("Å")| Radius of | | :--- | | gyration $(\AA)$ |

直径 回旋 ( Å ) ( Å )("Å")(\AA)Å
Diameter of
gyration ( ) ( ) ("Å")(\AA)
Diameter of gyration ("Å")| Diameter of | | :--- | | gyration $(\AA)$ |
1800000 6.26 662 1325
900000 5.95 440 881
600000 5.78 347 694
400000 5.60 273 546
220000 5.34 192 384
110000 5.04 128 255
50000 4.70 80 160
30000 4.48 59 119
17500 4.24 43 86
9000 3.95 29 58
5000 3.70 21 41
2500 3.40 14 27
"Molecular weight (Da)" "Log molecular weight" "Radius of gyration ("Å")" "Diameter of gyration ("Å")" 1800000 6.26 662 1325 900000 5.95 440 881 600000 5.78 347 694 400000 5.60 273 546 220000 5.34 192 384 110000 5.04 128 255 50000 4.70 80 160 30000 4.48 59 119 17500 4.24 43 86 9000 3.95 29 58 5000 3.70 21 41 2500 3.40 14 27| Molecular weight <br> (Da) | Log molecular <br> weight | Radius of <br> gyration $(\AA)$ | Diameter of <br> gyration $(\AA)$ | | :--- | :--- | :--- | :--- | | 1800000 | 6.26 | 662 | 1325 | | 900000 | 5.95 | 440 | 881 | | 600000 | 5.78 | 347 | 694 | | 400000 | 5.60 | 273 | 546 | | 220000 | 5.34 | 192 | 384 | | 110000 | 5.04 | 128 | 255 | | 50000 | 4.70 | 80 | 160 | | 30000 | 4.48 | 59 | 119 | | 17500 | 4.24 | 43 | 86 | | 9000 | 3.95 | 29 | 58 | | 5000 | 3.70 | 21 | 41 | | 2500 | 3.40 | 14 | 27 |

2.3. Data analysis 2.3.数据分析

The data, stored in Excel© spreadsheets (Microsoft, Redmond, WA), was processed for plotting using MATLABO (MathWorks, Natick, MA) version R2016a. Individual zone broadening estimates of plates and plates per unit time were obtained using the plates calculator in the Shimadzu HPLC instrument software utilizing the width at half-height method (full width at half maximum or FWHM). When interpolation is employed in the plots, the interpolation method is the cubic spline method native to MATLAB. Results were checked for spline artifacts with visual graph inspection.
数据存储在 Excel© 电子表格(Microsoft,Redmond,WA)中,使用 MATLABO(MathWorks,Natick,MA)R2016a 版本进行处理,以便绘图。使用岛津 HPLC 仪器软件中的板计算器,利用半高宽度法(半最大值全宽或 FWHM)获得了板和单位时间内板的单独区域展宽估计值。在绘图中使用插值时,插值方法是 MATLAB 原生的三次样条法。通过目测图检查结果是否存在样条假象。
It is well known that using the width at half height method can overestimate the number of plates [27-29] for peaks that are nonGaussian, typically with tailing that can be modelled with a peak model formed by the convolution of a Gaussian with an exponential tail [30]. Obtaining high accuracy plate count measurements is difficult because the peak shape is not Gaussian nor Gaussian with exponential convolution across the full molecular weight range.
众所周知,使用半高宽度法可能会高估非高斯峰的板数 [27-29],典型的非高斯峰有尾部,可以用高斯峰与指数尾部卷积形成的峰模型来模拟 [30]。由于在整个分子量范围内,峰形既不是高斯峰形,也不是指数卷积高斯峰形,因此很难获得高精度的板数测量结果。
The specific resolution was described by Yau et al. [31,32] and originally defined as:
特定分辨率由 Yau 等人描述[31,32],最初定义为:

