Implementation Science, including key topic clusters and the distribution of topics over time. A parallel study objective was to demonstrate how NLP can be used in research synthesis.Moving evidence-based practices into the hands of practitioners requires the synthesis and translation of research literature. However, the growing pace of scientific publications across disciplines makes it increasingly difficult to stay abreast of research literature. Natural language processing (NLP) methods are emerging as a valuable strategy for conducting content analyses of academic literature. We sought to apply NLP to identify publication trends in the journal Implementation Science abstracts published from February 22, 2006, to October 1, 2020. We retrieved the study data using PubMed’s Application Programming Interface (API) to assemble a database. Following standard preprocessing steps, we use topic modeling with Latent Dirichlet allocation (LDA) to cluster the abstracts following a minimization algorithm.We examined 1711 We examined 30 topics and computed topic model statistics of quality. Analyses revealed that published articles largely reflect (i) characteristics of research, or (ii) domains of practice. Emergent topic clusters encompassed key terms both salient and common to implementation science. HIV and stroke represent the most commonly published clinical areas. Systematic reviews have grown in topic prominence and coherence, whereas articles pertaining to knowledge translation (KT) have dropped in prominence since 2013. Articles on HIV and implementation effectiveness have increased in topic exclusivity over time.We demonstrated how NLP can be used as a synthesis and translation method to identify trends and topics across a large number of (over 1700) articles. With applicability to a variety of research domains, NLP is a promising approach to accelerate the dissemination and uptake of research literature. For future research in implementation science, we encourage the inclusion of more equity-focused studies to expand the impact of implementation science on disadvantaged communities. The classic implementations science axiom says that it takes an average of 17 years for evidence-based practices (EBPs) to be routinized in real-world settings . This stThe Interactive Systems Framework for Dissemination and Implementation was developed as a model to bridge the gap between research and practice . It deliImplementation Science—the flagship journal for peer-reviewed research on the scientific study of methods to promote the uptake of research findings into routine healthcare—the number of yearly submissions has increased eightfold, from 100 submissions in 2006 [Implementation Science Communication was launched in 2020, which accepts a broader range of methodological studies. Two other implementation-specific journals were established in 2020 , signifying the expanding appetite for implementation research. This does not include the multitude of journals that publish implementation science-adjacent literature .The accumulation of academic literature has grown exponentially across disciplines in the last 20 years , includi in 2006  to over  in 2006 . In respWith the rapid growth in publications, an emerging challenge to bridging the research-practice gap is the time required to synthesize research studies. The average time to complete and publish a systematic review is 67.3 weeks (1.3 years) . By the To accelerate the synthesis of research literature, scientists have increasingly turned to the use of natural language processing (NLP). NLP is a set of methods and computer-aided algorithms designed to detect patterns in textual data. By treating words and clustering of words as meaningful, NLP extracts concepts and relationships from texts more efficiently than humans are capable of doing. NLP is emerging as a valuable strategy for conducting content analyses of academic literature through bibliometric and scientometric studies. These studies involve the examination of digital data objects  to quantify study characteristics within a publication dataset , 12. TheImplementation Science, including the distribution of content areas and association among concepts. Specifically, this study aims to answer the following research questions:Implementation Science?What is the composition of content areas published in How have these content areas changed over time?The substantial growth of implementation science, and a personal affinity for this field, sparked our interest in how to leverage these advances to examine how the field has evolved over time. We utilize an NLP approach to identify publication trends in the flagship journal in the field, While answering these questions, we also have a methodological aim: to demonstrate how NLP can be used in research syntheses.Implementation Science articles published between February 22, 2006, (Welcome to Implementation Science [This study includes the abstracts and associated meta-data for  Science ) and OctImplementation Science website and when they appear in PubMed. Additionally, articles published in 2008 and 2009 were not indexed until 2010 for unknown reasons. We chose to preserve the PubMed indexing date because that is when a consumer of research would have come across the studies in PubMed.As of October 1, 2020, there were 1711 unique article abstracts available in PubMed, which we used in this analysis. We note that there appears to be a day or two lag time between when articles are uploaded to the Bag of Words approach. However, a fair amount of preprocessing is needed to prepare the text for analysis. We first removed stop words, or words that add little informative value to the overall text, such as “and,” “the,” and “with.” We then lemmatized the remaining words. Lemmatization involves converting a word down to its root form [stemming, is sometimes used because it is less computationally expensive than lemmatization. However, our dataset was not large enough for this to be an issue.We used several NLP methods to review the content areas in the abstracts. In this article, we treat “content areas” as the conceptual groups of constructs that are presented in the text. The core of NLP is tokenization, which treats the words as “tokens” that are meaningful information units in and of themselves. Generally, the more frequently that a token occurs, the more relevant it is in descriptive analysis. This is known as the oot form . In sometopic is a cluster of words that co-occur together, and that documents (in this case: abstracts) are clusters of topics [k), researchers want to find the lowest perplexity score across a number of possible topics. Perplexity is a metric that looks at how well a probability distribution predicts a sample. To arrive at this minimum, we tested potential topic clusters from two up to 60. While our goal was to minimize the perplexity score, we also wanted to maintain some topic interpretability. This can be a tradeoff; while we may show mathematical improvements, several of the emergent topics may strain a clear interpretation. We decided on a parameter of k = 30 topics. While we were continuing to show improvements in the perplexity at higher ks, too many of the emergent topics did not appear to add conceptual value. This is akin to finding the “bend in the curve” in principal components analysis when we begin to see diminishing returns in improvement in the metric of choice. Topic modeling is a Bayesian process [To examine study question 1, we used a topic modeling approach using Latent Dirichlet allocation (LDA). The use of “topic” here has a technical definition. LDA assumes that each f topics , 21. Topγ (“gamma”) value [γ value to assign a document to a topic, with the caveat that this statistic is a likelihood, rather than a classification.An LDA yields a matrix where each document (in this case: an abstract) is assigned a likelihood of belonging to a specific topic: the ”) value . ResearcTopic size: Total number of tokens by topic. There is a strong relationship between topic size and topic quality because common topics are generally represented in many documents, and are not as susceptible to being “diluted” by smaller topics [r topics .Mean token length: The average number of characters for the top tokens in a topic.Prominence: How many unique documents in which a topic appears. In this type of analysis, we generally find that the methodology-specific topics have higher prominence because they are more likely to feature in many articles.Coherence: How often the top tokens in each topic appears together in the same document.Exclusivity: How unique the top tokens in each topic are when compared to the other topics.Topic modeling is a primarily data-driven process, though there are methods to seed the model with specific words . Becauseγ value, and then plotted the number of articles in that topic over time against the years indexed in PubMed. To decide which articles to visualize, we used the above metrics to filter the results so we did not end up with a time-series plot with 30 lines, all along a grayscale.We assigned each article to a topic based on its maximum Implementation Science’s website.All statistics were computed in R 4.0.2 using a number of open-source packages . We used what are known as “out-of-the-box” algorithms, meaning we did not need to make any substantial changes to the underlying mathematics. All data that we used in this analysis are available, either directly as a database from the authors, through PubMed, or on Implementation Science, along with a steady volume of articles since 2014  coefficient indicates the likelihood that a word would appear in a particular topic