R s p = 2 ( V R 2 V R 1 ) ( W 1 + W 2 ) 1 log 10 ( M W 1 / M W 2 ) R s p = 2 V R 2 V R 1 W 1 + W 2 1 log 10 M W 1 / M W 2 R_(sp)=(2(V_(R2)-V_(R1)))/((W_(1)+W_(2)))*(1)/(log_(10)(M_(W1)//M_(W2)))R_{s p}=\frac{2\left(V_{R 2}-V_{R 1}\right)}{\left(W_{1}+W_{2}\right)} \cdot \frac{1}{\log _{10}\left(M_{W 1} / M_{W 2}\right)}
where W 1 W 1 W_(1)W_{1} and W 2 W 2 W_(2)W_{2} are the volume-based peak widths of adjacent peaks obtained by the tangent drop method. In addition, V R 1 V R 1 V_(R1)V_{R 1} and V R 2 V R 2 V_(R2)V_{R 2} are the retention volumes of adjacent peaks and M W 1 M W 1 M_(W1)M_{W 1} and M W 2 M W 2 M_(W2)M_{W 2} are the molecular weights of adjacent solute peaks.
其中, W 1 W 1 W_(1)W_{1} W 2 W 2 W_(2)W_{2} 是通过正切液滴法得到的相邻峰的基于体积的峰宽。此外, V R 1 V R 1 V_(R1)V_{R 1} V R 2 V R 2 V_(R2)V_{R 2} 是相邻峰的保留体积, M W 1 M W 1 M_(W1)M_{W 1} M W 2 M W 2 M_(W2)M_{W 2} 是相邻溶质峰的分子量。
In this paper we use a very similar approach. However, we use a continuous measure of the specific resolution which is constructed using interpolated time t t tt, standard deviation σ σ sigma\sigma and molecular weight M w M w M_(w)M_{w} so that:
在本文中,我们使用了一种非常类似的方法。不过,我们使用的是一种连续的特定分辨率测量方法,该方法使用内插时间 t t tt 、标准偏差 σ σ sigma\sigma 和分子量 M w M w M_(w)M_{w} ,因此:

R s p i = ( t i + 1 t i ) 2 ( σ i + 1 + σ i ) 1 log 10 ( M W i + 1 / M W i ) R s p i = t i + 1 t i 2 σ i + 1 + σ i 1 log 10 M W i + 1 / M W i R_(sp_(i))=((t_(i+1)-t_(i)))/(2(sigma_(i+1)+sigma_(i)))*(1)/(log_(10)(M_(Wi+1)//M_(Wi)))R_{s p_{i}}=\frac{\left(t_{i+1}-t_{i}\right)}{2\left(\sigma_{i+1}+\sigma_{i}\right)} \cdot \frac{1}{\log _{10}\left(M_{W i+1} / M_{W i}\right)}
where the 1 and 2 subscripts are replaced by the i th i th  i^("th ")\mathrm{i}^{\text {th }} and i th + 1 i th  + 1 i^("th ")+1\mathrm{i}^{\text {th }}+1 numbers in the interpolation vector of t , M w t , M w t,M_(w)t, M_{w} and of σ σ sigma\sigma. The standard deviation σ σ sigma\sigma is obtained from the FWHM measurement assuming a Gaussian zone so that σ = F W H M / 2 2 ln 2 σ = F W H M / 2 2 ln 2 sigma=FWHM//2sqrt(2ln 2)\sigma=F W H M / 2 \sqrt{2 \ln 2}. The specific resolution here is not normalized for comparison of columns of different lengths [31] because all of the compared columns are of the same 50 mm length.
其中,1 和 2 下标由 t , M w t , M w t,M_(w)t, M_{w} σ σ sigma\sigma 的插值向量中的 i th i th  i^("th ")\mathrm{i}^{\text {th }} i th + 1 i th  + 1 i^("th ")+1\mathrm{i}^{\text {th }}+1 数字代替。标准偏差 σ σ sigma\sigma 是根据假设为高斯区的 FWHM 测量值得出的,因此 σ = F W H M / 2 2 ln 2 σ = F W H M / 2 2 ln 2 sigma=FWHM//2sqrt(2ln 2)\sigma=F W H M / 2 \sqrt{2 \ln 2} 。这里的具体分辨率没有为比较不同长度的色谱柱而进行归一化处理 [31],因为所有比较的色谱柱都具有相同的 50 毫米长度。
The specific resolution R sp R sp  R_("sp ")R_{\text {sp }} is similar to other measures of separation for polymers and colloids, for example Giddings’ mass-based fractionating power [33,34] which is nearly identical to Eqs (2) and (3) and is expressed in compact form as:
比分辨率 R sp R sp  R_("sp ")R_{\text {sp }} 与聚合物和胶体的其他分离度量相似,例如吉丁斯的质量分馏能力 [33,34],它与公式 (2) 和 (3) 几乎相同,并以紧凑的形式表示为:

F m = R s δ ( M W ) / M W F m = R s δ M W / M W F_(m)=(R_(s))/(delta(M_(W))//M_(W))F_{m}=\frac{R_{s}}{\delta\left(M_{W}\right) / M_{W}}
where δ δ delta\delta is the differential operator.
其中 δ δ delta\delta