across all the abstracts in our data [We examined the composition of content areas published in article abstracts through the association of words within topics. Figure our data . First, Ã¢, nÃ¢, 95, increase, and lt. This type of clustering is not uncommon in high-k LDA models.We observe that the topics reflect qualities of research  and domains of practice . Two clinical health topics emerge as highly published topics: HIV (topic 19), stroke (topic 11). Prevalent settings for research are community (topic 21) and healthcare  Of note, we eliminated topics 4 and 25 from this analysis because these topic clusters appeared to collect terms that had no significant meaning. For example, the key terms of topic 4 included β coefficient, along with five LDA core statistics. Of note, we used the full model including all words to assess topic quality, not just the top five words as in Figure Topic size refers to the number of tokens, or meaningful units of information, per topic. The largest topic sizes pertain to patient safety . The topic with the longest average token length, that is the longest words, was #7 (about implementation effectiveness). Topic prominence is the number of abstracts in which a topic appears. The most prominent topics were #30 (pertaining to systematic reviews) and #12 . Topic coherence measures how the top tokens in each topic appear together in the same document. Values closer to zero indicated greater coherence. The most coherent topics were ##5 (pertaining to RCTs) and #30 (systematic reviews). Topic exclusivity is how unique the tokens in the topic are compared to other topics. Even after we scaled the data, we found the top exclusive topics had identical values. These were topics 5, 7, 15, 17, 19, and 29. Figure While we can intuitively identify concepts across the topics , there needs to be a better method to determine the overall quality of the topic apart from a prima facie glance. Table prominence, coherence, and exclusivity . We see topic 30 rising in prominence and coherence. This indicates that a growth in the appearance of systematic reviews over time and that the terms in this topic cluster tightly.Because a linear plot with 30 topic clusters is not visually pragmatic, we focused on the top five topics as ranked by We observed a drop-off in coherence for topic #5 (RCTs) since 2016. This indicates there are fewer abstracts that have the set of tokens representing these particular topics. We also see a very large drop in the prominence of knowledge translation articles (#8) after 2013, meaning the KT article appears in fewer unique abstracts overall. For exclusivity, there is a drop in primary care practice management (#29) since 2013, and a rise in HIV-specific (#19) and implementation effectiveness (#7) articles.Implementation Science across the tenure of the journal, and (ii) to demonstrate how NLP can be used as a method of synthesis and translation to support analyses of a large volume of publication content. Below we discuss the implications for both of these aims.We intended this article to serve two purposes: (i) to highlight publication trends in Implementation Science between 2006 and 2020. Analyses of 28 key topics (per research question 1) revealed that published articles largely reflect characteristics of research or domains of practice. Topic clusters encompass key terms that are relevant and common to implementation science . HIV and stroke represent the most commonly published clinical areas.Implementation science is a growing field of study. Consistent with publication trends reported by Sales et al. , we founIn examining trends over time (research question 2), we found that systematic reviews have grown in topic prominence and coherence in the journal. The prominence of systematic review is consistent with trends reported by Boyd-Graber and colleagues , indicatImplementation Science publications.In reflecting on the observed trends, we asked: Could these trends relate to specific editorial changes? We believe it is possible. The editor-in-chief position turned over in 2012  and 2019 . We see changes in the LDA core metrics of prominence, exclusivity, and coherence in 2013 and 2020, years which follow changes in editorial leadership. We do not have information about how much influence the editors-in-chief have over types of articles published but recognize that it is customary for journal editors writ large to encourage topic-focused publications akin to special issues. Observed trends may also be shaped by highly published authors . HoweveImplementation Sciences’ readers and editors.Full-time, academic researchers can find it arduous to keep pace with the scientific literature given the precipitous rise in academic publications in the recent decades . The chaNLP is not a panacea for a deep understanding of qualitative data. There is no substitute for deeply engaging with ideas, questioning results, or contemplative analysis of issues to inform research and practice. At this time, we do not see NLP-aided methods as replacing the need to actually engage with articles. After all, there is still a large gap between NLP and natural language understanding (NLU). However, cutting-edge research such as OpenAI’s GPT-3 is moving closer to those benchmarks by using huge amounts of text-based data (175 billion parameters) to better capture nuance in language usage and how people process information . We alsoImplementation Science. This method can be ported to other search terms like, “decision-support,” “barrier and facilitators,” and “COVID prevention.” A more robust and refined clustering approach can assist the dissemination of scientific findings. For instance, it can help to better organize the results of a literature search process above and beyond what is normally returned with publicly available search engines. In this way, the NLP method can reduce the time required to scour the literature and provide more articles that are more representative a consumer’s interest. Additionally, NLP methods can be applied to examine correspondence between research literature and societal goals [More efficient and effective synthesis and translation methods are needed in order to reduce lag in uptake of evidence-based practices. The NLP method we described addresses a barrier at the very beginning of the research-to-practice paradigm. For this article, we looked at the journal al goals , implicaal goals , along wal goals , 35.Several limitations are important to note in the context of the descriptive findings. First, the observed trends are informed by the parameters of our analysis: the use of 30 topics, and our decision to focus on trends among the top five topics. While establishing threshold limits is a common practice for NLP methods, these thresholds can shape reported trends because there is a level of analyst discretion involved .Implementation Science the journal, not “implementation science” as a search term. This limits the generalizability of the study findings. It would be possible to replicate this analysis with all 3353 articles returned by PubMed with implementation science as a search term or by pulling in the gray literature. Indeed, looking at this larger corpus would be an interesting next step for our research, which could include more deep semantic modeling with word vectorization models that depend on data-rich inputs [We bounded our search to just h inputs .Also, we clustered topics, not findings. This is a critical distinction. We used topic modeling to cluster what the articles were about, not what articles reported as outcomes. Many researchers are working on developing better extractive methods to pull findings and sources of bias out of the article and into a summarization algorithm .Further, topic modeling accounts for co-occurrence of words, and not the syntax or semantics of the abstract text data. The approach presented in this study does not analyze relations among words (synonyms and similar terms). This precludes in-depth insight into how particular implementation science concepts have evolved, which are better suited for literature reviews. In this study, a temporal decline in a specific topic area  might indicate that a concept is going out of favor, or simply that the term used to reference the concept has evolved. Analytic approaches involving deep semantic modeling with word vectorization are a step forward though are not without their own limitations .Fundamentally, we recognize the inherent challenges in human interpretation of machine-learned models. NLP has not advanced to the point where it can replace human understanding. Because the algorithms are data-driven that can lead to poor interpretations and messy results. A pure summarization-and-synthesis solution is not available yet. We acknowledge that more advanced language models are yielding incredible results. And yet, these results are only as good as the training data, and we have much work to do ensure our processes yield accurate, actionable, and ethical results , 40, 41.Lastly, we identified health equity as a specific topic important for the future of implementation science. As community psychologists, equity is an anchoring value and objective. Our reflection may bear a disciplinary bias pre-conditioned by our professional training. We encourage additional reflection and group conversation about other concepts and issues that could expand the value of implementation science.Implementation Science . Our methods have captured the rise in prominence of systematic reviews, and the clustering of health and methodological-related content. We believe that similar methods can be used in a synthesis and translation process to further the bottom-line goals of Implementation Science. We can leverage advances in technology to accelerate the dissemination and uptake of good ideas.By using NLP, we have demonstrated the ebbs and flows of


Mikania micrantha is a notorious invasive weed that causes substantial economic losses and negative ecological consequences in southern China. However, the contributions of gene flow, environmental variables, and functional genes, all generally recognized as important factors driving invasive success, to its successful invasion of southern China are not fully understood. Here, we utilized a genotyping-by-sequencing approach to sequence 306 M. micrantha individuals from 21 invasive populations. Based on the obtained genome-wide single nucleotide polymorphism (SNP) data, we observed that all the populations possessed similar high levels of genetic diversity that were not constrained by longitude and latitude. Mikania micrantha was introduced multiple times and subsequently experienced rapid-range expansion with recurrent high gene flow. Using FST outliers, a latent factor mixed model, and the Bayesian method, we identified 38 outlier SNPs associated with environmental variables. The analysis of these outlier SNPs revealed that soil composition, temperature, precipitation, and ecological variables were important determinants affecting the invasive adaptation of M. micrantha. Candidate genes with outlier signatures were related to abiotic stress response. Gene family clustering analysis revealed 683 gene families unique to M. micrantha which may have significant implications for the growth, metabolism, and defense responses of M. micrantha. Forty-one genes showing significant positive selection signatures were identified. These genes mainly function in binding, DNA replication and repair, signature transduction, transcription, and cellular components. Collectively, these findings highlight the contribution of gene flow to the invasion and spread of M. micrantha and indicate the roles of adaptive loci and functional genes in invasive adaptation.A long-standing and unresolved issue in invasion biology concerns the rapid adaptation of invaders to nonindigenous environments.  Biological invasion has been reported at increasing rates and on expanding scales since the 20th century, gaining the attention of ecologists and evolutionary biologists . A long-Ambrosia artemisiifolia individuals from 37 locations and found that SNPs active in geographic differentiation occurred more often within pathways related to growth and defense responses than in other pathways. Van Boheemen et al.  [A. artemisiifolia individuals from North America, Europe, and Australia, and found multiple introductions and adaptive signatures in response to environmental selection pressure. These studies have shown that the GBS approach can provide insight into the genetic basis underlying population invasion and contribute to the exploration of the environmental variables and genes involved in population adaptation.Early studies using anonymous markers  and amplified fragment length polymorphisms (AFLPs)) may have included bias in their assessment of genetic variation, while the utility of these markers to detect adaptive loci associated with environmental variables, especially when using few molecular markers, is limited. Furthermore, studies using anonymous markers cannot be used to determine the gene function of candidate loci. However, with the development of sequencing technology, reduced-representation genomic sequencing has provided an opportunity to obtain genome-wide sequence information across populations, greatly promoting research into population expansion and adaptive evolution in invasive plants . Genotyp7, 2020) ,15 analyMikania micrantha H.B.K. (Asteraceae), a tropical American perennial creeping vine, is one of the top 10 most problematic noxious weeds worldwide [M. micrantha covers a relatively wide array of biome types, including tropical and subtropical moist and dry broadleaf forests, tropical and subtropical coniferous forests, tropical and subtropical grasslands, deserts and xeric shrublands, savannas, and temperate broadleaf and mixed forest [M. micrantha plants can grow in a variety of soil types, in both dry and cold areas, in aquatic habitats, and in broad light niches [M. micrantha indicates its strong adaptability to different environmental conditions. The population of M. micrantha can not only adapt well to climate conditions similar to those of its native range, but also invade novel climatic niches [M. micrantha into different climatic niches demonstrate its wide ecological amplitude and strong environmental adaptability. In addition, M. micrantha can propagate sexually via thousands of lightweight seeds [M. micrantha populations. Considering its high dispersal potential and strong adaptability, M. micrantha is an ideal candidate for investigating the contribution of gene flow to population expansion, the response of genetic loci to environmental variables, and the role of functional genes in invasive adaptation.orldwide ,20. The orldwide ,22. The d forest . M. micrt niches ,25,26. Tc niches . Lineagec niches . The expht seeds . Long-diht seeds . FrequenM. micrantha spread rapidly throughout southern China following its introduction to the Hong Kong Zoological and Botanical Gardens in 1884 [M. micrantha in southern China has been investigated in several genetic studies, all of which used a small number of anonymous genomic markers [M. micrantha populations. Some key issues have not been resolved in these genetic studies. First, different molecular markers detected obviously different levels of genetic differentiation across populations distributed in similar geographic ranges , hindering the understanding of how gene flow contributes to the invasive success of M. micrantha in southern China. Secondly, genetic loci that respond to environmental variables have not been detected; consequently, the effect of environmental variables on genetic variation and population adaptation remains unresolved. Third, candidate functional genes related to colonization and population adaptation have not yet been identified. Population genomic data for natural populations of M. micrantha are needed to address these issues. in 1884 . The rol markers ,33,34,35M. micrantha individuals from 21 populations in six regions, covering the main part of its invasive range in southern China. Based on the SNP data, we detected the effect of gene flow on the genetic variation in the introduced populations of M. micrantha. We also identified outlier SNPs associated with environmental variables. Furthermore, the gene family clustering analysis of the genomes of M. micrantha and 12 other plants revealed unique gene families in M. micrantha that are important for adaptation. We also detected candidate genes that showed a significant positive selection signature in M. micrantha and explored their functions. The current study provides new insights into the population expansion and adaptation of M. micrantha.In the current study, we used the GBS approach to generate a large number of SNPs to analyze 306 M. micrantha individuals were sampled from 21 invasive populations distributed in six regions in southern China, including Hong Kong, Macao, Shenzhen, Neilingding Island, Dongguan, and Zhuhai  method [Genomic DNA from 306 ) method . DNA quaM. micrantha DNA samples using MseΙ  and EcoRΙ restriction enzymes. Briefly, the DNA samples were incubated with MseΙ at 37°C. The digested DNA samples were ligated to MseΙ Y adapter N containing individual-specific barcodes with T4 DNA ligase. To further reduce the complexity and increase genome coverage, the MseΙ-digested samples were redigested with EcoRΙ. EcoRΙ-digested DNA fragments were purified using Agencourt AMPure XP (Beckman) and PCR-amplified using Phusion Master Mix (NEB), universal primer, and index primer. Amplified DNA from 306 M. micrantha individuals was purified and screened for size. Size-selected fragments (insert sizes of 265–315 bp) were purified and diluted for sequencing. Samples for all M. micrantha individuals were divided into 12 pooled libraries, which were sequenced using the Illumina NovaSeq platform  to generate 150 bp paired-end reads.GBS libraries were constructed for 306 M. micrantha individual were sorted based on the individual-specific barcode within the raw reads. Raw reads for each individual were quality-controlled with in-house Perl scripts. Clean reads were obtained by removing reads that contained more than 10% ambiguous bases (N), more than 50% low-quality  bases, and bases that aligned to adapters exceeding 10 nt.Sequences from each M. micrantha reference genome (unpublished genome data from our laboratory) using the Burrows-Wheeler Aligner (BWA) v0.7.17 [M. micrantha genome, ANNOVAR [Clean reads from 306 individuals were aligned to the  v0.7.17  with the v0.7.17 . mpileup v0.7.17  was also ANNOVAR  was usedHO), gene diversity (HS), allelic richness (AR), and inbreeding coefficients (FIS), were calculated for each population and region using the R package hierfstat [p < 0.05).The genetic diversity indices, including the observed heterozygosity /4FST.To determine the level of genetic differentiation in the study areas, pairwise Weir and Cockerham’s age in R . To testage in R . Hierarcage in R  to identM. micrantha, PLINK v1.9 [2) = 0.8. Filtered SNPs (5966 SNPs) were used for genetic structure analysis. ADMIXTURE v1.23 [K) was set from 2 to 22. Cross-validation (CV) error was used to select the most likely number of clusters. The K value with the lowest CV error was determined to be the optimal number of clusters. GCTA v1.26 [M. micrantha. Based on the pairwise genetic distances among individuals, the unweight pair group method with the arithmetic mean (UPGMA) method implemented in MEGA v5.10 [To investigate the genetic structure of INK v1.9  was usedRE v1.23  was usedTA v1.26  was usedGA v5.10  was usedGA v5.10 .http://www.worldclim.org (accessed on 5 March 2019)) at a 2.5-arc-minute resolution. The soil factors, including fresh water content; air-dried water content; pH; electrical conductivity; organic matter content; total N, C, K, Ca, Na, Mg, Al, P, S, Si, Fe, Mn, Zn, Cu, Pb, Cr, As, Se, Ni, and Cd were measured following the method of Shen et al. (2021) [http://lpdaac.usgs.gov (accessed on 8 March 2019), recorded in 2006–2016). The yearly mean value for each ecological variable derived from the MODIS dataset was computed using the maximum value composites function [The environmental data comprised 19 bioclimatic variables, 25 soil factors, and 6 ecological variables. The bioclimatic variables corresponding to historic conditions (from 1960 to 1990) for each population were obtained from WorldClim v1.4 (. (2021) . The ecofunction . To redufunction . EnvironFST-based outlier approach was used with the BayeScan software [FST coefficient (α) shared by all populations and a population-specific FST coefficient (β) shared by all loci [10 (PO) > 2 were considered outlier SNPs under selection. Given that purifying or balancing selection is more prone to elevated false-positive rates when identifying outlier loci in BayeScan under the context of range expansion [To determine whether there were SNP loci potentially under selection in the 21 populations, an software . It has all loci . A positxpansion , only ou−5.The genes containing outlier SNPs identified by BayeScan were functionally annotated. Annotation based on the Pfam protein database was performed using Hmmscan implemented in HMMER v3.1 . Based oK) was selected using the snmf function, which uses a Bayesian clustering algorithm to estimate the admixture coefficients of individuals. Based on the results of the snmf analysis obtained with 10 repetitions and 20,000 iterations for each K value from 1 to 22, the K value with the minimum cross-entropy was identified. This K value was used to perform LFMM analysis with the lfmm function. The lfmm function was implemented using 50,000 burn-in, 100,000 iterations, and 10 replicates. The p-values were used to test the significance of the correlation, with a p-value < 0.005 representing a significant association.The latent factor mixed model (LFMM) was used to detect the associations of outlier SNPs with environmental variables in the R package LEA . LFMM haFST matrix using a Mantel test in the R package ade4 [The Bayesian method implemented in the BAYENV v2.0 package  was alsoage ade4  with 100M. micrantha and 12 other plants were identified using the OrthoMCL program [M. micrantha proteins, protein sets from 12 other plants were obtained from public sources  enrichment analysis was performed using KOBAS [p-values were used to determine the significance of the KEGG pathway enrichment, with adjusted p-values < 0.05 considered significant. In addition, ANNOVAR [M. micrantha have developed nonsynonymous amino acid mutations at the population level, overlaps between genes containing nonsynonymous SNPs  were annotated and used for KEGG enrichment analysis. Furthermore, the overlaps between genes containing nonsynonymous SNPs and PSGs in M. micrantha were detected.Based on the gene family clustering data for g MUSCLE . To iden of PAML  was usedM. micrantha individuals from 21 invasive populations was performed. The analysis yielded a total of 3320.63 million raw reads, with a mean of 10.85 million reads per individual  varying from 1.743 (HK3) to 1.886 (HK1), observed heterozygosity (HO) ranging from 0.280 (HK4) to 0.387 (DG4), and gene diversity (HS) ranging from 0.363 (MA1) to 0.458 (SZ1) (FIS) values were positive in all 21 populations. The distribution of genetic diversity across 21 populations was not limited by longitude and latitude (p > 0.05), except for a weak correlation between AR and latitude   . The inblatitude . In addi > 0.05) .FST value was highest between HK3 and NLD2 (0.133) and lowest (0.004) between NLD2 and NLD6  to 0.043 (SZ versus NLD)  between populations ranged from 1.63 to 62.25 and from 5.564 to 31 between regions  . These r regions .M. micrantha, genetic clustering of the 306 individuals was conducted using ADMIXTURE, which also provided evidence of an admixed structure at the optimal clustering level (K = 15)  revealed an admixed distribution pattern in which individuals from different populations clustered randomly . To furt(K = 15) . Consiste branch .p < 0.001, FST = 0.042; p < 0.001). The Mantel test showed that the genetic differentiation between populations included a weak genetic relatedness caused by isolation by distance (IBD) .The analysis of molecular variance (AMOVA) revealed that 95.84% of the total variation occurred within populations, while only 4.16% occurred among populations (10 (PO) > 2  showed an FST < 0.1. Outlier SNPs also exhibited low FST values, with a mean value of 0.151, implying that the 21 populations did not show significant adaptive differentiation at the outlier loci. The outlier SNPs were located in different regions of the M. micrantha genome, with 11 outlier SNPs residing in exon regions of 11 genes  > 2 . These 211 genes . The 11 11 genes . In addi11 genes . Under tM. micrantha populations. These two approaches have great power for the identification of outlier loci associated with environmental variables. We identified 31 associations between 18 outlier SNPs and six environmental variables in the LFMM analysis , “biosynthesis of unsaturated fatty acids” (ko01040), “photosynthesis” (ko00195), “other glycan degradation” (ko00511), and “ether lipid metabolism” (ko00565)—all of which are important for the growth, metabolism, and defense response of M. micrantha  and “integral to membrane” (GO: 0016021), respectively.Further, we found that eight genes unique to ectively . These cM. micrantha and the 12 other plants. Following the likelihood ratio test, 41 genes were considered candidate genes that had undergone positive selection in M. micrantha  were significantly enriched , and lower than the interpopulation genetic differentiation level reported by previous genetic studies based on a small number of molecular markers (8–483 markers) [M. micrantha. Low genetic differentiation levels may occur because M. micrantha plants can produce enormous numbers of lightweight seeds that are dispersed over long distances by the wind [M. micrantha mainly depends on clonal propagation, which may also be an important factor leading to low genetic differentiation. Human activities can result in seeds or cloned fragments being carried to different locations, increasing genetic exchange between individuals from different populations and weakening population divergence. In addition, the homogenization effect of frequent gene flow would reduce adaptive differentiation between populations. Consistent with this, we detected a high rate of gene flow (1.63 ≤ Nm ≤ 62.25) among M. micrantha populations. Similarly, a high rate of gene flow has been found among populations within the Asian invasion range [M. micrantha population invasion is accompanied by a high rate of gene flow.The genetic differentiation levels in markers) ,31,34. Ithe wind , promotiM. micrantha populations [M. micrantha exhibits patterns of population genetics that tend to enhance its invasion potential and that gene flow and multiple introductions (see the following paragraph) may mitigate the negative influence of the founder effect, genetic bottleneck, or inbreeding on genetic diversity and mediate genetic diversity distribution in its invasive range.Generally, the range limit linked to species invasions affects the genetic diversity of invasive populations . The genulations ,34,75, hulations , and notM. micrantha was also confirmed by the weak correlation between genetic differentiation and geographic distance. Geographical isolation can explain genetic differentiation during the early period of population invasion, when populations are geographically dispersed. The effect of geographical isolation on population differentiation becomes weaker with a high rate of gene flow during population expansion. The continued invasion of M. micrantha in southern China (more than 130 years) and high rate of gene flow among populations can explain the weak correlation between genetic differentiation and geographic distance. Overall, M. micrantha was introduced multiple times and subsequently experienced a rapid range expansion with recurrent high rates of gene flow.ADMIXTURE and PCA analysis revealed multiple genetic clusters and an admixed population structure in the study area, which was consistent with the genetic structure previously detected in populations that invaded Asia ,69. MultM. micrantha invasion from tropical America to southern China, the new environments encountered likely induced candidate-selected loci that respond to environmental conditions. The LFMM and BAYENV analysis revealed significant associations between 38 outlier SNPs and 13 environmental variables, including temperature, precipitation, soil, and ecological variables, all of which may play roles in the invasive adaptation of M. micrantha. Specifically, (i) rapid adaptation to climate is conducive to range expansion of invasive plants. Temperature and precipitation are important determinants of the growth, adaptation, and potential distribution of M. micrantha [M. micrantha genome responds to temperature and precipitation fluctuations. (ii) The significant effect of soil composition on the invasion of M. micrantha has been confirmed [M. micrantha depends on accelerated activation of soil K [M. micrantha. We found significant associations between outlier SNPs, the percentage of non-vegetation cover, and the percentage of tree cover. M. micrantha often forms a dense thicket above nearby plants [M. micrantha genome may respond positively to tree or vegetation cover. Taken together, these results reveal that the M. micrantha genome already has selective signatures that respond to the changing environment to promote its invasive adaptation.The detection of association between SNPs and environmental variables helps us to recognize the bioclimatic, soil, and ecological variables that contribute to population adaptation. During the process of icrantha . Consisticrantha ,78 and ionfirmed . The soionfirmed . C depleonfirmed . Ca playonfirmed . Furtheronfirmed . Indeed,onfirmed  and it hf soil K . Finallyf soil K . In addif soil K . (iii) Ey plants  and escay plants . These cM. micrantha. Unique gene families were also observed in the genomes of Artemisia annua [Chrysanthemum nankingense [M. micrantha were enriched in functional pathways related to important physiological processes and stress response, such as oxidative phosphorylation, the biosynthesis of unsaturated fatty acids, and photosynthesis. Oxidative phosphorylation is an important metabolic pathway that generates ATP required for plant growth and metabolism. There is evidence that oxidative phosphorylation plays a crucial role in abiotic stress responses. For instance, oxidative phosphorylation is enriched by genes associated with the cold tolerance in Anthurium andraeanum [M. micrantha, providing the energy supply needed for rapid vegetative growth [M. micrantha. Similarly, the upregulation of genes related to photosynthesis promotes the adaptation of Solidago canadensis to a shaded habitat, greatly improving the plant’s ability to adapt to different light conditions [M. micrantha with functional importance will provide a valuable resource for further research on the adaptive mechanisms of this species.Based on gene family clustering analysis, we identified gene families unique to ia annua  and Chrykingense . Gene fadraeanum . Unsaturdraeanum . High phe growth . The enrnditions . The ideM. micrantha to drought and biotic stress. We also found unique genes related to ‘integral to membrane’. This functional category is related to the defense response of apple root to pathogen infection [Unique genes that harbor mutations leading to nonsynonymous amino acid substitutions among populations are worthy of attention. These unique genes encode functional proteins and are involved in important function categories. For example, GSTT3 is a member of GST superfamily and is related to antioxidants that respond to drought and biotic stress ,95. We dnfection . AlthougM. micrantha would enable us to better understand the molecular associations underpinning its invasive adaptation. Accordingly, we identified 41 genes showing a significant positive selection signature in M. micrantha. These PSGs were found to be involved in (i) binding, (ii) DNA replication and repair, (iii) signature transduction, and (iv) transcription and cellular components  [M. micrantha to cold stimulus. Finally, key cellular components  and transcriptional regulation are essential for plant growth and environmental adaptation. The positive selection of genes related to the nucleus and transcription suggests their potential roles in M. micrantha. In addition, two functionally important PSGs harbored mutations that resulted in nonsynonymous amino acid substitutions among 21 populations, which implies that they may contribute to the population adaptation of M. micrantha.Genes under evolutionary pressure (such as positive selection) may be important for the survival and spread of various species . The idemponents . Functionis seed ,99. DNA nis seed . Plants nis seed . Exonuclnis seed . The posl change . For exavuglare) . This paM. micrantha, providing insights into M. micrantha adaptation. We detected 11 outlier SNPs distributed in 11 genes as candidate targets of adaptive importance. The GO annotation analysis of the 11 genes revealed that they were related to stress response, which should be considered in future population genomic research. Under the biological process category, “metabolic process”, “oxidation–reduction process”, and “response to stimulus” were highly representative GO terms important for the environmental adaptation of invasive and non-invasive plants. The metabolic process may contribute to the cold tolerance in the northward invasion of Alternanthera philoxeroides [Amorpha fruticosa seedlings [Kandelia obovata [Phragmites karka [Tamarix ramosissima in response to water deficiency [During rapid population invasion, genes that regulate important biological processes are expected to function in xeroides . The oxieedlings  and in t obovata . Effecti obovata . In the  obovata  and the es karka . In termficiency .M. micrantha in southern China. Genome-wide SNP analysis revealed, for the first time, the effects of gene flow, environmental variables, and functional genes on the population variation and adaptation of M. micrantha. The presented findings demonstrate the importance of a high rate of gene flow coupled with genetic admixture for the invasion success of M. micrantha. We identified 38 outlier SNPs associated with environmental variables. Soil composition, temperature, precipitation, and ecological variables were important determinants affecting the invasive adaptation of M. micrantha. Candidate genes with outlier signatures were related to abiotic stress responses. Gene family clustering analysis revealed 683 gene families unique to M. micrantha that may have significant implications for the growth, metabolism, and defense response in M. micrantha. Forty-one genes with significant positive selection signatures were identified. These PSGs mainly function in binding, DNA replication and repair, signature transduction, transcription, and cellular components. In addition, eight genes unique to M. micrantha and two PSGs harbored mutations leading to nonsynonymous amino acid substitutions across 21 populations; these may play crucial roles in the population expansion and adaptation of M. micrantha. Overall, the presented population genomic analysis indicates that the sustained high rate of gene flow, response of adaptive loci, and regulation of functional genes have enabled the invasion and adaptation of M. micrantha in southern China. These findings can be used to guide prevention and monitoring efforts for this invasive species.In the present study, we used population genomics to characterize the range expansion and invasive adaptation of


In addition, there are pure spin currents for CuPc and FePc molecular devices in the PC, and an almost pure spin current for FePc molecular devices in the APC at some temperature. Meanwhile, there is a high SFE of about 99.99585% in the PC and a reserved SFE of about −19.533% in the APC and a maximum MR ratio of about 3.69 × 108% for the FePc molecular device. Our results predict that the monolayer and bilayer MPc  molecular devices possess large advantages in designing high-performance electrical and spintronic molecular devices.The spin related electrical and thermoelectric properties of monolayer and bilayer MPc  molecular devices in a parallel spin configuration (PC) and an anti-parallel spin configuration (APC) between the V-shaped zigzag-edged graphene nanoribbon electrodes and the center bilayer MPc molecules are investigated by combining the density functional theory and non-equilibrium Green’s function approaches. The results show that there is an ultrahigh spin filter efficiency exceeding 99.99995% and an ultra-large total conductance of 0.49996G Molecular spintronic devices have been extensively studied in the past decade, and are a hopeful approach to downsize spintronic devices and are expected to be widely used in high-density data storage and quantum computing ,2. A lar7%/2.58 × 107%/3.069 × 108%, respectively. The SFE of the bilayer CoPc and FePc devices exceed 99.9998% and the SFE of the bilayer CuPc molecular device is over 99.99996%, which exceeds those of the bilayer FePc, bilayer CoPc and FePc-CoPc molecular devices. In addition, the SFE of the CuPc-FePc and CuPc-CoPc devices exceed 99.999997%, which are perfect SFEs. In addition, the spin-down and total conductance and SFE of our bilayer Co-/Fe-/Cu-phthalocyanine molecular devices in the PC are bigger than the ones in our monolayer Co-/Fe-/Cu-phthalocyanine molecular devices, the spin-down channel dominates the transmission and density of state at Fermi energy in the PC. There are large pure spin currents in the monolayer CuPc and FePc molecular devices in the PC at some temperatures. Additionally, the spin-dependent charge Seebeck coefficient and the spin Seebeck coefficient are almost equal for CoPc, CuPc-CuPc, CoPc-CoPc, FePc-FePc, CuPc-CoPc, CuPc-FePc, and CoPc-FePc molecular devices in the PC due to the spin-down Seebeck coefficient being close to zero and there are usually large charge Seebeck coefficients at high temperatures for the abovementioned molecular devices in the APC. Physical mechanisms are proposed for these phenomena. The calculated transmission spectra and the real-space scattering states of the modeled mono- and bilayers can be used for future experimental photoemission spectroscopy and other studies. These theoretical calculations predict new monolayer and bilayer phthalocyanine-based molecular devices with high MR and Seebeck coefficients. Based on the calculated spin-dependent conductance and other characteristics, ultrahigh SFE, GMR, and Seebeck coefficients for Co-/Fe-/Cu-phthalocyanine molecular devices are predicted for the first time. This makes the results motivating for further experimental studies of phthalocyanine- and TM-layered structures. The obtained large SFE of the parallel spin configuration can be utilized in various molecular spintronic devices.In this paper, we studied the spin-related electrical and thermoelectric properties of monolayer and bilayer MPc  molecular device in the PC and APC with two V-shaped zigzag-edged graphene nanoribbons (ZGNRs) electrodes by employing LDA, LDA + U and the non-equilibrium Green’s function (NEGF) in combination with the density functional theory (DFT) ,15. ThesWe investigated the spin transport properties of monolayer and bilayer MPc  molecular devices by combining the density functional theory and non-equilibrium Green’s function approach, as implemented by the NanoDCAL transport package ,26. FiguSpin-polarized zero-bias conductance is given by the Landauer–Buttiker formula (1)Gσ=e2At zero bias, SFE is defined as:,Considering the spin direction of the lead, there is a giant magnetoresistance (MR) effect and the MR ratio for the device for the PC and APC are defined as:C) and the spin Seebeck coefficient (SS) are defined as SC = (S↑ + S↓)/2 and SS = (S↑ − S↓)/2, where S↑ and S↓ are the SU and SD Seebeck coefficients, respectively [Considering the spin-dependent thermoelectric properties of the monolayer and bilayer MPc  molecular devices, the usual charge Seebeck coefficient  and the nearest nitrogen (N) atoms for Fe-/Co-/Cu-phthalocyanines in the gas phase in our NanoDCAL program LDA (for CuPc) and LDA + U (for FePc and CoPc) results with calculations for the same systems done with the SIESTA + SMEAOGOL programs and the PBE and PBEh functionals in Reference  as well For monolayer CoPc, FePc, CuPc, bilayer CuPc-CuPc, FePc-FePc, CoPc-CoPc, FePc-CoPc, CoPc-CuPc and CuPc-FePc molecular devices (MDs) at zero bias, 8% for the monolayer FePc MD, and the MR ratio of FePc-FePc MD is largest in the bilayer MPcs MD.For CoPc, CuPc, FePc, CuPc-CuPc, FePc-FePc, CoPc-CoPc, FePc-CoPc, CuPc-CoPc, and FePc-CuPc molecular devices at zero bias, the SFE in the PC are all close to 100% and there is a maximum SFE of 99.999998% for CuPc-CoPc MD in To better understand the transport properties of the MPc molecular devices, we investigated transmission and the projected density of state  in the SU and SD channels among FePc and FePc-FePc molecular devices in the PC and APC under equilibrium states, as shown in σ in the left column, the spin-dependent charge Seebeck coefficient SC and the spin Seebeck coefficient SS in the right column of CuPc, CoPc, FePc MD in the PC and APC versus the temperature at zero bias. Generally, shown in ↓ of CuPc and CoPc in the PC, the latter is quite small, S↓ ≈ 0. The S↑ of CuPc, CoPc, FePc MD in the PC first decreases, then increases and then decreases with T. SC = 0 at T ≈ 307 and 482 K for CuPc MD and SC = 0 at T ≈ 218 K for FePc MD in the PC, a pure spin current produced by the temperature gradient. For CoPc MD in the PC, SC ≈ SS duo to S↓ is close to zero and described by the formula of SC and SS. As shown in σ is linear in T [↑ and S↓ around T = 160 K for CuPc MD, large S↑ and S↓ around T = 211 and 261 K for CoPc MD, and a large S↓ around T = 207 K and a large and negative S↑ around 207 K for FePc MD in the APC. There is a pure spin current with SC = 0 at T ≈ 100 K for CuPc MD, at T ≈ 143 K for CoPc MD, at T≈99 K for FePc MD and a large and negative SC and a small and negative SS at T = 161 K for CuPc MD, a large SC and a small SS at T = 500 K for CoPc MD, a large and negative SS and a small and negative SC at T = 206 K for FePc MD in the APC.ear in T  and therσ in the left column, the SC and SS in the right column of CuPc-CuPc, CoPc-CoPc, FePc-FePc, CuPc-CoPc, CuPc-FePc, CoPc-FePc MD in the PC and APC versus the temperature at zero bias. The S↓ of the bilayer MPc MD is close to zero and SC≈SS in the PC in S is close to zero in most regions of the temperature range. There is a tendency to produce a pure charge current with the increasing temperature and S↓≈S↑ for CoPc-CoPc, CuPc-CoPc, CuPc-FePc, CoPc-FePc MDs in the APC, as shown in 6% at zero bias. There are pure spin currents for CuPc and FePc molecular devices in the PC, and an almost pure spin current for FePc molecular device in the APC at some temperatures. Meanwhile, there is a high SFE of about 99.99585% in the PC and a reserved SFE of about −19.533% in the APC as well as a maximum MR ratio of about 3.69 × 108% for monolayer FePc molecular device. In addition, the spin-dependent charge Seebeck coefficient SC and the spin Seebeck coefficient SS are almost equal for CoPc, CuPc-CuPc, CoPc-CoPc, FePc-FePc, CuPc-CoPc, CuPc-FePc, CoPc-FePc molecular devices in the PC duo to the spin-down Seebeck coefficient is close to zero and there are usually large charge Seebeck coefficients at high temperatures for the above molecular devices in the APC.Therefore, we can obtain an ultrahigh spin filter efficiency exceeding 99.99995% and ultra-large total conductance of 0.49996G0 for FePc-CoPc molecular device in the PC and a nearly pure charge current at the high temperature in the APC as well as a giant MR ratio exceeding 9.87 × 108% for monolayer FePc molecular devices. These transport phenomena could be well understood by analyzing the transmission spectra, projected density of state and scattering states. The calculated transmission spectra and the real-space scattering states of the modeled mono- and bilayers can be used for future experimental photoemission spectroscopy and other studies. These theoretical calculations predict new monolayer and bilayer phthalocyanine-based molecular devices with high MR and Seebeck coefficients. This makes the results motivating for further experimental studies of phthalocyanine and transition metal layered structures. The obtained large spin-filter efficiency of the parallel spin configuration can be utilized in various molecular spintronic devices.In conclusion, we investigated the spin-dependent conductance, spin filter efficiency, giant magnetoresistance ratio, Seebeck coefficient, charge Seebeck coefficient and spin Seebeck coefficient by analyzing the projected density of state, transmission spectrum and scattering state of monolayer and bilayer MPc  molecular devices in the parallel and anti-parallel spin configurations by employing LDA, LDA + U and nonequilibrium Green’s function approaches. These methods have their benefits over other approaches for nanomaterials with transition metals, Fe in particular. The results show that the spin filter efficiency in the parallel spin configuration, spin-down and total conductance of the bilayer MPc molecular devices were superior to the monolayer MPc molecular device. There are large pure spin currents in the CuPc and FePc molecular devices in the parallel spin configuration at some temperatures. Meanwhile, there is a high SFE of about 99.99585% in the parallel spin configuration and a reserved SFE of about −19.533% in the anti-parallel spin configuration and a maximum MR ratio of about 3.69 × 10


Molecular diameters are an important property of gases for numerous scientific and technical disciplines. Different measurement techniques for these diameters exist, each delivering a characteristic value. Their reliability in describing the flow of rarefied gases, however, has not yet been discussed, especially the case for the transitional range between continuum and ballistic flow. Here, we present a method to describe gas flows in straight channels with arbitrary cross sections for the whole Knudsen range by using a superposition model based on molecular diameters. This model allows us to determine a transition diameter from flow measurement data that paves the way for generalized calculations of gas behaviour under rarefied conditions linking continuum and free molecular regime. For unification, the size is often generalized as the diameter of an equivalent sphere. However, a molecule is neither a rigid object nor has, in most cases, a spherical shape. Its volume is made up of electron orbitals representing the probability of finding an electron at a specific position. It is, therefore, impossible to state one definite diameter of a molecule. As a result, different operationally defined diameters are used for various scenarios.In many natural processes and technical applications, the size of a gas molecule is a key property that influences the rate at which lungs can absorb oxygen5. The van der Waals diameter is obtained by crystallographic experiments such as X-ray diffraction, zero point density data5 or the molar volume of solids6. This diameter is also applied in the van der Waals equation of state to account for the nonideal behaviour of gases under high pressure8.The van der Waals diameter is a prominent measure and is typically used to calculate the properties of condensed matter4.Transport processes such as the absorption of oxygen in the lungs or gas transport through a membrane are strongly influenced by diffusion effects. The diffusion coefficients of molecules can be estimated based on their diameter. Here, the kinetic diameter of a molecule is frequently used, which is obtained from molecular sieving experiments9, which plays a crucial role in gas kinetics:The drag on wind turbines is mainly a result of gas viscosity. Viscosity is also an effect resulting from the molecular diameter of a gas. Measured viscosity, usually obtained at normal pressure, can therefore be used to estimate molecular diameters. These diameters are typically utilized to calculate the mean free path λ10. The movement of a gas molecule is strongly affected by its interaction with its environment involving collisions with other gas molecules or the surrounding geometry. The frequency of intermolecular collisions is determined by the mean free path of a molecule. The lower the pressure is, the less likely a collision is between two molecules within a given time. In contrast, the smaller the surrounding geometry is, the more likely a collision is between gas molecules and the surface. The ratio between the mean free path and the characteristic length of the geometry in question is represented by the dimensionless Knudsen number Kn11.The kinetic theory of gases can be utilized to describe the behaviour of a gas12. At ultrahigh vacuum or in relatively small geometries, however, the Knudsen number becomes very large. This is an important property of scanning electron microscopy in which gas molecules do not interfere with the electron beam. Another example where the Knudsen number tends to infinity is molecular sieves. Here, gas molecules diffuse through a pore structure where pore sizes are on the same order of magnitude as the molecular diameter13. Both extremes are well described by existing models, such as the Navier Stokes equations for continuum conditions11 or the Smoluchowski equation for free molecular flow14. Each of these models has analytical solutions for simple geometries.Under elevated pressure conditions or in larger geometries, the Knudsen number is close to zero. This so-called continuum regime is present in many technical applications. Examples are everyday phenomena under standard conditions, such as the abovementioned viscosity effects, as well as high-pressure reaction engineering15 including models using superposition17, fitting coefficients are needed, which lack a clear physical meaning. For generalizability and understanding of the involved effects, distinct physical meaning is desirable. Recent works are presented by Xie et al.18 and Groll et al.19 and a comprehensive overview of macroscopic modelling techniques for rarefied gas flows is given by Struchtrup20.In many applications, however, Knudsen numbers are neither close to zero nor tend to infinity. Examples for applications that are based on gas flow under these so-called rarefied conditions are swing-adsorption processes, catalysis, and space propulsion systems with typical Knudsen ranges depicted in Fig. Kn → 0 or at Kn → ∞. Because the effective diameter of a molecule depends on the surrounding conditions, those diameters are most likely not sufficiently accurate for covering the full Knudsen range. A diameter relating to rarefied conditions with emphasis on the transitional regime11 could fill this gap.When modelling such scenarios, one has been limited to choosing between molecular diameters obtained at either To determine molecular diameters under rarefied gas conditions, we propose a method using measurements of rarefied gas flows in straight channels. Based on well-established models for convective, diffusive and free molecular flow, a superimposed form is introduced covering the whole Knudsen range. The resulting model is predictive and applicable to any channel cross section. We show that this analytical description can be utilized to obtain molecular diameters corresponding to rarefied conditions.DH = 4A/Π, where A is the channel cross sectional area, and Π is the perimeter of the cross section. This mass flow recovers Knudsen’s original expression for free molecular mass flow in circular channels23. Normalizing yields the nondimensional flow:The proposed analytical description for rarefied gas flow is composed of models for convective and diffusive flow. Mass flow components and total mass flow for an exemplary case are depicted in Fig. L is the length of the channel, Δp is the pressure difference between the inlet and outlet, R is the gas constant, T is the temperature, and M is the molecular weight. The Knudsen number used for plotting is computed using the mean free path calculated from viscosity; see the Methods section for details. The mean free path calculated from viscosity is used to ensure comparability between the different data sets and the viscosity is an objective measure. This does not influence the results of the model, as explained in detail in the Supplementary Text 22 is clearly visible at approximately Kn = 1. This minimum is the result of the vanishing influence of convection and the rising contribution of diffusion at large Knudsen numbers.Here, 24, and more details are given in the Methods section. In the literature, a slip coefficient is used to adjust the boundary condition to experimental data. This slip coefficient is then usually used to extract the tangential momentum accommodation coefficient (TMAC)25. The TMAC describes the ratio of diffuse to specular reflection of molecules at the surface 26. Although it has been shown that specular reflection cannot play a role in diffusion on technical surfaces and only starts to dominate when the roughness is very small27, fitting TMACs is widely used, and values typically range between 0.8 and 128. Here, we show that it is not necessary to adjust the TMAC and therefore the slip coefficient for channels with circular cross sections. When comparing similar geometries with similar surface properties in terms of roughness and material, the TMAC should be equal. Therefore, the adjustment typically applied to the slip coefficient is more likely covering an effect arising from molecular diameters because the slip expression is dependent on the molecular diameter. Considering technical surfaces with TMAC equal to unity, the equation for slip flow through a circular cross section is:pm is the mean pressure between the inlet and outlet, D is the diameter of the circular channel and Kn(d) = λ(d)/D) is the mean Knudsen number in the channel. The influence of the molecular diameter d is caused by the mean free path λ(d); see Eq. ,,4), the A detailed derivation of the model and the explicit formulation of Eq.  are give4, the van der Waals diameter31 and the diameter calculated from viscosity9. For the nonspherical molecules nitrogen and carbon dioxide, two different van der Waals diameters are given for the transversal and longitudinal orientations, respectively. Figure transition diameter introduced in this work by least squares fitting on experimental data of rarefied flow in microchannels is also included.Figure The mass flow according to Eq.  is calcu32, nitrogen33 and carbon dioxide22.Figure Kn → 0 becomes identical, as in this range, the flow is dominated by the macroscopic property viscosity. Additionally, the free molecular mass flow for Kn → ∞ is equal because free molecular flow does not depend on molecular diameters but on the interaction with the geometry. However, different diameters show distinct mass flows at approximately Kn = 1 because here, the influence of slip is strong while the convective term is still present and self-diffusion is the dominating mechanism for diffusive flow. Both these terms are affected by the mean free path being in turn highly affected by the molecular diameter: the larger the molecule is, the smaller the mean free path. A smaller mean free path results in smaller diffusion and slip contributions and, therefore, in a smaller mass flow.For all diameters, the convective mass flow for Kn = 1. For argon, the deviation from data is less distinct. For the nonspherical molecules nitrogen and carbon dioxide, the smaller transversal diameter reproduces the data better than the larger longitudinal diameter.This influence of molecular diameters is demonstrated by the mass flows resulting from the van der Waals diameters , which are larger than the others see Fig. , leadingThe kinetic diameter is neither the largest nor the smallest diameter. It is close to the van der Waals diameter for helium and thus also predicts a very distinct minimum. In contrast, the kinetic diameter of carbon dioxide is close to the small transversal van der Waals diameter and overpredicts experimental data. For argon, the kinetic diameter yields a slightly lower mass flow than the data, while it results in a good prediction for nitrogen.The diameter calculated from viscosity gives reasonable results for helium and nitrogen. The mass flow for argon and carbon dioxide is slightly underestimated. Of all the literature diameters, this diameter seems to be the most suitable one even though showing deviations to experimental data.transition diameter. A comparison between molecular diameters from the literature and the transition diameters is depicted in Fig. Because of the predictive nature of the model, it can be applied to experimental data of mass flows using the molecular diameter as a free parameter, yielding a new diameter. Since the model is most sensitive in the transitional regime, we use the expression Kn range, the transition diameter-based model predicts the experimental mass flow data very well. Especially around the Knudsen minimum at Kn = 1, the transition diameter shows superior agreement with experimental data compared to the literature values of the established molecular diameters. For Kn → 0, reasonable agreement with experimental data was expected because the main transport mechanism, convection, is influenced mainly by viscosity which is constant over pressure. The viscosity is taken from the literature data where the values are obtained under continuum conditions, rendering it a suitable property for describing gas flow under these conditions. In addition, the mass flow for Kn → ∞ is convincingly predicted as a result of the dominating Smoluchowski expression in the free molecular regime, which is influenced only by the TMAC and the geometry of the channel and not by the molecular diameter, indicating that the TMAC is indeed unity for technical surfaces. A smaller TMAC would result in a higher mass flow for the free molecular regime, which is not represented by the experimental data. The good prediction for Kn = 1 is the result of the superposition of the different transport mechanisms at work. This agreement allows for the determination of a valid molecular diameter in this region.For the whole transition diameter values obtained before are used as well as data for helium34 and for argon and nitrogen25. Comparing literature data shows that the TMAC is found to be unity exclusively for circular cross sections, while the TMAC is 0.8–0.9 for rectangular cross section28, in agreement with our model, which provides good results when adjusting the TMAC to 0.9 for the slip expression. Since TMAC is defined as the ratio between diffuse and specular reflection, the values should be equal for surfaces with similar roughness made from a similar material. Apparently, the TMAC for the slip expression is also a function of cross-sectional shape, whose expression has not yet been established. To account for this cross-section effect, the TMAC in the slip expression is adjusted, while the TMAC for the free molecular flow is still unity. A clear indicator that the TMAC should in fact be unity for technical surfaces is the mass flow in the free molecular regime for Kn → ∞, as discussed above. See Supplementary Fig. 27. This contradiction could point to a flaw in the slip expression for rectangular channels.Furthermore, the validity of the model can be confirmed by transferring it to rectangular cross sections, as shown in Fig. Kn = 1 is properly predicted by the model. The minimum is more pronounced for high-ratio rectangular channels than for circular channels, which is a phenomenon widely known in the literature15. The reason for this phenomenon is the geometry-dependent behaviour of free molecular flow as expressed by the Smoluchowski Equation: the flow through an infinitesimal cross-sectional element is proportional to the mean of its distances to equiangular spaced points on the boundary at the element’s axial position, since these distances are proportional to the distances molecules are free to travel. For rectangular channels, the sum of the mean distances of all cross-sectional elements is larger than the sum of the mean distances of a circular channel with equal hydraulic diameter and thus yields a larger nondimensional mass flow in the free molecular regime. This effect is the more enhanced the larger the channel’s aspect ratio becomes.The well-known Knudsen minimum at transition diameters, resulting in the best agreement with the data, demonstrating the predictive capability of the model with a known molecular diameter.Although rectangular geometries are more intricate than circular geometries, the model represents the experimental data well with To analyse the quality of the model, the deviations between the modelled mass flow and experimental data are investigated. For a model explaining the data well, these deviations should occur only due to noise in the measurement data. This is the case for the circular channels, as seen in Fig. Kn < 3, the modelled mass flow is smaller than the experimental data. For regions of Kn > 3, the modelled mass flow is larger but then approaches the experimental data again. For argon, the same behaviour is observed, with an additional offset. A possible reason for this deviation pattern is the pressure regimes that are covered by different pressure sensors. Each pressure sensor exhibits a linear increase or decrease in deviation, so the systematic error of the model is likely due to measurement uncertainty, which is naturally largest at the outer region of each sensor measurement domain.The systematic deviations for the rectangular channels presumably result from measurement uncertainties of pressure sensors, as shown in Fig. transition diameter, however, is smaller than the diameter calculated from viscosity for helium, argon and carbon dioxide but only smaller than the kinetic diameter for helium and argon. The transition diameter is almost equal to the kinetic diameter and the diameter calculated from viscosity for nitrogen and larger than the kinetic diameter for carbon dioxide. The smallest deviation from the transition diameter shows the viscous diameter for helium, nitrogen and carbon dioxide and the kinetic diameter for argon. Bondi states that the van der Waals diameter changes less for heavy atoms5, which could explain why argon shows the smallest range of diameters.An interesting observation on the order of the size of molecular diameters can be made from Fig. transition diameter. The kinetic diameter, however, follows this trend up to nitrogen but is smaller for carbon dioxide than for nitrogen and argon. This can be explained by the measurement principle for kinetic diameters, which are obtained by molecular sieving experiments. Carbon dioxide can align lengthwise to pass very small pores, but this size is not representative of diffusion because it does not account for the longitudinal extent of the molecule.There are different trends when comparing the same diameter across gases. The van der Waals diameters and the diameters calculated from viscosity increase from helium via argon and nitrogen up to carbon dioxide. The same applies to the Kn = 1, both of these effects in sum are minimal. The transition diameter is, as all the other singular diameters, an effective diameter describing the flow and diffusion behaviour and is therefore capable of describing non-spherical molecules as well.These inconsistencies can be attributed to different molecules behaving very differently in certain situations. On the one hand, the shape of molecules, which may be nonspherical, can contribute to this. On the other hand, because of their nonrigid nature, the diameter can be interpreted as an apparent size that changes with environmental conditions and is not exclusively geometrical. These environmental conditions are dominated by intermolecular interactions for the viscous regime, which become less significant for higher Knudsen numbers, and by molecular-surface interactions for measurements of the kinetic diameter, which are less considerable for smaller Knudsen numbers. In the transitional regime of approximately Kn → 0 or Kn → ∞ does not reflect the different surrounding conditions. The method presented here can determine a molecular diameter that is sensitive to the region around Kn = 1 and therefore represents a diameter that is valid in the transitional regime. This transition diameter is suitable to explain the behaviour of gas flow in this region, where the proposed model is most sensitive to the molecular diameter. This can be a valuable addition to the existing diameters, which are corresponding to conditions at Kn → 0 or Kn → ∞. For the continuum regime, using the viscosity is a reasonable approach. However, as soon as rarefaction effects become important, that is, for Kn > 0.01, using the transition diameter allows for describing gas flows more accurately. For the many relevant applications in the field of rarefied gases, we propose to use the presented method to extract the corresponding transition diameter and apply it to calculations for gases under rarefied conditions.To describe the behaviour of gases, the molecular diameter is a crucial parameter. The diameters found in the literature are either acquired under continuum conditions or under conditions that are dominated by gas-surface interactions. However, many applications, such as swing-adsorption processes, catalysis and space propulsion systems, operate at Knudsen numbers between 0.1 and 10. In these scenarios, the molecular diameter acquired at The model for calculating the mass flow consists of convective and diffusive parts.35:By assuming low Mach and Reynolds numbers, a gas can be treated as incompressible, and the mass flow through a circular channel can be expressed by the Hagen-Poiseuille law24:us is the velocity difference between the wall and the gas directly at the wall. du/dr is the directional derivative of the gas velocity normal to the wall. λ is the mean free path of the gas and depends on the molecular diameter via Eq.   eff and ing Eqs. , which i29 is used. Kno from the original expression is replaced by Lc = h, yielding Eq. , and λm,visc is the mean free path calculated by:pm = (pin + pout) / 2 is the mean pressure, and μ is the dynamic viscosity.Here, transition diameter, the free parameter of the model is fitted to the experimental data by least square optimization. The data are used in their nondimensional form to avoid overweighting of high absolute mass flows for small Knudsen numbers. Theoretically, it would be sufficient to have data points just at Kn = 1 for determining the molecular diameter. However, it is important to be sure that the TMAC is set correctly, which can be validated by measurements in the free molecular regime. Additionally, the more data points are available, the smaller the uncertainty becomes. The prediction interval shown in Fig. s and N are the variance and number of data points, respectively. tα is the percentile of Student’s t-distribution corresponding to a confidence level of α38. In this case, α = 0.05 is used for a two-sided 95% confidence interval. The degree of freedom is N – 1 when fitting one parameter, the diameter in this case.For determination of the Supplementary Information.